sorin mitran applied mathematics university of north carolina at chapel hill

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