sorin mitran applied mathematics university of north carolina at chapel hill
TRANSCRIPT
Self-organization of ciliary motion:
beat shapes and metachronicity
Sorin Mitran
Applied Mathematics
University of North Carolina at Chapel Hill
Overview
Detailed cilia mathematical modelBeat shape (dynein synchronization)Metachronal wave (cilia synchronization )Coarse graining – a lung multiscale model
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
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
Axoneme mechanical modelX Y
Z
Axoneme mechanical modelX Y
Z
X Y
Z
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
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
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
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
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
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
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
Velocity field around cilium
X
Y
Z
Beat shapes
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
Begining of recovery stroke
Detail of moment near tip
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
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
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
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
Force exerted on fluid
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
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
Metachronal waves
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
Start from random dynein phase
Allow phase to adjust
Metachronal wave results
Large-scale simulation
Effect of structural defects
Mucociliary transport
Coarse graining
Full computation of cilia induced flow is expensive
Extract force field exerted by cilia and impose on ASL model without cilia
Motivation
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
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