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

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

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

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 model

Axoneme mechanical model

Axoneme mechanical model

Axoneme mechanical model

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

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)

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

Equations Stokes Oldroyd-B

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

Velocity field around cilium

Beat shapes

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

Detail of moment near tip Begining of recovery stroke

MT pair forces – begin power stroke 1 9 2 8 3 7 4 6 5

MT pair forces – mid power stroke 1 9 2 8 3 7 4 6 5

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

Cilium motion

Force exerted on fluid

Modify ASL height

Structural defects Microtubule stress Normal axoneme Axoneme with defect

Metachronal waves

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

Start from random dynein phase

Allow phase to adjust

Metachronal wave results

Large-scale simulation

Effect of structural defects

Mucociliary transport

Coarse graining

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

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

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