Multiscale modeling of Cilia Coordination

Feng Ling

Kanso Bio-inspired Motion Lab, USC

April 23, 2022

SoCal Fluids

Why cilia?

Motile cilia are microscopic hair-like protrusions that direct fluid flow for many Eukaryotic life.

5μm

Daghlian (2006)

K.Y. Wan (2018)

10μm

Quadflagellate

Wan (2019)

30μm

Hamel, et al. (2010)

Paramecium

Coordination for diverse functions

30μm

Squid Light Organ

Nawroth, Guo, et al. (2017)

Lechtreck (2000)

Mouse Airway

10μm

Multi-synchrony for distinct swimming gaits

Coherent coordination for specialized functions

Nawroth, et al. (unpublished)

healthy

inflamed (IL13)

structure

function

less cilia

less coherent beat pattern

more eddies

slower flow

in-vitro human airway tissue

More than scientific curiosities

Understanding tissue- and organ-level functions of cilia have direct consequences on human health and disease!

The key is to enable functional prediction from structural observations!

Multiscale models for multiscale physics

To obtain principled understanding of cilia requires studying them across many length scales

Multiscale models for multiscale physics

1 mm\(^2\) of tissue can have a million cilia, and each driven by thousands of motor protein complexes!

Different abstraction levels are both necessary and desirable to truly understand cilia

Cilia Coordination

Guo, Man, et al., R. S. Interface (2020)

Man, Kanso, PRL (2020)

Kanale, Ling, et al., In preparation (2022)

Cilia Oscillation

Ling, Guo, Kanso, R. S. Interface (2018)

Man, Ling, Kanso, Phil. Trans. B (2020)

Ling, Kanso, In preparation (2022)

Ciliary Function

Guo, Ding, et al., JFM,PRE… (2014-2017)

Nawroth, Guo et al., PNAS (2017)

Nawroth, et al., Phil. Trans. B (2020)

Nawroth, Ling, et al., Submitted (2022)

Multiscale models for multiscale physics

1 mm\(^2\) of tissue can have a million cilia, and each driven by thousands of motor protein complexes!

Different abstraction levels are both necessary and desirable to truly understand cilia

Cilia Coordination

Guo, Man, et al., R. S. Interface (2020)

Man, Kanso, PRL (2020)

Kanale, Ling, et al., In preparation (2022)

Cilia Oscillation

Ling, Guo, Kanso, R. S. Interface (2018)

Man, Ling, Kanso, Phil. Trans. B (2020)

Ling, Kanso, In preparation (2022)

Ciliary Function

Guo, Ding, et al., JFM,PRE… (2014-2017)

Nawroth, Guo et al., PNAS (2017)

Nawroth, et al., Phil. Trans. B (2020)

Nawroth, Ling, et al., Submitted (2022)

Phenomenological model of cilia oscillation

Postulate: exact driving mechanism of cilia matter little when studying cilia coordination
e.g., symmetric cilia beats reproduced by follower force driven filament (Man 2020)
- Condense to two non-dimensional ratios:
  \(F=\|\bm{f}\|/(B/L^3)\) \(\rightarrow\) activity level
  \(\gamma=\ln(L/h)/\ln(L/a)\) \(\rightarrow\) coupling strength
  with an emergent time scale \(t_o=\zeta L^4/B\)

Man, Kanso, PRL (2020)

\(a\)

\(h\)

\(L\)

\(\bm{f}\)

Phenomenological model of cilia oscillation

Postulate: exact driving mechanism of cilia matter little when studying cilia coordination
e.g., symmetric cilia beats reproduced by follower force driven filament (Man 2020)
- Condense to two non-dimensional ratios:
  \(F=\|\bm{f}\|/(B/L^3)\) \(\rightarrow\) activity level
  \(\gamma=\ln(L/h)/\ln(L/a)\) \(\rightarrow\) coupling strength
  with an emergent time scale \(t_o=\zeta L^4/B\)
reveals multi-synchrony and bistability for a pair of hydrodynamically coupled cilia

initially in-phase

initially anti-phase

Man, Kanso, PRL (2020)

in-phase

anti-phase

Phenomenological model of cilia oscillation

Postulate: exact driving mechanism of cilia matter little when studying cilia coordination
e.g., symmetric cilia beats reproduced by follower force driven filament (Man 2020)
- Condense to two non-dimensional ratios:
  \(F=\|\bm{f}\|/(B/L^3)\) \(\rightarrow\) activity level
  \(\gamma=\ln(L/h)/\ln(L/a)\) \(\rightarrow\) coupling strength
  with an emergent time scale \(t_o=\zeta L^4/B\)
reveals multi-synchrony and bistability for a pair of hydrodynamically coupled cilia

initially in-phase

initially anti-phase

Man, Kanso, PRL (2020)

in-phase

anti-phase

Stability analysis as phased oscillators

Assuming coupled filaments always follow the same trajectory but at a different phase (Kuramoto reduction*), we can write an evolution equation of the phase difference \(\Delta\theta\):\[\partial_t(\Delta\theta) = \gamma\omega\Psi(\Delta\theta)\]

Man, Kanso, PRL (2020)

Kawamura, Tsubaki, PRE (2018)

in-phase

anti-phase

non-trivial

pitchfork bifurcations

saddle-node bifurcations

(weak hydrodynamic coupling)

Stability analysis as phased oscillators

Assuming coupled filaments always follow the same trajectory but at a different phase (Kuramoto reduction*), we can write an evolution equation of the phase difference \(\Delta\theta\):\[\partial_t(\Delta\theta) = \gamma\omega\Psi(\Delta\theta)\]

Man, Kanso, PRL (2020)

Kawamura, Tsubaki, PRE (2018)

in-phase

anti-phase

non-trivial

pitchfork bifurcations \(\rightarrow\)

saddle-node bifurcations \(\rightarrow\)

weak hydrodynamic coupling

How much can phased oscillator theory help us in understanding synchronization of many, many cilia?

\(\theta_i\)

⤸

Effective strokes / rotor tilt
\(\rightarrow\) first-harmonic forcing
Planar beats / rotor eccentricity
\(\rightarrow\) second harmonic forcing
Planar circular rotor with first and second harmonic forcing
- matches far-field flow
- do not produce net flow without hydrodynamic interactions