Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
Biophysical Journal Volume 109 July 2015 1–14
1
Article
A Computational Model of Dynein Activation Patterns that Can Explain
Nodal Cilia Rotation
Duanduan Chen1,* and Yi Zhong1
1
School of Life Science, Beijing Institute of Technology, Beijing, China
ABSTRACT Normal left-right patterning in vertebrates depends on the rotational movement of nodal cilia. In order to produce
this ciliary motion, the activity of axonemal dyneins must be tightly regulated in a temporal and spatial manner; the specific activation pattern of the dynein motors in the nodal cilia has not been reported. Contemporary imaging techniques cannot directly
assess dynein activity in a living cilium. In this study, we establish a three-dimensional model to mimic the ciliary ultrastructure
and assume that the activation of dynein proteins is related to the interdoublet distance. By employing finite-element analysis
and grid deformation techniques, we simulate the mechanical function of dyneins by pairs of point loads, investigate the
time-variant interdoublet distance, and simulate the dynein-triggered ciliary motion. The computational results indicate that,
to produce the rotational movement of nodal cilia, the dynein activity is transferred clockwise (looking from the tip) between
the nine doublet microtubules, and along each microtubule, the dynein activation should occur faster at the basal region and
slower when it is close to the ciliary tip. Moreover, the time cost by all the dyneins along one microtubule to be activated can
be used to deduce the dynein activation pattern; it implies that, as an alternative method, measuring this time can indirectly
reveal the dynein activity. The proposed protein-structure model can simulate the ciliary motion triggered by various dynein activation patterns explicitly and may contribute to furthering the studies on axonemal dynein activity.
INTRODUCTION
All vertebrates show distinct left-right (L-R) asymmetry in
the patterning and positioning of their internal organs; in humans, abnormal L-R patterning is strongly associated with
congenital heart disease (1,2). In the mammalian embryo,
L-R symmetry is first broken when posteriorly polarized
motile monocilia, in a pitlike structure called the node,
rotate clockwise (observed from the distal end toward the
basal plane) so as to drive extracellular fluid toward the
left of the embryonic node (nodal flow). In response to
this flow, L-R asymmetric gene expression is initiated
(3,4). The monocilia in the embryonic node are primary
9þ0 cilia. Their characteristic architecture is based on a cylindrical arrangement of nine doublet microtubules (MTs),
as shown in Fig. 1 a. Axonemal dynein molecules are
located between adjacent MT doublets. By converting the
chemical energy of ATP hydrolysis into mechanical work,
these motor proteins drive sliding of the doublet MTs (5),
resulting in rotational movement of the nodal cilia (6).
The effective movement of nodal cilia is therefore crucial
for correctly directing L-R specification (4,7–11). However,
the mechanism underlying the correct spatio-temporal activation of the axonemal dynein molecules has yet to be
described.
Much effort from both experimental and theoretical science has been invested into studying axonemal dynein funcSubmitted March 18, 2015, and accepted for publication May 18, 2015.
*Correspondence: [email protected]
tion and regulation. However, most of the existing studies
focus on the 9þ2 cilia. Major experimental advances,
among others, include the observation of dynein-driven
MT sliding in isolated axonemes (12–14), structural analysis of dyneins (15–17), mechanical functions of the inner
and outer dynein arms (18–20), the force-generation mechanism of dyneins (21,22), measurements of dynein force
(23–25), and the coordination of dynein activity along
MTs (26,27). Theoretical works have mainly focused on
dynein activation patterns, something that remains beyond
the scope of contemporary observation. Pioneer studies,
among others, include geometric clutch theory (28–30)
and its improvement to a continuum model (31), a switchpoint model (32–34), and a curvature-controlled model
(35–37). These studies, simulating the ciliary planar
beating, are based on a switching theory, which has then
been confirmed by experiments conducted by Shingyoji
et al. (23). Sugino and Naitoh (38), Sugino and Machermer
(39), Gueron and Levit-Gurevich (40), and Gueron and
Liron (41) expand the research to the three-dimensional
movement of cilia in Paramecium. This is of significant
importance to promote our understanding regarding the
dynein activity in nonplanarlike movement of cilia.
Nodal cilia present a different ultrastructure than the conventional 9þ2 cilia/flagella. Precise observation on them is
more challenging. Most of the existing studies on nodal cilia
focus on three issues—how do the cilia generate the unidirectional flow, how does the nodal flow induce signal transmission, and what is the force-generation mechanism of
Editor: Kazuhiro Oiwa.
2015 by the Biophysical Society
0006-3495/15/07/0001/14 $2.00
http://dx.doi.org/10.1016/j.bpj.2015.05.027
BPJ 6589
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
2
Chen and Zhong
nodal cilia. The first two issues were studied at multilevels,
both experimentally (4,42–46) and theoretically (47–50);
however, due to the inherent difficulties in experiments,
direct evidence of the third issue is hard to achieve. Computational simulations can provide alternative solutions to the
force-generation mechanism. However, due to the difference in ultrastructure and moving patterns, most of the
previous models designed for 9þ2 cilia/flagella cannot be
directly applied. Existing theoretical models simulating
the particular dynein-triggered rotation of nodal cilia, to
the best knowledge of the authors, include the study by Hilfinger and Jülicher (51) and the model proposed by Brokaw
(37). The former simulated the ciliary dynamics by performing an analytical study of linearly unstable modes and the
latter proposed a three-dimensional model with active force
regulated by sliding velocity. Both of the simulations have
reproduced the circling motion of nodal cilia.
In this article, we establish a three-dimensional model
that mimics nodal ciliary ultrastructure. The sliding forces
of the activated dyneins are embedded in the model at a prescribed spatial position and time. The material surrounding
MTs serves as the passive elastic component that provides
transverse force between the doublets and maintains the
ciliary overall structure. The dynein-triggered ciliary motion is computed by the finite-element method. Various
dynein activation patterns are assessed in this model and
the rationality of each pattern is evaluated by comparing
the simulated dynein-driven ciliary motion to the observed
data of nodal cilia. We particularly focus on two issues: 1)
the direction in which the initiation of dynein activity is
transferred; and 2) the pattern of dynein activation along a
doublet MT.
MATERIALS AND METHODS
Establishment of the three-dimensional model
The ciliary body is modeled as a composite of a cylindrical body (1.8 mm in
height and 0.3 mm in diameter) with a parabolic tip (0.2 mm in height). The
MT doublets consist of a complete A-tubule and an incomplete B-tubule.
The cross section of a doublet MT is thus modeled as an ellipse with
0.03 mm in the major diameter and 0.017 mm in the minor diameter.
Fig. 1 b displays this model. The spatial resolution of the ciliary model
comprises 153 points in the vertical direction of the cylindrical body and
MTs. This indicates an ~12-nm distance between the adjacent grid nodes
in the longitudinal direction. Reports regarding the exact location of dyneins in nodal cilia are lacking. However, it is generally believed that the
outer dynein arms, in eukaryotic cilia/flagella, are bound to MTs with an
interval of 24 nm (52,53), while the interval of inner dynein arms varies
according to its species (54). Recent study indicates that the inner-arm
dynein can only generate beating by cooperating with the central pair/
radial spokes and the nodal cilia may lack the inner-arm (55). Giving the
9þ0 architecture of nodal cilia, in this model, we only consider the mechanical effects of the outer-arm dyneins, which are believed to produce
more force than the inner-arm, and assume their interval is the same as
that previously reported for other cilia types (24 nm). Thus, the distance
between adjacent grid nodes along the MTs in the model (12 nm) is just
one-half of this reported interval. The tip of the ciliary model is discretized
using the unstructured tetrahedral grid and the cylindrical body, including
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the doublet MTs that are extruded by triangular prisms. The grid configuration of the model consists of a total of 651,732 cells. Fig. 1 c displays the
grid resolution of the model.
Mechanical properties
The elastic modulus of MTs is estimated by either applying bending force
on the MTs (56–59) or studying the thermally induced shape fluctuations
(58,60). The reported elasticity of individual MTs ranges between 0.1
and 1.2 GPa. On the other hand, Ishijima and Hiramoto (61) reported that
the flexural rigidity of doublets ranges between 1.4 and 38 1023 Nm2
in 0.1 mM ATP medium and suggested that the minimum value (1.4 1023 Nm2) may indicate the rigidity of a single doublet MT. Considering
the elliptical cross section of the doublets in our model, the Young’s
modulus of them is calculated to be 0.62 GPa, which is also within the range
of the reported stiffness of single MTs (0.1–1.2 GPa).
Literature on the elastic properties of ciliary cytoplasm is lacking. The
elastic properties of the cellular cytoplasm ranges between 25 and 5 105 Pa (62,63); however, due to the difference in components (the mechanical properties of cell cytoplasm depend on the concentrations of actin filaments, myosin filaments, intermediate filaments, and microtubules), this
value cannot be directly applied to ciliary cytoplasm and the actual value
for ciliary cytoplasm should be smaller. In order to determine the proper
elastic moduli of the ciliary cytoplasm for nodal cilia, a parameter study
is conducted over the possible range of the Young’s modulus for the cytoplasmic component (5–103 Pa). The displacement magnitude of the ciliary
tip is recorded and its variation corresponding to the stiffness of the cytoplasm is shown in Fig. S1 in the Supporting Material. The ciliary bending
direction is same at different cytoplasmic stiffness conditions; however, the
deformation magnitude varies. The maximum deformation on the ciliary tip
occurs when the stiffness of the cytoplasm is ~100 Pa. This is because cytoplasm functions as the passive elastic component of the ciliary body, assisting in maintaining the overall structure of the cilium; when the stiffness of
the cytoplasm is too small, it cannot properly maintain the ciliary structure
and the deformation is mainly incurred on the dynein-force affected doublets. However, when the stiffness of the cytoplasm is too high, it resists
MT sliding and thus the deformation of the overall ciliary body is also
small. For the particular Young’s modulus of the doublet MTs (0.62
GPa), this elastic modulus of the cytoplasm (100 Pa) allows the most flexibility of the ciliary movement, which is therefore employed in the computational model.
When activated, the dynein motors located on one doublet MT forms a
transient attachment to its adjacent counterpart and pushes the target MT
tipwards (23,64). At the same time, the reaction force drives the dyneinlocated MT toward the ciliary base, thereby inducing MT sliding. This mechanical function of the dyneins allows us to model them as pairs of point
loads, with the same magnitude but working in opposite directions, along
the doublets. As shown in the magnified image in Fig. 1 b, along each
MT doublet, the grid discretization interval is 12 nm in the longitudinal direction. This indicates a distance of 24 nm between every other grid points
along the doublet, which is precisely the spatial interval of dyneins
observed in nodal cilia. Thus, the point loads that mimic the dynein forces
are added at every second grid point along the doublet MTs in the model
(displayed by the orange spheres in Fig. 1 b). The force exerted by a single
dynein arm is reported to be 1–10 pN (24,25,65–67).
During computation, the mechanical functions of the activated dyneins
are modeled as pairs of point loads with the magnitude of 5 pN, the intermediate value of the reported force magnitude (1–10 pN). The point loads
work in opposite directions, acting on corresponding sites located on the
dynein-located doublet and the target adjacent doublet, respectively. The
forces of other deactivated dyneins are assigned to zero.
It should be noted that this work focuses on simulating the dynein-driven
ciliary moving pattern, which is mainly determined by the configuration of
the ciliary ultrastructure and the location of dynein forces. The lack of precise mechanical parameters of the model may reduce the accuracy of the
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
Dynein Activation in Nodal Cilia
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FIGURE 1 Nodal ciliary ultrastructure and the
three-dimensional model. (a) Ciliary ultrastructure
by a transmission electron microscopy image and
an illustration diagram of the ciliary cross section;
(b) geometry of the computational model; (c) grid
resolution of the model. To see this figure in color,
go online.
calculated ciliary bending magnitude but the bending direction (or ciliary
moving pattern) could be reasonably simulated. A test on the ciliary deformation with different elasticity and dynein force is shown in Fig. S2. It
confirmed that the uncertainty of mechanical properties of the model
does not affect the calculated ciliary bending direction, and thus ensures
the feasibility to apply this model to study dynein activation pattern.
the displacement between the moved boundaries, and is based on the solution of the equations of linear elasticity. In this method, the motion of the
mesh u is governed by the equilibrium equations of elasticity,
V , s þ f ¼ 0;
(3)
where f is the prescribed body force. The stress tensor s is related to the
strain tensor ε as
Numerical methods
s ¼ lTrðεÞI þ 2mε;
The motility of cilia is calculated by embedding the dynein forces on
the discretized elements in the model (properly located in space and time
according to the scenarios proposed in the Results). The problem is solved
by using a stress solver in CFD-ACEþ (ESI CFD, France; www.esi-group.
com), based on the finite-element method. The equilibrium of the system is
described as
Kd ¼ P;
(1)
where d denotes the node displacement vector, K is the stiffness matrix, and
P denotes the load vector. The global stiffness matrix K is assembled by the
stiffness matrix of each element, which is described as
where Tr is the trace, and l and m are the Lamé constants. The strain tensor
is related to the displacement gradients as
ε ¼
BT DBdU;
Ue
(5)
u ¼ g on Gg ;
n , s ¼ h on Gh ;
(2)
where Ue indicates the domain of each element, B is the strain-displacement
matrix, and D is the elasticity matrix containing the material properties.
The finite-element method allows explicit input of the point loads on particular grid nodes by modification on the load vector P, thus, facilitating the
addition of the transient dynein forces into the model. Because the cytoplasm and MTs are treated as the elastic continuous materials, during
dynein activation, these materials produce passive elastic resistance, induce
internal transverse forces, and maintain the doublet distance and ciliary
structure.
The solid-body elasticity analogy method (68) is employed to update
mesh deformation, where the remeshing problem is solved by calculating
1
Vu þ VuT :
2
Consider the elastic body occupying a bounded region U with boundary G.
The Dirichlet and Neumann-type boundary conditions are imposed by the
motion of the boundaries and interfaces, respectively, as
Z
ke ¼
(4)
(6)
where Gg and Gh are complementary subsets of G. The finite-element function spaces are constructed as
n
Uh ¼ uh uh ˛H h ðUÞ ; uh ¼ gh on Gg ;
n
Fh ¼ fh fh ˛H h ðUÞ ; fh ¼ 0 on Gg :
(7)
The mesh moving scheme can thus be constructed as follows: find uh ˛Uh
such that cfh ˛Fh , giving
Z
U
ε fh : s uh dU Z
Z
f , f dU ¼
h
U
Gh
fh , hdG:
(8)
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Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
4
Chen and Zhong
The computational platform we used allows the concurrent embedding of
the force-generating elements of the axoneme, along with the passive
deformable elements of the structure into a unified framework, permitting
the computation of self-induced deformation and motion for this proteinstructure system.
RESULTS
When the dynein motor is activated, a thin stalk is extended
from the globular dynein head, which is a cross bridge to the
adjacent doublet MT, pushing the target MT toward the
plus-end (69). To effectively establish the dynein bridge,
the dynein must span the interdoublet gap; however, during
MT sliding, the interdoublet distance may change. This
implies that the binding and dissociation of dynein arms
to the target MT may be affected by the interdoublet distance: the reduction of distance facilitates the formation of
dynein bridges, while the enlargement of distance induces
dynein detachment. In other words, the initiation of dynein
activity among the nine doublet MTs might be triggered by
the variant interdoublet distance during ciliary motility. In
fact, this distance-controlled hypothesis is supported by
many theoretical studies (28,30,31) and by the experiments
of Shingyoji et al. (23), which found that a singlet MT
crossing a doublet MT at the angle of 45–90 could interact
with only 2–4 dynein arms, and by decreasing this angle, the
number of dynein arms that could possibly interact with the
MT increases.
To investigate the time-variant interdoublet distance during nodal ciliary movement, we first computed the ciliary
bending induced by the dyneins between a pair of doublet
MTs (doublets 1 and 2). The dynein bridges between two
adjacent doublet MTs are established in a sequential manner
(27) and because the base cell supplies energy to the dynein
activity, the dynein binding is likely incurred from the
ciliary base and climbs toward the ciliary tip. In this study,
we define the time cost by the dynein binding between two
adjacent doublet MTs as TMT, and in this section, we assume
this value is equal to 1/9 of the cycle, which means the dyneins finish attachment to the target doublet before the
dynein activity between the next pair of double initiates.
To investigate the interdoublet distance change during MT
sliding, we considered the linear dynein-bridge climbing
manner, i.e., the activation of each dynein along one MT
initiates from the base and climbs up toward the ciliary tip
with equal time intervals, and assigns it to be completed
in a 1/9 cycle (TMT ¼ 1/9 ciliary beating cycle). Other
dynein activation patterns and dynein-acting-time periods
will be discussed in the next section.
The rotational cycle of nodal cilia is 10 Hz (70). If
the dynein activation between two adjacent doublet MTs
is completed in a 1/9 cycle, the height of the dynein
bridges between these doublets can be represented by h ¼
163.64,t mm, where t denotes time. Fig. 2 displays the simuBiophysical Journal 109(1) 1–14
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The initiation of dynein activity between doublets
FIGURE 2 Ciliary deformation triggered by dynein forces between doublets 1 and 2. (Left) Longitudinal stress distribution along doublets 1 and 2,
when all of the dyneins finish attachment between them, are shown; (right)
stress distribution along doublets 2 and 3. The cross-sectional image shows
the change of the interdoublet distance. The distance between doublets 1
and 2 (DMT12) is enlarged while the distance between doublets 2 and 3
(DMT23) is reduced. To see this figure in color, go online.
lation results at t ¼ 0.011s (1/9 cycle), when all of the
dynein bridges between doublets 1 and 2 have established.
Doublets 1–3 are contoured by the stress along the longitudinal direction of the MTs (szz). The established dynein
cross links produce a positive szz on the targeting doublet
2 (toward the plus-end) and negative szz on the dyneinlocated doublet 1. The cilium bends approximately along
the plane determined by doublets 1 and 2 and toward
doublet 2. Detailed discussion regarding this bending direction is presented in the Supporting Material. As shown in the
left panel of Fig. 2, it is exactly the sliding forces that induce
MT bending and thus generate ciliary bending. At this time
point, distance between doublets 1 and 2 (DMT12) gradually
increases from the base to the tip, while the distance between doublets 2 and 3 (DMT23) reduces. A cross section
at the height of 1.78 mm (99% of the MT length) is extracted
from the computed result (the right panel of Fig. 2), where
the reduction of DMT23 can be clearly revealed.
To further quantify the distance variations, the DMT mn
(where m and n indicates the doublet MT numbers according to Fig. 2) is calculated along the MTs at different time
points. Fig. 3, a–c, shows the distance variations at 3.65,
7.31, and 11.11 ms, respectively, which are approximately
the 1/3, 2/3, and end of the first 1/9 cycle (11.11 ms).
With the dynein bridges formed between doublets 1 and 2,
the interdoublet distance is changing from its original value
(30.82 nm). At the 1/3 of the first 1/9 cycle (Fig. 3 a), the
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
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Dynein Activation in Nodal Cilia
FIGURE 3 (a–f) Variation of interdoublet distance during the dynein-driven ciliary motion. To see this figure in color, go online.
maximum reduction (2.5 nm) and the maximum increase
(2.33 nm) of interdoublet gaps occurred on DMT23 and
DMT91, respectively, indicating that the dynein bridge establishment between a pair of adjacent doublet MTs (doublets
1 and 2 in this case) would mostly affect the interdoublet
distance of their higher-numbered (doublets 2 and 3) and
lower-numbered (doublets 9 and 1) neighbors. At the 2/3
(Fig. 3 b) and the end (Fig. 3 c) of the first 1/9 cycle, the distance variation continues, with the dynein binding climbing
up along doublets 1 and 2. The position where the maximum
increase of the interdoublet gap occurred moves from
1302.62 to 1800 nm of the ciliary height, corresponding to
the linear dynein activation pattern employed in this case.
At the end of the first 1/9 cycle (Fig. 3 c), all the dynein
arms along doublet 1 finish attachment to doublet 2 and
the cilium bends to the maximum. The maximum increase
of the interdoublet distance still occurs on DMT91, which is
22.89 nm, 74% of the original interdoublet gap. And the distance between doublets 2 and 3 (DMT23) is the only interdoublet gap that decreases during the ciliary bending, the
maximum reduction of which is 4.2 nm, 14% of the original
interdoublet gap. Based on the proposed distance-controlled
hypothesis, this indicates that a) after the dynein establish-
ment completed between doublets 1 and 2, the initiation
of dynein activity should be transferred to the dyneins
located between doublets 2 and 3; and b) because the interdoublet gap increases most at the ciliary tip, the dynein
bridges between doublets 1 and 2 should dissociate in the
direction from the ciliary tip to the base.
Fig. 3, d–f, shows the distance variations at the 1/3, 2/3,
and end of the second 1/9 cycle, respectively, when the
dynein bindings are initiated between doublets 2 and 3
and at the same time the dynein dissociation occurs between
doublets 1 and 2. During this process, DMT34 is gradually
reduced, while the distances among other doublet increase.
The position of the maximum reduction of DMT23 moves
from the ciliary tip to the base (Fig. 3, c–e, indicated by
arrows) until vanished (Fig. 3 f), leaving the distance between the dynein-binding doublet (2,3) greater than the
original value all over the MT length. At the end of the second 1/9 cycle, the DMT12 presents the maximum increase
and the DMT34 is the only distance reduced, indicating
dynein activity is going to be transferred to the dyneins between doublets 3 and 4. In general, this computational test
implicates that the initiation of dynein activity among doublets is occurring in a clockwise manner (observed from
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Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
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Chen and Zhong
above). More specifically, when the dynein bridges finish establishing between doublets N and Nþ1, the dynein activity
between doublets Nþ1 and Nþ2 is triggered by the reduced
interdoublet distance, and the dynein detachment is started
between doublets N and Nþ1, from the ciliary tip to the
base, due to the increased interdoublet gap.
The dynein activation pattern along individual
doublet MTs
The interdoublet distance study proposes the clockwise
initiation sequence of dynein activity among doublets; however, the dynein activation pattern along individual doublet
MTs remains unanswered. Previous experimental studies
suggested that, during activation, dynein activity progresses
along MTs systematically, as one active dynein arm stimulates the next; and suggested that the dynein activity along
individual doublet MTs would complete in ~1/9 cycle (27)
(TMT z 1/9 ciliary beating cycle). They provide basic understandings about the dynein activation along MTs; however, the specific pattern of dynein progression and the
exact time period to complete dynein activation along one
MT has not been discussed in detail. To date, experimental
observations on dynein activity of nodal cilia in real time are
very challenging; to investigate these two issues (dynein
progression pattern and acting time), computational studies
could be an alternative solution.
In this work, we investigate two types of acting time of
the dynein activation along individual doublet MT. The first
scenario suggests the time period is exactly 1/9 cycle, i.e.,
TMT ¼ 1/9 cycle, named the 1/9 cycle scenario, which indicates the dynein activation between doublets Nþ1 and Nþ2
initiates after the dynein bridge establishment finishes between doublets N and Nþ1. On the other hand, the second
scenario, named the time-lapped scenario, suggests the
dynein activation between doublets Nþ1 and Nþ2 could
start before the dynein binding between doublets N and
Nþ1 finishes; in other words, the completing time of dynein
activation along one doublet MT could be longer than the
1/9 cycle (TMT > 1/9 cycle). In each scenario, we study
four dynein activation patterns—the linear, sine, circle,
and ellipse patterns. The rationality of each pattern is discussed in the 1/9 cycle scenario and a proper dynein activation pattern that is able to generate the smooth rotational
movement of the cilium is proposed, and in the time-lapped
scenario, the proper time periods for each pattern that can
produce a circular trajectory of ciliary motion are proposed.
Detailed results are presented as follows.
The ciliary moving patterns in the 1/9 cycle
scenario
In this scenario, the dynein bridges climb up along the
doublet MT in exactly 1/9 cycle, and in the next 1/9 cycle,
the dynein arms established in the original pair of doublets
Biophysical Journal 109(1) 1–14
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(doublets N and Nþ1) dissociate while the dynein binding
starts in the next pair of the doublets clockwise (doublets
Nþ1 and Nþ2). As shown in Fig. 4, if we draw a diagram
putting time on the horizontal axis and the dynein binding
height on the vertical axis, the curve that describes the
dynein activation pattern along doublets N and Nþ1 must
pass the points [0, 0], [0.011, 1.8], and [0.022, 0] (indicated
by circles) and the curve that describes the dynein activity in
doublets Nþ1 and Nþ2 must pass the points [0.011, 0],
[0.022, 1.8], and [0.033, 0].
To satisfy these conditions, we therefore propose four
dynein activation patterns—the linear, sine, circle, and
ellipse patterns (Fig. 4 a)—and simulate their-induced
ciliary motion. The functions of the linear, sine, circle,
and ellipse patterns are shown in Eqs. 9–12, respectively,
8
T
>
>
kL , t
0<t%
>
<
9
hlinear ¼
(9)
>
T
T
2T
>
>
: hMT kL , t <t%
;
9
9
9
hsine ¼ hMT , sinðkS , tÞ;
1
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2
T
A , 106 mm=s;
¼ @kC1 þ kC2 t 9
(10)
0
hcircle
and
hellipse
vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
u
2 , !
u
T
t
¼ hMT ,
kE ;
1s2 t 9
(11)
(12)
where t denotes time with the unit of seconds and h denotes
the dynein activation height with the unit of micrometers.
The parameters of Eqs. 9–12 are displayed in Table 1. In
the linear pattern, the dynein climbing velocity is constant
(162 mm/s). However, in other patterns, the dynein activation velocity is time-variant; the dynein binding is formed
faster at the basal region of the MTs and slower near the
ciliary tip (Fig. 4 b). In the sine, circle, and ellipse patterns,
the maximum velocity values are 254.72, 322.86, and
2410.7 mm/s, respectively, occurring at the beginning of
dynein activation or ending of dynein detachment. It should
be noted that the dynein activation velocity discussed here,
describing how fast one active dynein stimulates the next, is
different from the sliding velocity investigated in other
studies (71).
Fig. 5 b displays the tip trajectory of the dynein-driven
ciliary motion simulated under these proposed dynein activation patterns. All of the four dynein activation patterns,
in general, are capable of generating a clockwise rotational
movement of the cilium (Fig. 5 a); however, the specific tip
trajectories are different. As shown in Fig. 5 b, the cilium is
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
7
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Dynein Activation in Nodal Cilia
FIGURE 4 Dynein activation pattern along a doublet microtubule. (a) Linear, sine, circle, ellipse, and polynomial dynein activation patterns in the 1/9
cycle scenario; (b) climbing velocities of the dynein activation patterns; (c) linear, sine, circle, and ellipse patterns in the time-lapped scenario. To see
this figure in color, go online.
rotating in a wavelike fashion. At the end of each 1/9 cycle,
all of the dynein bridges finish establishment between the
two adjacent doublet MTs while the dyneins on other
MTs have not been activated yet; therefore, the ciliary
deformation at these time points is the same in all of the
four dynein activation patterns (indicated by red circle in
Fig. 5 b). During other times, due to the different dynein
activation velocities, the dynein binding heights are
different in the four patterns and thus induce various wave
crests in the ciliary tip trajectory. The wave crests in
Fig. 5 b are actually consistent with the curve slopes in
Fig. 4 a.
Ideally, the nodal cilia should present circular tip trajectory during the rotation, as indicated by the black curve
(arrow pointed) in Fig. 5 b. This implicates, in this 1/9 cycle
scenario, that the curve describing the time-variant dynein
activation height should lie between the linear function
and the sine function in Fig. 4 a. We therefore employ a
piecewise quadratic polynomial function to describe the
dynein activation pattern. Based on various computational
tests, the function is determined as Eq. 13, where the parameters are listed in Table 1. This polynomial function that
regulates dynein activation is presented in Fig. 4 a by the
orange curve (arrow pointed), and the ciliary tip trajectory
under this dynein activation pattern is shown by the orange
squares in Fig. 5 b, which indeed presents a circular movement of the nodal cilium as observed by experiments. Fig. 6
a shows three snapshots of the stress (szz) distribution along
TABLE 1
Parameter
(Unit)
hMT (mm)
T (s)
kL (mm/s)
kS (rad/s)
Parameter list
Value
1.8
0.1
162.162
141.513
Parameter
(Unit)
Value
kC1 (s)
kC2 (s2)
kE (s2)
kP1 (mm/s2)
34.225
1171.351
0.000123
9000
Parameter
(Unit)
2
kP2 (mm/s )
hE (mm)
tE1 (s)
tE2 (s)
Value
64,696
1.907
0.0098
0.0121
the MTs during ciliary motion under the proposed polynomial function. The solid arrows indicate dynein attachment
while the open arrows indicate dynein dissociation. At time
point T1, the dynein motors finish attachment between doublets I and II; at T2, the dynein bridge activation initiates between doublets II and III, while detachment occurs between
doublets I and II; and at T3, the dynein establishment finishes between doublets II and III, while detachment finishes
between doublets I and II. The stress distribution indicates
that the dynein forces produce the rotational movement of
the ciliary body:
8
>
kP1 , ðt 0:0146sÞ2 þ hE
ð0<t%tE1 Þ
>
>
>
>
>
2
>
T
<
kP2 , t þ hMT
ðtE1 <t%tE2 Þ
hpolynomial ¼
9
>
>
>
>
>
2T
2
>
>
:
tE2 <t%
: kP1 ,ðt 0:00762sÞ þ hE
9
(13)
The ciliary moving pattern in the time-lapped
scenario
The dynein activation time has not been reported precisely.
In the previous section, we assume that it costs 1/9 of the
ciliary beating cycle to complete the dynein activation along
one doublet MT, which means the dynein binding in doublets Nþ1 and Nþ2 initiates exactly after the dynein attachment finishes between doublets N and Nþ1. In this section,
other time periods are considered. It means the dynein activation in doublets Nþ1 and Nþ2 can be initiated before the
dynein binding is completed in doublets N and Nþ1; in
other words, there could be an overlapped time when the
dynein activation in doublets N and Nþ1 and that in doublets Nþ1 and Nþ2 (the overlapped time is defined as TL
from here on) takes place. Various time periods for the
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BPJ 6589
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
Chen and Zhong
Web 3C
8
FIGURE 5 Simulated ciliary motion. (a) Rotation of the ciliary body under the polynomial dynein activation pattern in the 1/9 cycle scenario; (b) ciliary tip
trajectories of the linear, sine, circle, ellipse, and polynomial dynein activation patterns in the 1/9 cycle scenario; (c) ciliary tip trajectories of the sine, circle,
and ellipse patterns in the time-lapped scenario at k ¼ 0.6. To see this figure in color, go online.
tivity, and it is the same as the condition in the 1/9 cycle scenario; when k ¼ 1, all of the dyneins on different doublet
MTs initiate acting together, which would generate torsion
in the ciliary body (as shown in Movie S1 in the Supporting
Material), rather than the rotational movement as observed
in experiments. In fact, apart from the extreme cases (k ¼
0 and k ¼ 1), the overlap time TL should also be smaller
than 7/9 of the dynein activation time along an entire MT
(TMT), i.e., k % 7/9; when k > 7/9, the dynein attachment
and detachment would occur at the same doublet MT
at the same time, which is not likely to happen under
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overlapped time (TL) are investigated for the linear, sine, circle, and ellipse patterns (Fig. 4 c), and the criterion to select
the appropriate TL for each pattern is whether the proposed
dynein activation pattern under the particular time-control
could induce the circular tip trajectory of the ciliary motion.
In fact, the overlapped time can be defined by TL ¼
k,TMT, where the parameter k should lie in the range of
0–1, i.e., 0 < k < 1, and the full ciliary rotating cycle T
(0.1 s) is equal to 9,(1k),TMT, indicating the parameter
k and TMT should be correspondingly related. In the extreme
cases, when k ¼ 0, there is no time overlap of the dynein ac-
FIGURE 6 Stress distribution of the doublet microtubules. (a) Three snapshots of the ciliary motion under the polynomial dynein activation pattern in the
1/9 cycle scenario; (b) two snapshots of the ciliary motion under the sine pattern at k ¼ 0.6. (Solid arrows) Dynein activation; (open arrows, a) dynein detachment. The dynein activation/detachment in the time-lapped scenario involves multiple doublet microtubules. It is different to show all the doublets with activated dyneins during both the attachment and detachment processes in (b). More detailed dynein activation/detachment process is presented in Movie S2. To
see this figure in color, go online.
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BPJ 6589
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
Journal (2015), http://dx.doi.org/10.1016/j.bpj.2015.05.027
Dynein Activation in Nodal Cilia
9
‘‘real world’’ conditions. Thus, the actual testing range of k
is 0–0.77.
For each dynein activation pattern, we simulate the ciliary
motions when k ¼ 0.1–0.7 with 0.1 intervals and when
k ¼ 0.77 (eight cases in total). Fig. 7, a–d, shows the tip trajectories of the dynein-driven ciliary motion for the linear,
sine, circle, and ellipse cases, respectively; in each subfigure, the ciliary tip trajectories at k ¼ 0 (1/9 cycle scenario),
0.2, 0.4, 0.6, and 0.77 are displayed. In the linear case, the
higher value of k induces smoother tip trajectory of the
ciliary rotation; however, even if the k value reaches to its
maximum (0.77), it still cannot produce the smooth circular
tip trajectory of the ciliary motion, indicating the dynein
activation velocity along each MT should not be constant.
In the sine and circle patterns, smoother rotation of the cilia
are found when k ¼ 0.6 (red-colored curves in Fig. 7), while
higher values of k (>0.6) induce angular configurations on
the calculated ciliary tip trajectory, which is possibly due
to the more linear feature of the sine and circle curves at
the initiation of dynein establishment. In the ellipse pattern,
angular configuration of the ciliary tip trajectory is also
found when k > 0.6. Moreover, the maximum deformation
of the cilium is reduced when k > 0.6, as shown by the blue
curve in Fig. 7 d at k ¼ 0.77. In all other cases, higher k induces larger deformation of the cilium, which is because the
larger overlapping time of the dynein activation among MTs
would involve more dyneins being activated at the same
time, generating more sliding force and therefore producing
larger ciliary deformation. In fact, as shown in Fig. 5 c, at
k ¼ 0.6, the maximum ciliary deformation of the ellipse
case is larger than the circle and sine cases, and this is
due to faster dynein activation in the ellipse case, which
would involve more motor proteins to generate the internal
force for the ciliary bending. However, as mentioned before,
if we let all the dyneins along each of the MTs be activated
together, they would produce a torsion on the ciliary body
with much smaller radial deformation than the rotational
movement. This indicates the dynein climbing velocity
and the overlapped time TL (or k) must work in a coordinated way to produce the effective ciliary rotation, or in
other words, find a balance between bending deformation
and body torsion of the cilium. In the ellipse case, the dynein
climbing velocity is larger than all other cases at the initiation of dynein activity; thus the higher k (>0.6) makes the
body torsion dominate the ciliary movement and then produces the ciliary rotation with smaller deformations.
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FIGURE 7 Ciliary tip trajectory in the time-lapped scenario. (a–d) Computed ciliary tip trajectories in the linear, sine, circle, and ellipse
patterns, respectively. To see this figure in color,
go online.
Biophysical Journal 109(1) 1–14
BPJ 6589
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
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10
Chen and Zhong
The dynein-driven ciliary motion under the sine pattern at
k ¼ 0.6 is shown in Movie S2 and two snapshots of the stress
distribution of the MTs is shown in Fig. 6 b. In contrast to
Fig. 6 a (the 1/9 cycle scenario), dynein attachment occurs
along multiple doublets (solid arrows indicated) and, during
the time T1-T2, the dynein activation is transferred clockwise. More dynein forces are involved in this case, thus
the generated ciliary deformation is greater than that in
the 1/9 cycle scenario. However, in comparison to the observation by Okada et al. (72), this deformation is still smaller.
This is because of the mechanical properties of the MTs
and cytoplasm assigned in this model, which affect the
simulated ciliary bending magnitude but not direction
(Fig. S2). Although these values were selected according
to previous studies and the parameter test, the smaller simulated ciliary deformation indicates that more accurate measurements of the elastic moduli of the particular nodal cilia
should be needed.
It should be noted that the clockwise transfer direction
of dynein activation among the nine doublets is proposed
by the distance-controlled hypothesis and is tested when
k ¼ 0. To ensure its rationality for use in this scenario,
the time-variant interdoublet distances is calculated again.
Fig. 8, as a representation, displays the results for the sine
pattern at k ¼ 0.6. Fig. 8, a–c, shows the distance variations
at 3.66, 7.33, and 10.99 ms, respectively, when the dynein
activation only occurs between doublets 1 and 2 (TMT ¼
27.78 ms, k ¼ 0.6). It shows that, during this period, the distance between doublets 2 and 3 (DMT23) is the only interdoublet gap being reduced. Fig. 8, d–f, shows the distance
variations at 14.65, 18.32, and 21.98 ms, respectively,
when the dynein activation between doublets 2 and 3 starts
to take place. These subfigures show that DMT23 is gradually
increased and the distance between doublets 3 and 4
(DMT34) is reduced. This confirmed the rationality of the
distance-controlled hypothesis to be applied in this scenario. Moreover, when the dynein establishment between
doublets 1 and 2 is nearly finished at the ciliary tip
(Fig. 8 f), greatest increase of interdoublet gap (DMT12) occurs at the ciliary tip, indicating the dynein detachment initiates from the ciliary tip and proceeds toward the ciliary
base.
DISCUSSION
The sliding of adjacent MTs during ciliary beating has been
known for a long time (73) and a variety of experimental
studies have confirmed that the axonemal dyneins, which
are large motor complexes converting chemical energy
into mechanical force between the MTs, are responsible
for ciliary motility (74,75). It is believed that, in order
to produce effective ciliary motion, the dynein motors
must be tightly regulated in a temporal and spatial manner;
however, due to the inherent difficulties in experimental
observations, the specific temporal and spatial patterns of
Biophysical Journal 109(1) 1–14
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dynein activity have not been reported, especially for nodal
cilia. The two main issues that remained unanswered are 1)
how is dynein activation transferred between the nine
doublet MTs, and 2) what is the dynein activation pattern
along each MT doublet.
For the first issue, it is proposed that the dynein activity
might depend on the separation between adjacent doublets
(28,30). Here we applied the same hypothesis to the
studies of nodal cilia: smaller interdoublet distance facilitates dynein activation, while larger interdoublet distances
reduce the possibility of effective dynein bridge establishment. This is actually not in contradiction to the curvature-based theory; the imposed bend of the axoneme
varies ciliary curvature and, at the same time, changes the
interdoublet distance between the split two doublet bundles.
Because experimental observations on separated MTs
of nodal cilia are currently not available, we conducted
computational studies to simulate ciliary movement based
on the distance-controlled hypothesis. The results indicate
that the dynein-triggered microtubule sliding between doublets N and Nþ1 would reduce the distance between
doublets Nþ1 and Nþ2 while increasing the distance between other adjacent pairs of doublet MTs. Moreover,
when all the dyneins between doublets N and Nþ1 are activated, the largest interdoublet distance between doublets N
and Nþ1 is found at the ciliary tip. In other words, among
the nine doublets, dynein activation should be transferred
clockwise, and along each individual doublet, the detachment of dynein bridges should occur from the ciliary tip
to the base. Indeed, further simulation on ciliary motility
confirms that this distance-controlled theory plays an
important role in directing the clockwise rotational movement of nodal cilia (Fig. 5). This also implies that, to
generate the rotational movement, the dyneins activation
should be transmitted from one pair of the neighboring doublets to another pair. This is different from the planar
beating cilia, where the movement is induced by two alternately activated dynein (or MT) sets and the switching
mode is directed by the central structure. Nodal cilia lack
a central structure and inner dynein arms (55), implying
that their unique movement may be due to a different
type of dynein regulation pattern, and also possibly to a special ultrastructure.
For the second issue, a key question is how long is TMT?
Previous studies assume this to be ~1/9 cycle of ciliary
beating (27), but the precise time period has not been
confirmed and no direct information is reported for nodal
cilia. Thus, we assessed two types of TMT—the 1/9 cycle
and other time periods (the time-lapped scenario). The results show that, to generate ciliary rotation with a circular
trajectory, dynein activation must occur faster near the
ciliary base and slower at the ciliary tip. If the TMT value
is equal to 1/9 of the ciliary moving cycle (0.011 s), the
possible dynein activation pattern along each MT could be
presented as Eq. 5. However, if the actual dynein activation
Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
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11
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Dynein Activation in Nodal Cilia
FIGURE 8 (a–f) Variation of interdoublet distance under the sine dynein activation pattern at k ¼ 0.6. To see this figure in color, go online.
pattern is similar to the sine, circle, or ellipse function, the
TMT would need to be ~0.028 s (k ¼ 0.6, TL ¼ 0.0168 s)
to produce the circular trajectory of nodal ciliary motion.
Indeed, Sugino and Naitoh (38) have proposed a similar
time-lapped dynein activation pattern in Paramecium,
which also presents a three-dimensional movement. The
overlap time (TL) plays an important role in this scenario.
Considering the 9þ0 structure of nodal cilia, a possible
structure regulating the timing of dynein activation could
be the nexin-dynein regulatory complex (defined in Heuser
et al. (16)), although the mechanism of the nexin-dynein
regulatory complex that regulates or influences MT sliding
has not been fully understood and its existence in nodal cilia
has not been confirmed. Future observation on this structure
in actively beating or rotating cilia would contribute to
revealing the overlap time and thus improve our understanding of the dynein activity in nodal cilia.
Compared to previous theoretical studies on the dynein
activation pattern in flagella/cilia, this work proposes a
possible new approach to model the mechanics of the
ciliary axoneme, which is able to mimic the ultrastructure
of the cilium realistically and allows us to evaluate
different dynein activation patterns by positioning the
stress vectors explicitly and treating the cilium as a complex elastic matrix that is deformable. As a conceptual
experiment, the computational results propose a possible
dynein activation pattern for nodal cilia. It should not
rule out other possibilities and the validation relies on
more advanced observation techniques on nodal cilia.
Indeed, Aoyama and Kamiya (26) have proposed an experimental method to separate doublet MTs in Chlamydomonas reinhardtii, observe their attachment and detachment,
and measure the attachment/detachment time. The ultrastructure, geometry, and beating pattern of this flagellum
is different from nodal cilia, thus their conclusions can
hardly be directly adopted in this study. However, the
experimental method they proposed may improve the observations on living cilia, making future direct validation
of this computational work or the measurement of the overlap time (TL) possible. Validations of this work could also
be conducted by studying mutant nodal cilia, in which the
dysfunction of the dyneins and/or positional change of the
MTs induce abnormal moving patterns of cilia. The proposed in silico work is able to model the ciliary ultrastructure realistically and the dynein forces are embedded
explicitly into the particular grid points, thus the modeling
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12
Chen and Zhong
method can be further applied to mimic the morphological
and positional change of the MTs as well as the dysfunction of dyneins. By comparing the simulated ciliary motion
to the lab-observed results, the rationality of the proposed
dynein activation patterns could also be evaluated. As of
this writing, experimental studies on the dynein activation
pattern or time in nodal cilia or the ultrastructural change
of mutant nodal cilia are lacking; the computational work
cannot be validated so far. However, we believe the distance-controlled theory determining the switch-on and
-off of the dyneins and the proposed dynein activation
pattern under different time conditions may provide insight
into the field and assist to explain the axonemal dynein
working mechanism in nodal cilia.
Limitations
This work suffers from two main limitations. First, precise
measurements of the mechanical properties of the doublet
MTs and the cytoplasm for nodal cilia are not available.
Although a parameter study has been conducted to determine the elastic moduli, more accurate simulation results
could only be achieved when better measurements of the
Young’s modulus of the isolated doublets and cytoplasm
for nodal cilia are available. Second, considering the
length of the nodal cilium and its slow rotating cycle,
the viscous drag induced by the surrounding fluid is expected to be small (an estimated comparison between the
drag torque and active elastic torque has been described
in the Supporting Material); thus it is not considered in
this model.
SUPPORTING MATERIAL
Supporting Materials and Methods, four figures, and two movies are available
at http://www.biophysj.org/biophysj/supplemental/S0006-3495(15)00539-1.
AUTHOR CONTRIBUTIONS
D.C. designed the study, performed the analysis on the interdoublet distance, and wrote the manuscript. Y.Z. performed the computation on the
dynein activation models.
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ACKNOWLEDGMENTS
The transmission electron microscopy image in Fig. 1 a is provided by Dr.
Patricia Goggin from the Primary Ciliary Dyskinesia Centre, University
Hospital Southampton National Health Service Foundation Trust and University of Southampton, Southampton, UK. D.C. thanks Dr. Kyosuke Shinohara from the Developmental Genetics Group, Graduate School of
Frontier Biosciences, Osaka University, Japan for providing the geometric
information about the ciliary length and also thanks Dr. Dominic Norris
from Medical Research Council Harwell, UK for reading and commenting
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Please cite this article in press as: Chen and Zhong, A Computational Model of Dynein Activation Patterns that Can Explain Nodal Cilia Rotation, Biophysical
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