Feng Ling
Postdoc @ Helmholtz Pioneer Campus, studying biophysics, cilia, and mucus flow. Looking for Universality from Diversity.
Flow Physics Explains Morphological Diversity of Ciliated Organs
Feng Ling
Nawroth Mechanobiology
Helmholtz Pioneer Campus
2023 APS March Meeting - PP08.08
Motile cilia are microscopic, hair-like cellular protrusions that direct fluid flow for almost all Eukaryotic life forms
Understanding how ciliated organ function has direct consequences on human health and diseases (airway, cerebral flow, reproduction, ...)
What is a ciliated organ?
Cilia manifest diverse motion despite their conserved structures
Cilia can coordinate in large numbers and organize in different spatial patterns for specialized flow functions
Roth, et al. (unpublished)
airway cilia
Larvacean funnel
Nawroth, Ling, et al.
20μm
Ciliated ducts that direct internal luminal flows are integral to animal physiology
Motile cilia are microscopic, hair-like cellular protrusions that direct fluid flow for almost all Eukaryotic life forms
Understanding how ciliated organ function has direct consequences on human health and diseases (airway, cerebral flow, reproduction, ...)
What is a ciliated organ?
Cilia manifest diverse motion despite their conserved structures
Cilia can coordinate in large numbers and organize in different spatial patterns for specialized flow functions
Roth, et al. (unpublished)
airway cilia
Larvacean funnel
Nawroth, Ling, et al.
20μm
Ciliated ducts that direct internal luminal flows are integral to animal physiology
Motile cilia are microscopic, hair-like cellular protrusions that direct fluid flow for almost all Eukaryotic life forms
Understanding how ciliated organ function has direct consequences on human health and diseases (airway, cerebral flow, reproduction, ...)
What is a ciliated organ?
Cilia manifest diverse motion despite their conserved structures
Cilia can coordinate in large numbers and organize in different spatial patterns for specialized flow functions
Roth, et al. (unpublished)
airway cilia
Larvacean funnel
Nawroth, Ling, et al.
20μm
Ciliated ducts that direct internal luminal flows are integral to animal physiology
Motile cilia are microscopic, hair-like cellular protrusions that direct fluid flow for almost all Eukaryotic life forms
Understanding how ciliated organ function has direct consequences on human health and diseases (airway, cerebral flow, reproduction, ...)
What is a ciliated organ?
Cilia manifest diverse motion despite their conserved structures
Cilia can coordinate in large numbers and organize in different spatial patterns for specialized flow functions
Roth, et al. (unpublished)
airway cilia
Larvacean funnel
Nawroth, Ling, et al.
20μm
Ciliated ducts that direct internal luminal flows are integral to animal physiology
Motile cilia are microscopic, hair-like cellular protrusions that direct fluid flow for almost all Eukaryotic life forms
Understanding how ciliated organ function has direct consequences on human health and diseases (airway, cerebral flow, reproduction, ...)
What is a ciliated organ?
Cilia manifest diverse motion despite their conserved structures
Cilia can coordinate in large numbers and organize in different spatial patterns for specialized flow functions
Roth, et al. (unpublished)
airway cilia
Larvacean funnel
Nawroth, Ling, et al.
20μm
Ciliated ducts that direct internal luminal flows are integral to animal physiology
Ciliated ducts come in different ‘shapes’
- Many animals have both "ciliary carpets," where short cilia grow perpendicular to the duct lumen, and "ciliary flames," where bundles of long cilia fill the duct lumen longitudinally.
- Are these morphological differences related to their function?
Ciliary Carpet
Ciliary Flames
Esophagus
Ciliated Funnel
Nawroth, Ling, et al (2023)
Curious case of the Giant Marine Larvacean
Ciliary Carpet
Ciliary Flames
- Both types of ciliated ducts produce significant luminal fluid flow.
- How can we probe the functional differences between these organs?
Nawroth, Ling, et al (2023)
Unified fluid model for all Ciliated Ducts
- Cilia generate coordinated waves to drive luminal fluid flow.
- We propose a quasi-1D duct model the can capture the essential flow physics for diverse duct morphologies AND enables the fair comparison of of their performance!
Nawroth, Ling, et al (2023)
Ciliary Carpet
Ciliary Flames
Motile cilia as Active porous media
- Specifically, we model cilia activity as a porous media with prescribed quasi-1D density wave that transfer ciliary waves to fluid motion via drag forces
- \(h/H\) - cilia-to-lumen ratio
- \(H/L\) - lumen diameter / ciliary wavelength
- Remaining inputs to the model are reference density of cilia \(\bar\rho_c\) and the adverse pressure \(\Delta P\).
Nawroth, Ling, et al (2023)
Motile cilia as Active porous media
Cilia-to-lumen Ratio h/H
Lumen Diameter H [μm]
Nawroth, Ling, et al (2023)
- We fix the total amount of input activity by constraining the total amount of active material present inside a channel by
- \((\bar\rho_cL)(h/2r_{pore})=\bar\rho_ch=\text{const.}\)
- This implies that carpet-like designs will need to be more densely distributed to be comparable to flame-like designs with the SAME lumen diameter
Quasi-1D metachronal wave
Prescribing the 1D wave by setting
\(\bm{v}_c = \begin{cases} \bm{0} & y\in (h/2, H-h/2)\\ v_c(x,t) {\bm{e}}_x & \text{otherwise} \end{cases}\)
\(\zeta_c =\begin{cases} 0 & y \in (h/2, H-h/2)\\ \bar\zeta_c\left(\partial_x x_c(x,t)\right)^{-1} & \text{otherwise} \end{cases} \)
with \(\bar\zeta_c =\bar\rho_c\dfrac1{Lh}\dfrac{F_{drag}}{v_c}=3\pi\mu\bar\rho_c^2\), and
\(F_{drag}=(\bar\rho_cL)( h/{2r_{pore}})(6\pi\mu r_{pore}v_c)\)
Since \(x_c=x_c(kx-\omega t)\) is a traveling wave, only need to solve for \(\bm{u}\) at \(t=t_o\)
\(v_c(x,t)\)
\(x_c(x,t)\)
\(x_c(x,t) = x + \epsilon \cos(2\pi x/L+\omega t)\)
\(\langle v_c\rangle_x=\frac{1}{L}\int_0^L v_c(x,0)\,\mathrm{d} x=\pi \omega\epsilon^2/L\)
\(v_c(x_c,t) = \dfrac{\partial x_c}{\partial t}, \rho_c(x_c,t) = \bar\rho_c \left( \dfrac{\partial x_c}{\partial x} \right)^{-1}\)
Taylor (1951) & Blake (1971)
Pak, Lauga, Proc. R. S. A (2009)
Nawroth, Ling, et al. ???
Brinkman-Stokes equation
For simplicity, we only consider one-way coupling of cilia driven velocity as drag forces on to the fluid
Resulting equations are solved numerically via mixed FEM
\(p(0,y,t)=p(L,y,t)\)
\(\bm{u}(0,y,t)=\bm{u}(L,y,t)\)
\(\bm{u}(x,0,t)=\bm{u}(x,H,t)={0}\)
\(-\nabla^2 \bm{u} + \nabla p + \zeta_c(\bm{u}-\bm{v}_c) = \Delta p/L\cdot\bm e_x\)
\(\nabla \cdot \mathbf{u} = 0\)
Nawroth, Ling, et al. ???
\(\mu\)
\(\omega\)
\(L\)
\(\epsilon\)
\(h/H\)
\(H\)
\(\Delta P\)
\(\bar\rho_ch\)
\(U=\int_0^L\bm u\cdot\bm e_x\)
\(10^{-3}\) [Pa\(\cdot\)s]
\(15\sim30\) [Hz]
\(100\) [μm]
\(L/2\pi\)
\(0-1\)
\(5-200\times10^{-6}\) [m]
\(0-100\) \(Pa\)
\(1\sim1000\)
\(\leq \omega L /(4\pi)\)
fluid viscosity
wave frequency
imposed wavelength (wave number \(=1\))
wave amplitude \(\in(0,\lambda/2\pi)\)
cilia-to-lumen ratio
duct lumen diameter
adverse pressure gradient
material constraint constant
mean flow speed
Model parameters
Nawroth, Ling, et al. ???
Ciliary carpet Maximizes flow rate
In absence of adverse pressure,
- Low cilia-to-lumen ratio h/H and large lumen diameter H (ciliary carpet) produces the largest possible flux
- In the limit of thin, dense layer of cilia, we recover the theoretical upper bound for flow as predicted by the classic envelope model of Black (1971)
Nawroth, Ling, et al (2023)
Ciliary carpet Maximizes flow rate
In stark contrast, even mild adverse pressure can reverse the net flow!
In absence of adverse pressure,
- Low cilia-to-lumen ratio h/H and large lumen diameter H (ciliary carpet) produces the largest possible flux
- In the limit of thin, dense layer of cilia, we recover the theoretical upper bound for flow predicted by the classic envelope model of Black (1971)
Nawroth, Ling, et al (2023)
Unless under Pressure
Ciliary flame pumps less fluid
In absence of adverse pressure,
- High cilia-to-lumen ratio h/H and small lumen diameter H (ciliary flame) produces less flux than carpets
- This is because the active ciliary layers also introduce material that adds resistance/restrictions to luminal flow
Nawroth, Ling, et al (2023)
Ciliary flame can Pump against Pressure
However, unlike the case of ciliary carpets, ciliary flames is much more robust against adverse pressure!
In absence of adverse pressure,
- High cilia-to-lumen ratio h/H and small lumen diameter H (ciliary flame) produces less flux than carpets
- This is because the active ciliary layers also introduce material that adds resistance/restrictions to luminal flow
Nawroth, Ling, et al (2023)
Pressure-Flow tradeoff for ciliary pumps
-
Just like any engineered pumps, ciliated duct with fixed morphology is most efficient (maximum \(\eta\)) when pumping at the 'shoulder' of their operating curve (solid lines)
- \(\eta=\dfrac{\text{power transmitted to fluid}}{\text{power input by cilia}}\propto\dfrac{QU^2}{\bar\rho_ch}\)
- Ducts with smaller \(H\) and higher \(h/H\) can pump against larger pressure without compromising efficiency
- This suggests \(h/H\) and \(H\) should be inversely related if natural ducts are flow efficient!
\(\mu\) - viscosity; \(\omega\) - cilia beat frequency; \(L\) - ciliary wavelength
Nawroth, Ling, et al (2023)
-
Just like any engineered pumps, ciliated duct with fixed morphology is most efficient (maximum \(\eta\)) when pumping at the 'shoulder' of their operating curve (solid lines)
- \(\eta=\dfrac{\text{power transmitted to fluid}}{\text{power input by cilia}}\propto\dfrac{QU^2}{\bar\rho_ch}\)
- Ducts with smaller \(H\) and higher \(h/H\) can pump against larger pressure without compromising efficiency
- This suggests \(h/H\) and \(H\) should be inversely related if natural ducts are flow efficient!
\(\mu\) - viscosity; \(\omega\) - cilia beat frequency; \(L\) - ciliary wavelength
Nawroth, Ling, et al (2023)
Pressure-Flow tradeoff for ciliary pumps
-
Just like any engineered pumps, ciliated duct with fixed morphology is most efficient (maximum \(\eta\)) when pumping at the 'shoulder' of their operating curve (solid lines)
- \(\eta=\dfrac{\text{power transmitted to fluid}}{\text{power input by cilia}}\propto\dfrac{QU^2}{\bar\rho_ch}\)
- Ducts with smaller \(H\) and higher \(h/H\) can pump against larger pressure without compromising efficiency
- This suggests \(h/H\) and \(H\) should be inversely related if natural ducts are flow efficient!
\(\mu\) - viscosity; \(\omega\) - cilia beat frequency; \(L\) - ciliary wavelength
Nawroth, Ling, et al (2023)
Pressure-Flow tradeoff for ciliary pumps
Morphometric spectrum of ciliated ducts
Cilia-to-lumen Ratio h/H
Lumen Diameter H [μm]
- Survey of 58 ducts from 26 phyla neatly reveals an inverse relationship between \(h/H\) and \(H\)!
- Where available, our data also show flow function correlates with the value of \(h/H\)!
- Can our model explain this correlation?
- Why aren't there ducts at the anti-diagonal corners?
more open space in the duct
less open space in the duct
Nawroth, Ling, et al (2023)
airway
tissue
Universal design rules for ciliary pumps
Convergence of ciliated duct designs follows functional and efficiency constraints rather than phylogenetic distance!
-
high \(h/H\) small \(H\) \(\rightarrow\) better for pressure generation (filtration/valve)
-
low \(h/H\) large \(H\) \(\rightarrow\) better for flux generation (bulk transport)
-
a continuous design spectrum emerges in accordance with the biological data.
Nawroth, Ling, et al (2023)
Efficiency criterion - no pressure
Nawroth, Ling, et al. (2022)
- low material constant
- large lumen diameter
- low cilia-to-lumen ratio
\(\dfrac{U^3H}{\bar\rho_ch}\)
Efficiency criterion - medium \(\Delta P\)
Nawroth, Ling, et al. (2022)
- medium material constant
- small lumen diameter
- medium cilia-to-lumen ratio
\(\dfrac{U^3H}{\bar\rho_ch}\)
Efficiency criterion - large \(\Delta P\)
Nawroth, Ling, et al. (2022)
- high material constant
- small lumen diameter
- high cilia-to-lumen ratio
\(\dfrac{U^3H}{\bar\rho_ch}\)
Janna Nawroth
Eva Kanso
Hanliang Guo
Anup Kanale
Yi Man
Sebastian Fuerthauer
Michael Shelley
David Stein
Yusheng Jiao
Sina Heydari
Chenchen Huang
Jingyi Liu
Doris Roth
Ayse Tugce Sahin
Tankut Gueney
Dorothea Kraft
Austin Kellogg
Kakani Katija
Tara Essock-Burns
Margaret McFall-Ngai
Kanso Bio-Inspired Motions Lab
Nawroth Mechanobiology
Amy L Ryan
Erik Quiroz
Lalit Gautam
Gregory Bonde
Denise Seabold
External Collaborators
Josh Merel Nicolas Heess
Funding Grants
EU ERC StG: MecCOPD S-701477-5100-350
NIH R01: 1R01HL153622-01A1
NSF INSPIRE: 1608744
ONR: N00014-17-1-2062
ARO: W911NF-16-1-0074
Flow Physics Explains Morphological Diversity of Ciliated Ducts
By Feng Ling
Flow Physics Explains Morphological Diversity of Ciliated Ducts
defense weeeeeeee
