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Self-organization of ciliary motion:
beat shapes and metachronicity
Sorin Mitran
Applied Mathematics
University of North Carolina at Chapel Hill
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Overview
Detailed cilia mathematical modelBeat shape (dynein synchronization)Metachronal wave (cilia synchronization )Coarse graining – a lung multiscale model
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Cilia mathematical modelGoals
Model all mechanical components in ciliumProvide a computational framework to test cilia
motion hypothesesInvestigate collective behavior of dynein
molecular motors, patches of ciliaModel features
Fluid-structure interaction modelFinite element model of cilium axonemeTwo-layer airway surface liquid
Newtonian PCLViscoelastic mucus
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Cilium axoneme – internal structure
Microtubule doublets – carry bending loads
Radial spokes, nexin, inner sheath, membrane – carry stretching loads
Dynein molecules – exert force between microtubule pairs
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Axoneme mechanical modelX Y
Z
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Axoneme mechanical modelX Y
Z
X Y
Z
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Axoneme mechanical modelX Y
Z
X Y
Z
Internal
Elastic Forc
Fl
A
D
e
ynein
Force
xoneme
Accelera
uid
F
t
o
io
r
n
ses
s
c
T
B
N
NV
NMBM
BV
TV
i
js u
vw
TM
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Axoneme mechanical modelX Y
Z
X Y
Z
T
B
N
NV
NMBM
BV
TV
i
js u
vw
TM
(
( )
( , )
)
fl
d
l
yn
e
F X X
F
X
XMX
F
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Dynein modelOne end fixedOne end moves at
constant speed + thermal noise
Force proportional to distance between attachment points
Advancing end can detach according to normal distribution centered at peak force 6pN
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Dynein modelObtain average speed from least
squares fit to experimental beat shapes
Here: 760±112 nm/sAccepted range 1020±320 nm/s(Taylor & Holwill, Nanotechnology 1999)
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
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Airway surface liquid modelBilayer ASL
Newtonian periciliary liquid (~6 microns)Viscoelastic (Oldroyd-B) mucus layer (~30
microns)Low Reynolds number (~10-4)
Computational approachOverlapping gridsMoving grid around each cilium – transfers
effect of other ciliaBackground regular grid – transfers effect of
boundary conditions
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Stokes
Oldroyd-B
Equations
0
2
u
uput
2
0
( )
2
t S
P
u
u u u p u
τ
τ τ D
X
Y
Z
-2
-1
0
1
2
3
0
X
Y
Z
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Moving grid formulation
cos1ln
cos1ln111
cos1
1
2
1
cos1
1
2
2
2
2
2
22
2
22
rrrrr
rrrr
rssr
Grid around cilium is orthogonal in 2 directions – efficient solution of Poisson equations through FFT
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Velocity field around cilium
X
Y
Z
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Beat shapes
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Bending moments in axonemeMaximum bending
moment in travels along axoneme
Out-of-plane beat shape results from fitted dynein stepping rate
During power stroke maximum bending moment is at 1/2-2/3 of length
During recovery stroke maximum at extremities
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Begining of recovery stroke
Detail of moment near tip
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MT pair forces – begin power stroke
12
3
456
7
8
9
-4 -3 -2 -1 0 1 2 3 40
1
2
3
4
5
6
x
y
Cilium beat shape
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MT pair forces – mid power stroke
12
3
456
7
8
9
-4 -3 -2 -1 0 1 2 3 40
1
2
3
4
5
6
x
y
Cilium beat shape
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Average forces on cilium are similar in power/recoveryPropulsion of ASL due to asymmetry of shape
Normal stress on cilium
X
Y
Z
P
3.43.232.82.62.42.221.81.61.41.210.80.60.40.2
X
Y
Z
P
3.43.232.82.62.42.221.81.61.41.210.80.60.40.2
Powerstroke
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Cilium motionX
YZ
P
3.43.232.82.62.42.221.81.61.41.210.80.6
X
YZ
P
3.43.232.82.62.42.221.81.61.41.210.80.6
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Force exerted on fluid
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Modify ASL height
-5-4-3-2-1012340
1
2
3
4
5
6
-0.20
0.2 y
x
Cilium beat shape
X
Y
Z
P
3.43.232.82.62.42.221.81.61.41.210.80.60.40.2
-4 -3 -2 -1 0 1 2 3 40
1
2
3
4
5
6
x
y
Cilium beat shape
X
Y
Z
P
3.43.232.82.62.42.221.81.61.41.210.80.60.40.2
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Structural defects
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
00.20.40.60.811.2
-0.4-0.2
00.2
0.40.6
-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
xy
z
Microtubule stressNormal axoneme
Microtubule stressAxoneme with defect
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Metachronal waves
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Hypothesis: minimize work done by cilium against fluid
How does synchronization arise?
Fdyneinm s, t pmscosk ms t m. #
W m ,n 0
1
0
Lpmscosk m s tn tn1 m ,n
x2n1s x1
n1s x2ns x1
ns ds d
m ,n1 12 m ,nW m ,n 2W m ,n m ,nW m ,n W m ,n
m ,n
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Start from random dynein phase
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Allow phase to adjust
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Metachronal wave results
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Large-scale simulation
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Effect of structural defects
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Mucociliary transport
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Coarse graining
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Full computation of cilia induced flow is expensive
Extract force field exerted by cilia and impose on ASL model without cilia
Motivation
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With cilia motion
Comparison of air-ASL entrainment
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
0 5 0 1 0 0 1 5 00
1 0
2 0
3 0
4 0
5 0
6 0
No cilia motion
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Detailed model of mucociliary transportBeat shape shown to result from simple
constant velocity + noise of dyneinMetachronal waves result from hydrodynamic
interaction effects and minimum work hypothesis
Conclusions