1. Self-organization of ciliary motion: beat shapes and metachronicity
Sorin Mitran
Applied Mathematics
University of North Carolina at Chapel Hill

2. Overview Detailed cilia mathematical model
Beat shape (dynein synchronization)
Metachronal wave (cilia synchronization )
Coarse graining – a lung multiscale model

3. Cilia mathematical model
Goals
Model all mechanical components in cilium
Provide a computational framework to test cilia motion hypotheses
Investigate collective behavior of dynein molecular motors, patches of cilia
Model features
Fluid-structure interaction model
Finite element model of cilium axoneme
Two-layer airway surface liquid
Newtonian PCL
Viscoelastic mucus

4. 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

5. Axoneme mechanical model

6. Axoneme mechanical model

7. Axoneme mechanical model

8. Axoneme mechanical model

9. Dynein model One end fixed
One 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

10. Dynein model
Obtain average speed from least squares fit to experimental beat shapes
Here: 760±112 nm/s
Accepted range 1020±320 nm/s
(Taylor & Holwill, Nanotechnology 1999)

11. Airway surface liquid model
Bilayer ASL
Newtonian periciliary liquid (~6 microns)
Viscoelastic (Oldroyd-B) mucus layer (~30 microns)
Low Reynolds number (~10-4)
Computational approach
Overlapping grids
Moving grid around each cilium – transfers effect of other cilia
Background regular grid – transfers effect of boundary conditions

12. Equations
Stokes
Oldroyd-B

13. Moving grid formulation
Grid around cilium is orthogonal in 2 directions – efficient solution of Poisson equations through FFT

14. Velocity field around cilium

15. Beat shapes

16. Bending moments in axoneme
Maximum 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

17. Detail of moment near tip
Begining of recovery stroke

18. MT pair forces – begin power stroke1 9 2 8 3 7 4 6 5

19. MT pair forces – mid power stroke1 9 2 8 3 7 4 6 5

20. Normal stress on cilium
Average forces on cilium are similar in power/recovery
Propulsion of ASL due to asymmetry of shape
Power
stroke

21. Cilium motion

22. Force exerted on fluid

23. Modify ASL height

24. Structural defects Microtubule stress Normal axoneme
Axoneme with defect

25. Metachronal waves

26. How does synchronization arise?
Hypothesis: minimize work done by cilium against fluid

27. Start from random dynein phase

28. Allow phase to adjust

29. Metachronal wave results

30. Large-scale simulation

31. Effect of structural defects

32. Mucociliary transport

33. Coarse graining

34. Motivation Full computation of cilia induced flow is expensive
Extract force field exerted by cilia and impose on ASL model without cilia

35. Comparison of air-ASL entrainment
With cilia motion
No cilia motion

36. Conclusions Detailed model of mucociliary transport
Beat shape shown to result from simple constant velocity + noise of dynein
Metachronal waves result from hydrodynamic interaction effects and minimum work hypothesis