Kinematics and resistive-force theory explain the swimming of
Spiroplasma
Jing Yang (杨靖),1 Charles W. Wolgemuth (吴格木),1, 2 and Greg Huber (胡伯乐)1, 2, 3, ∗
1
The Richard Berlin Center for Cell Analysis & Modeling,
University of Connecticut Health Center,
Farmington, Connecticut 06030, USA
2
Department of Cell Biology, University of Connecticut Health Center,
Farmington, Connecticut 06030, USA
3
Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269, USA
(Dated: September 1, 2008)
Abstract
Spiroplasma swimming is studied with a simple model based on resistive-force theory. Using
a generalized description of the shape changes that drive the swimming of this unique family of
bacteria, we explore the effect of this dynamic geometry on cell motility. Specifically, we consider
a bacterium shaped in the form of a helix that propagates traveling-wave distortions which flip the
handedness of the helical cell body. We treat cell length, kink velocity, and distance between kinks
as parameters and calculate the swimming velocity that arises due to the distortions. We find that
scaling collapses the swimming velocity (and the swimming efficiency) to a universal curve that
depends only on the ratio of the distance between kinks to the cell length. A ratio ' 0.3 (agreeing
with experimental observations) gives the maximum efficiency.
1
Helices are profoundly involved in the swimming of many types of bacteria. For example,
the classical picture of bacterial swimming involves long helical filaments that extend out
from the inner membrane of the cell [1]. Tiny rotary motors spin these helices, pumping fluid
away from the bacterium, which, by Newton’s third law, drives the bacterium forward. But
helices are also used in other more ingenious ways. In Treponema pallidum, the spirochetal
bacterium responsible for syphilis, rotation of helical filaments encased between the bacterial
cell wall and the outer membrane leads to periodic undulations that drive this cell through
water and other viscous fluids. (This is not, presumably, how it crossed the Atlantic Ocean
[2, 3].) The Leptospiraceae rely on a helical body plan and rotation of tightly-coiled filaments
to drive their motility [4–7]. Generally speaking, it is the dynamics and shape together, the
“swimming strategy”, that is the key to understanding the motility of helical bacteria. For
the flagella-less family of plant pathogens represented by Spiroplasma, the precise propulsive
strategy was shrouded in mystery until careful observations [8, 9] revealed that these tiny
organisms processively flip the chirality of regions of their helical bodies, generating pairs
of kinks moving down the cell body. The internal propulsive mechanism underlying this
peculiar swimming strategy is still unclear, as is the quantification of its motility.
Spiroplasma swimming is realized by the propagation of a pair of kinks along the body
axis of the cell, and the bacterium is propelled by the hydrodynamic force as the fluid
associated with the kinks moves rearward with the “body wave”. Kinks start at the same
end of the cell (front), and travel toward the other end (back). As the kink propagates, the
cell changes direction through an angle related to the pitch angle of the helix [10]. Due to
the unbalanced viscous drag between different portions of the cell body separated by kinks,
eventually the cell swims in a zigzag path. The helical transformation requires a cell to rotate
about its body axis during kink propagation. In the cell-body frame, there are two possible
rotations: “crankshafting” [11], in which one helical section rigidly pivots about the axis
of the other, and “speedometer-cable motion,” where each helix rotates about its own axis
[11, 12]. Both forms are observed, but speedometer-cable motion appears predominantly
[9].
In this Letter we use the experimentally-observed geometry of Spiroplasma to explore its
swimming behavior for a range of kinematic deformations. We consider a helically-shaped
cell of fixed pitch angle and helix diameter. To swim, the bacterium flips the handedness
of its helicity beginning at one end. This change in helicity propagates toward the back
2
D+d
!
p
d
"
(a)
(b)
FIG. 1: (a) Spiroplasma in differential interference contrast (Image courtesy of J. W. Shaevitz).
(b) Schematic of helical geometry. D is the helix diameter, D + d is the outer coil diameter
(D = 0.138µm, d = 0.20µm). Here the bend angle α and pitch angle ψ satisfy α = π − 2ψ.
of the cell with a uniform velocity, Uk . After a time tk , the front of the cell reverts to
its original handedness and this change in helicity also propagates toward the back of the
cell with the same kink velocity. The distance between the kinks is Lk = Uk tk , and only
an integer number of turns is considered. We treat Uk , Lk , and the cell length, Lc , as
parameters and use resistive-force theory to calculate the swimming speed and trajectory
as a function of these parameters. This model is complementary to the work done in [10]
where slender-body hydrodynamics was used to explore the dependence of swimming speed
on the pitch angle. It was found experimentally that the inter-kink distance Lk shows a
remarkable consistency – what determines this preference? Our results demonstrate that
the maximum propulsive efficiency is achieved when the ratio between inter-kink length and
cell-body length is around 0.3, which is consistent with most experimental observations and
suggests that the kinematics of Spiroplasma have been evolutionarily tuned.
Any helix can be described by its pitch, p, and helix radius, r. The arclength, s, is the
distance from one end to any point along the helical curve. One helical repeat in arclength is
√
` = p2 + 4π 2 r2 . When two helices of differing pitch are smoothly concatenated, such that
the tangent and normal vectors are continuous, a kink is formed [11, 13]. The bend angle
α between the two axes in this construction is α = π − 2ψ, where ψ is the pitch angle (see
Fig. 1). The parameters employed in the calculations are p = 0.62 µm, r = D/2 = 0.069
µm, and ψ = 35◦ [8, 9].
For the numerical integration, the cell body is discretized into straight-line elements.
3
According to resistive-force theory [14, 15], the resistance force acting on a unit length of
rod is given by
"
#
h
i
2πµ
(2I − λλ) · U + O ln−2 (2/)
(1)
f =−
ln(2/)
where U is the local velocity, I is the unit tensor, is the slenderness ratio of rod diameter
to length, λ is the unit tangent vector, and µ is viscosity. The total force and momentum on
the unrestrained cell body is zero, i.e.,
R Lc
0
f (s)ds = 0,
R Lc
0
M(s)ds = 0. The translational
and rotational velocities of the cell body can be solved for from the resulting system of
linear equations. To verify that the implementation of the model has been properly made,
the translational velocity of a rigid helix with angular velocity ω around its body axis is
calculated with the same numerical code. Since the total force on the rotating helix is
zero, i.e.,
RL
0
f (s)ds = 0, the translational velocity of the helix is U = B`ω, where L is the
h
i
total length of the rigid helix, B = sin2 ψ cos ψ/ 2π(1 + sin2 ψ) [10]. The numerical results
match this analytical solution with relative errors less than 0.05% under our typical spatial
discretization. Based on the aforementioned model, a number of dynamical questions on the
swimming of Spiroplasma can be formulated and addressed.
Typical experimental and simulated displacement curves during one swimming cycle of
Spiroplasma are shown in Fig. 2 (a). Here the displacement is defined as the movement
of the center of mass along the direction heading from initial position to end position.
For comparison with the experimental data obtained by Shaevitz et al. [9], our numerical
simulation results are also smoothed with a 175-ms boxcar filter [16]. Using the parameters
given above, our simulations agree well with the corresponding experimental results. The
average velocity in our simulations is about 3.03 µm/s, which is close to the cell velocity
observed in the experiment, 3.3±0.2 µm/s [9]. An interesting point: the simulated results fit
best with the corresponding experimental data obtained under higher viscosity rather than
low viscosity. This suggests an explanation for the puzzling phenomena that Spiroplasma
swims with higher speed in more viscous media. At low viscosity, Brownian motion randomly
alters the orientations along the cell body, which could reduce the net propulsive force
generated by the traveling kinks. The effect of these fluctuations on swimming speed may
be reduced in higher-viscosity media.
In the simulation shown in Fig. 2 (a), the second kink is generated at the front of the cell
around 0.16 s after the start of the first kink. Thereafter, the first and second kink both exist
and travel toward the end of cell until they disappear. It is observed experimentally that
4
Cell Position, S [ µm ]
2.0
1st Kink
1st + 2nd Kink
2nd Kink
1.5
1.0
Simulation
Experiment
(Shaevitz et al.)
0.5
(a)
0.0
0
0.1
0.2
0.3
0.4
Time, t [ s ]
0.5
0.6
8.0
Cell Velocity, Uc [ µm / s ]
7.0
(b)
6.0
5.0
4.0
3.0
2!
3!
4!
5!
2.0
1.0
0.0
5
10
15
20
Kink Velocity, Uk [ µm / s ]
25
FIG. 2: (a) Typical displacement curves (cell position versus time). Lk is 2`, Lc is 7`, and Uk is 12`
per second in the simulation. (b) Cell velocity as a function of kink velocity at different inter-kink
lengths.
the speed is decreased when two kinks are present as compared with when only one kink
is present. This key feature of the displacement curve is well reproduced in the numerical
simulations (Fig. 2 (a)).
The cell velocity of Spiroplasma as a function of the kink velocity at different Lk (2`, 3`,
4` and 5`) is shown in Fig. 2 (b) (The cell body length is 7` in Fig. 2 (b).). The cell velocity
increases linearly with kink velocity, and the ratio of cell velocity to kink velocity is constant
for fixed inter-kink length, with the proportionality constant varying with inter-kink length.
For the cases listed in Fig. 2 (b), the proportionality constant varies from 0.26 to 0.33, which
is consistent with the experimental reports [8, 9].
In order to delineate the effect of geometry on Spiroplasma motility, two dimensionless
parameters are introduced. One is the ratio of cell velocity Uc to kink velocity Uk ; the other
is the ratio of inter-kink length Lk to cell body length Lc . Remarkably, we find that all the
5
100
0.35
Envelope Angle
80
0.25
60
0.20
0.15
4!
5!
6!
7!
8!
9!
" = 35 ˚
0.10
0.05
0.00
10!
20!
40!
50!
60!
Eq. (2)
40
20
Envelope Angle, ! [ ˚ ]
Velocity Ratio, Uc / Uk
0.30
0
0
0.2
0.4
0.6
Length Ratio, Lk / Lc
0.8
1
FIG. 3: Velocity ratio and the corresponding envelope angle, as a function of the length ratio for
different body lengths.
data fall on a single curve after the results from different cell-body lengths and inter-kink
lengths are taken together (Fig. 3). The curve describes a velocity ratio that increases with
length ratio at small values, and then begins to decrease with length ratio after reaching a
peak value.
From Fig. 3, the maximum velocity ratio of the data-fitted curve is around 0.32 when the
length ratio is about 0.3. An interesting observation is that the velocity ratios and length
ratios observed in the experiment fall close to the aforementioned value, which suggests that
wildtype Spiroplasmas have optimized their geometry to achieve the “fastest state” in the
course of evolution.
The velocity ratio can be calculated analytically in the limit that the length ratio approaches one. As the length ratio goes to one, there is effectively only one kink propagating
along the cell body at any given time. In this case, Spiroplasma can be treated as a filament
consisting of two helical sections, one right handed and the other left handed. The two helical
sections rotate around their respective axes with the opposite sense. The total translational
velocity is the resultant of the vector sum of these two components with an interior angle
2ψ. As mentioned previously, the translational velocity of a rigid helix with angular velocity
ω around its axis is B`ω [10]. From geometrical considerations, the kink velocity Uk can be
represented as a function of rotational velocity ω, i.e., Uk = 2ωr/ tan ψ. Therefore the total
velocity can be roughly estimated to be 2B`ω cos ψ, and the corresponding velocity ratio is
sin2 ψ cos ψ/ 1 + sin2 ψ . For a pitch angle of 35◦ , the velocity ratio is around 0.202, which
is close to the simulation result of 0.24. (Since the rotation of the whole cell body is not
6
considered in this estimate, the velocity ratio is slightly smaller than the simulation result.)
!"
#"
$"
%"
)"
'"
&"
("
FIG. 4: The trace swept out by the cell-body axis at different inter-kink lengths 2-9 ` (for cell
length 10 `). The envelope angle is indicated.
The computed velocity-ratio data (Fig. 3) appears to flatten as the length ratio approaches one: why? In this limit, the kinematic asymmetry between the left-handed interkink length and the right-handed extra-kink length vanishes. This motivates the use of a
chirality variable χ = (Lc − Lk )/(Lc + Lk ) to study the velocity ratio on both sides of χ = 0.
Even though Lk > Lc is physically nonsensical per se, the solutions of the Stokes equations
behave smoothly as “inter-kink” and “extra-kink” labels are exchanged at Lk = Lc , i.e.
under the mapping χ → −χ. But this sign change is equivalent to a spatial inversion, under
which the symmetries of the Stokes equations imply that the cell velocity is unchanged.
Hence, the velocity ratio is a smooth, even function in χ, and about χ = 0 a best fit gives
Uc /Uk = 0.24 + 0.71χ2 − 1.99χ4 + O(χ6 ). We note in passing that the entire curve is in
excellent empirical agreement with a quartic polynomial
Uc ∼
= 0.24 + 0.711φ2 − 1.513φ4
Uk
(2)
(the solid curve in Fig. 3) where φ = tan−1 χ.
Not only translation, but also rotation of the cell body must satisfy the dynamic balance,
i.e., zero force and torque. At the beginning of a kink cycle, one end of the cell-body axis
traces out a circular arc (see Fig. 4). The maximum angle swept out, we term the envelope
angle. We use this concept in Fig. 3, where the dashed curve shows the envelope angle as a
function of the length ratio. The envelope angle increases with length ratio, and eventually
saturates to a certain value. Fig. 3 also shows that the velocity ratio achieves its maximum
at the length ratio whose envelope angle equals the pitch angle (see the dot-dashed line).
For swimming cells, it is customary to compute the swimming efficiency. This value is
often defined as the ratio of the power dissipated by dragging the appropriate rigid cylinder
7
Fuel Mileage, ! [ nm/ATP ]
6!
8!
10!
12!
80
18!
20!
24!
30!
70
60
25
50
20
40
Fuel Mileage
( Lc = 10! )
Scaled Fuel Mileage
( Lc = 6 - 30! )
15
0
0.2
0.4
0.6
Length Ratio, Lk / Lc
30
20
0.8
Scaled Fuel Mileage, !"-# [arb. unit]
30
1
FIG. 5: Fuel mileage as a function of the length ratio. The solid curve and circle data points show
the case with Lc = 10` (left ordinate). Using the scaling factor −δ , the results from all cell lengths
can be collapsed. Here is the slenderness ratio and the exponent δ = 1.90 ± 0.01.
through fluid to the power dissipated by the actual swimmer [10, 15, 17]. For the swimming
of Spiroplasma, we propose an alternative quantity, the fuel mileage: the distance traveled
per cycle per energy dissipated. If fuel consumption is at all a concern, then we might
expect optimization of the total distance traveled per energy used. As ATP is a common
cellular energy source, we compute the fuel mileage in units of nanometers traveled per ATP
burned [18], where we assume that 1 ATP gives about 20 kB T worth of energy. Using our
simulations, we calculate the mileage as a function of the length ratio (the solid curve in
Fig. 5). We find that the fuel mileage is maximized when this ratio is around 0.3, which is
the experimentally-observed value [9]. Furthermore, we find that to propagate one double
kink down the length of Spiroplasma requires at least 50-100 ATPs worth of energy and the
cell moves about 26 nm/ATP, which is over 3 times further than a kinesin molecule moves
per ATP burned [19]. Interestingly, the fuel mileage from different cell lengths can all be
collapsed onto one curve by constructing a scaling factor from a power of the slenderness
ratio (= d/Lc ) as: −δ γ(Lk , Lc ). Fig. 5 shows the resulting data collapse for δ = 1.90±0.01
[20].
In this paper, we investigated via a numerical study the swimming of Spiroplasma with
a model based on resistive-force theory and the kinematic deformations of Spiroplasma.
The key features of Spiroplasma swimming are well reproduced with this model, and the
simulation results agree quantitatively with the observed experimental data. The important
effects of dynamic geometry on the motility of Spiroplamsa are explored, and the motility
8
efficiency is shown to assume the highest possible value given the kinematics observed in
experiments. One implication is the optimization of the geometry of Spiroplasma during
the course of evolution. By examining cells of different lengths, it should be possible to test
the predictions of the model.
We gratefully acknowledge National Institute of Health grants R01 GM072004 (CWW)
and U54 RR022232 (CWW, GH), and support from the Richard Berlin Center for Cell
Analysis & Modeling (JY, GH).
∗
Author to whom correspondence should be addressed (Email: [email protected])
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1
2σ
R t+σ
t−σ
S (t0 ) dt0 , where S is the displacement before filtering,
SBC is the displacement after filtering, and σ is the half width.
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9
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[20] A similar data collapse using (1/)2 / ln(a/) (where a is constant) rather than −δ also works
well.
10