Dynamics of Active Microfilaments

Mechanics of Flagella and Cilia in Motion

Feng Ling

May 9, 2018

Ciliary Actions

Cilia and Flagella are Everywhere

 

Cilia and Flagella are Important

 

Cilia and Flagella work in groups

Merismo (2013)

Zvonimir Dogic (2011)

Ciliary Actions in Us

For example, cilia and/or flagella

  • are in are in Trachea Lining
  • Brain Ventricles
  • Spermatozoa and Fallopian Tubes

Varied in 'macroscopic' functions

but similar in 'microscopic' structures!

Specifically, they serve crucial functions in

  • Keep us safe and healthy
  • Support our brain functions
  • Responsible for our reproduction

Structure of Cilia Network

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Tissue:

 

 

 

Wan and Goldstein (2016)

Machemer (1998)

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Lin, Jianfeng, et al. (2014)

Zvonimir Dogic (2011)

Saternos, AbouAlaiwi (2015)

Structure of Cilia Network

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Coordinated

    Motion:

 

 

 

Wan and Goldstein (2016)

Machemer (1998)

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Zvonimir Dogic (2011)

Lin, Jianfeng, et al. (2014)

Schouteden, Clementine, et al. (2015)

Structure of Cilia Network

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Coordinated

    Motion:

 

 

 

Wan and Goldstein (2016)

Machemer (1998)

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Zvonimir Dogic (2011)

Lin, Jianfeng, et al. (2014)

Lodish, Berk, et al. (2000)

Structure of Cilia Network

Lin, Jianfeng, et al. (2014)

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Coordinated

    Motion:

 

 

 

Wan and Goldstein (2016)

Machemer (1998)

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Zvonimir Dogic (2011)

BioVisions (2011)

Structure of Cilia Network

Lin, Jianfeng, et al. (2014)

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Coordinated

    Motion:

 

 

 

Zvonimir Dogic (2011)

Wan, K. Y., and Goldstein R. E. (2016)

Machemer, H. (1998)

Structure of Cilia Network

Lin, Jianfeng, et al. (2014)

Wan and Goldstein (2016)

Machemer (1998)

Internal Structures:

 

 

 

Motile Filament:

 

 

 

Coordinated

    Motion:

 

 

 

\(\text{[nm]}\)

\(\Downarrow\)

\(\text{[}\mu\text{m]}\)

\(\Downarrow\)

\(\text{[mm]}\)

\(\Downarrow\)

Zvonimir Dogic (2011)

Organization for Transport

  Ciliophora

Placozoa

Invertebrate

Mammalian Organs

Smith, et al. (2015)

Bobtail squid

Trichoplax

Nawroth, et al. (2017)

Trachea Lining

Stentor

Susumu Nishinaga (2013)

Merismo (2013)

Outline

Part I: Dynamics of a single Active Microfilament

    3-dimensional beating patterns of a single filament

Part II: Dynamics of Active Microfilaments

     Effectiveness of ciliary pumping in a channel

Part I: Dynamics of Active Microfilament

3-dimensional beating patterns of a single filament

Part I: Dynamics of Active Microfilament

how a simple instability drive complex motion

Flagella Beating in 3D and 2D

D. Meng, M. Cao,T.  Oda, and J. Pan, Journal of Cell Science (2014)

Flagella Beating in 3D and 2D

D. Meng, M. Cao,T.  Oda, and J. Pan, Journal of Cell Science (2014)

 What we want to know:​

   How are oscillations regulated?

   How to obtain varying modes of motion?

Cilia/Flagella as Microfilaments

Sleigh, Brokaw, and Gittes...

Geometric Clutch theories, Lindermann (1990s-2000s)

Sliding-Control theories, Jülicher et al., (2000s-)

...

Instability-driven:

Bayly and Dutcher (2016)

De Canio, Lauga, and Goldstein (2017)

Han and Peskin (2018)

 

Other communities have cared about follower forces,
  but not necessarily also for large deformations and life past instabilities

Elastic Rod Model - Kinematics

\(\text{d}s\)

Bending

Twisting

Torsion

Assumptions:

No Shear

\(\mathbf{d}_3\equiv\mathbf{t}\)

Elastic Rod Model - Kinetics

Balance of Linear Momentum

 

 

Balance of Angular Momentum

 

 

Constitutive Relations

Assumptions:

No Inertia

\(\sum_i{\mathbf{F}_i}=0\)

\(\sum_i\mathbf{T}_i=0\)

\(=\partial_s\mathbf{N}+\mathbf{f}^h+\mathbf{f}^a\)

\(=\partial_s\mathbf{M}+(\partial_s\mathbf{r})\times\mathbf{N}\)

\(\mathbf{M}_b=\mathbf{B}_b\mathbf{\Omega}_b\)

\(\mathbf{N}\cdot\partial_s{r}=\Lambda\)

Elastic Rod Model - Dynamics

Fluid structure interaction at low Reynolds number

1-dimensional slender filaments experience anisotropic local drag

\(\mathbf{f}^h=-\zeta_{iso}\left[\mathbf{t}\otimes\mathbf{t}+\alpha(\mathbf{d}_1\otimes\mathbf{d}_1+\mathbf{d}_2\otimes\mathbf{d}_2)\right]\mathbf{v}\)

\(~~~\,:=-\zeta_{aniso}\mathbf{v}\)

where \(\alpha>1\) and \(\zeta_{iso}=\frac{4\pi\mu}{\alpha\log(L/a)}\)

\(\mathbf{v}\cdot\mathbf{t}\)

\(\mathbf{v}\cdot\mathbf{d}_1+\mathbf{v}\cdot\mathbf{d}_2\)

\(\mathbf{v}\)

\(-\zeta_{iso}\mathbf{v}\)

\(-\zeta_{aniso}\mathbf{v}\)

Assumptions:

No Hydrodynamic
Interaction

Elastic Rod Model - Dynamics

Real motor proteins produce noisy, oscillatory forces, but

  they induce a ~constant compressive axial component

\(\quad\Rightarrow\mathbf{f}^a=-f^a\mathbf{t}\)

Finally we have the equation of motion:

\(\zeta_{aniso}\mathbf{v} = {B}\frac{\partial}{\partial s}\left[\mathbf{t}\times\frac{\partial}{\partial s}\mathbf{\Omega}\right] +\frac{\partial}{\partial s}\big(\Lambda\mathbf{t}\big) -{f}^a\mathbf{t}\)

where we assumed \(\mathbf{\Omega}=k_1\mathbf{d}_1+k_2\mathbf{d}_2\) twist-free

  and \(\mathbf{B}=B\)

BioVisions (2011)

Bending

Activity

Drag

Tension

Elastic Rod Model - Dynamics

BioVisions (2011)

Real motor proteins produce noisy, oscillatory forces, but

  they induce a ~constant compressive axial component

\(\quad\Rightarrow\mathbf{f}^a=-f^a\mathbf{t}\)

Finally we have the equation of motion:

\(\zeta_{aniso}\mathbf{v} = {B}\frac{\partial}{\partial s}\left[\mathbf{t}\times\frac{\partial}{\partial s}\mathbf{\Omega}\right] +\frac{\partial}{\partial s}\big(\Lambda\mathbf{t}\big) -{f}^a\mathbf{t}\)

where we assumed \(\mathbf{\Omega}=k_1\mathbf{d}_1+k_2\mathbf{d}_2\) twist-free

  and \(\mathbf{B}=B\)

There exists multiple types of

periodic solutions despite \(|\mathbf{f}^a|\neq|\mathbf{f}^a(t)|\)

Linearized Dynamics

Close to straight equilibrium \(\mathbf{r}=(0,0,z)\), we have

\[B \partial_z^4 K + F \partial_z^2 K + \alpha\zeta_{iso} \partial_t K= 0\]

\[\mathbf{q}(B\mathbf{K}+\mathbf{F}+\lambda\alpha\zeta_{iso}\mathbf{M})=\mathbf{0}\]

 

\(K=\sqrt{k_1^2+k_2^2}\)

\(F(z):=-\Lambda=-\int_{L}^{z}f^a(\hat{z})\mathsf{d}\hat{z}\)

It does not depend on 'torsion'!

Motion occurs only  in the plane of curvature.

We can safely exchange \(K(z,t)\) by \(y(z,t)\).

Assume \(y(z,t)=e^{\lambda t}\sum_iq_i\Phi_i(z)\), we now have

\[q_i\left[B\int_{0}^{L}\Phi''_i\Phi''_j\,\mathsf{d} z + \int_{0}^{L}\Phi_iF\Phi''_j\,\mathsf{d} z + \lambda\alpha\zeta_{iso}\int_{0}^{L}\Phi_i\Phi_j\,\mathsf{d} z\right]=0\]

\(\Phi_i(z)\) are modal solutions of cantilever beam

Bending

Activity

Drag

Tension

\(\text{Re}[\lambda]>0\) \(\rightarrow\) Instability

\(\text{Im}[\lambda]\neq0\) \(\rightarrow\) Oscillations

Linearized Dynamics

Close to straight equilibrium \(\mathbf{r}=(0,0,z)\), we have

\[B \partial_z^4 K + F \partial_z^2 K + \alpha\zeta_{iso} \partial_t K= 0\]

\(K=\sqrt{k_1^2+k_2^2}\)

\(F(z):=-\Lambda=-\int_{L}^{z}f^a(\hat{z})\mathsf{d}\hat{z}\)

Motion occurs only in the plane of curvature.

We can safely exchange \(K(z,t)\) by \(y(z,t)\).

Assume \(y(z,t)=e^{\lambda t}\sum_iq_i\Phi_i(z)\), we now have

\[q_i\left[B\int_{0}^{L}\Phi''_i\Phi''_j\,\mathsf{d} z + \int_{0}^{L}\Phi_iF\Phi''_j\,\mathsf{d} z + \lambda\alpha\zeta_{iso}\int_{0}^{L}\Phi_i\Phi_j\,\mathsf{d} z\right]=0\]

\(\Phi_i(z)\) are modal solutions of cantilever beam

\(\text{Re}[\lambda]>0\) \(\rightarrow\) Instability

\(\text{Im}[\lambda]\neq0\) \(\rightarrow\) Oscillations

\[\mathbf{q}(B\mathbf{K}+\mathbf{F}+\lambda\alpha\zeta_{iso}\mathbf{M})=\mathbf{0}\]

 

Hopf bifurcation brings system to limit cycle solutions with large enough \(|\mathbf{f}^a|\)

But Solutions are always planar!

Discrete Elastic Rod

Spatial Discretization: \(\mathbf{r}_i\) \(\Rightarrow\) \(\ell_i,\overline{\ell}_i;\)   \(\mathbf{t}_i,\mathbf{Q}_i;\)

\(\Rightarrow\)  \(e_i,\mathbf{s}_i;\)    \(\mathbf{\Omega}_{b,i}^\times=-\log\big(\mathbf{Q}_{i}\mathbf{Q}^\mathsf{T}_{i-1}\big)/\overline{\ell}_i+\text{twist}\)

Discrete Equations:

\(\mathbf{M}_{b,i} = \mathbf{B}_{b,i}(\mathbf{\Omega}_{b,i}-\mathbf{\Omega}_{b,i}^o)\)

\(\mathbf{N}_{b,i} = \mathbf{t}_{b,i}\times\mathcal{D}[\mathbf{M}_{b,i}] + \mathcal{A}[(\mathbf{\Omega}_{b,i}\overline{\ell}_i)\times(\mathbf{M}_{b,i})] + (EA)\mathbf{s}_{b,i}\)

\(\mathbf{v}_i = \frac{1}{\alpha\zeta}\Big(\mathbf{I}+(\alpha-1)\mathcal{A}\big[\mathbf{t}_i\big]\otimes\mathcal{A}\big[\mathbf{t}_i\big]\Big)\Big( \mathcal{D}\big[\mathbf{N}_i\big] - \mathbf{f}^a_i \ell_i\Big).\)

ODE Integrators (MATLAB ode15s)

Stiffness

Frame Transport

Validation: Cantilever Beam

Validates DER against linear Euler-Bernoulli beam theory

Validation: Euler Buckling

Buckling threshold consistent with linear beam theory predictions

  but also returns nonlinear buckling shapes

Validation: Twisting Instabilities

Actual algorithm also can take care of twists

Generic Axial Active Force

Linearly varying profiles

 

\(f^a_L=f^a_L(s;p,m)\)

 

 

 

Delta function profiles

 

\(f^a_D=f^a_D(s;p,s_o)\)

Motion Gallery - Linear Forces

Motion Gallery - Delta Forces

Possible Behaviors

Spinning vs Whipping

At smaller axial force, spinning is stable

At larger axial force, whipping is stable

Resolution: \(\delta s=L/160\)

\(|\mathbf{f}^a|<F_{crit}\)

\(|\mathbf{f}^a|>F_{crit}\)

'Phase' Diagram

Power law force profiles

\(f^a_P=f^a_P(s;p,n)\)

 

 

 

 

 

\(p\) measures total force magnitude

\(n\) measures tip dominance

Larger axial forces closer to the

tip induce the spinning to whipping transitions

 

GOAL:

  • Obtain a reduced order model
    to understand the cause of
    transition

'Phase' Diagram

Bead-Spring Model

Forces at the Beads

  • Tip Activation:

 

  • Drag Forces:

 

  • Torsional Springs:

Linearized system again has planar symmetry!

\(l\)

\(\mathbf{F}_{tip}=-\dfrac{F}{l}\overline{\text{CD}}\)

\(\mathbf{F}_{drag,i}=-\zeta\mathbf{v}_i\)

\(\mathbf{U}_{spring,i}=-\frac12k\theta_i^2\)

\(-\frac12k_T\phi_i^2\)

Bead-Spring Model

Two Links can reproduce both spinning and whipping

but spinning seems stable in general

Bead-Spring Model

Three Links reaches an almost planar solution in the limit of large forces

however the required tip force overwhelms the spring/drag force

Path To Transition

How do we obtain 3D to 2D transition?

Let's crank up the number of beads...

1 link: inextensibility \(\Rightarrow\) stable

2 links: spinning is stable at large \(\mathbf{F}_{tip}\)

3 links: vertical 'windmill' \(\Rightarrow\) spinning

4 links: horizontally precessing 'windmill'

5 links: whipping-like 'windmill'

6 links: Spinning to Whipping transition!

Reasons for Transition?

Why \(6\)?

  • need \(N>3\) to allow effects of torsion
  • under larger forces, 'equilibrium' \(\theta_i\) gets larger
  • at large forces, we have a flatter, more horizontal 'tip'
  • non-circular perturbations grow stronger as tip force points
    closer to tangent of the circle formed by the spinning tip
  • At \(N\geq4\), this brings instability to spinning trajectories
  • At \(N=6\), precession morphs into whipping

\(\theta_{tip}\)

Reasons for Transition?

So hwæt?

Biological significance:

  • This provides a simple yet effective way of generating very
    different types of oscillation as do flagella/cilia
  • Qualitatively consistent with known structural and dynamical models
    e.g. not necessarily contradictory to Jülicher's 2D sliding model
           \(\mathbf{f}^a=(aF_{ss}-aK^2F)\mathbf{n}+[-2aKF_{s}-aK_sF]\mathbf{t}\)
    Han and Peskin

Mechanical significance:

  • mechanism for soft robotic actuations

Moving Forward

  • Experimental Verifications
  • Asymmetric Activity
    • Curvature feedback
  • Elasticity Models
    • Natural Curvature, shear, & twist
  • Contacts and Interactions

Part II: Dynamics of Active Microfilaments

Effectiveness of ciliary pumping in a channel

Organization for Transport

  Ciliophora

Placozoa

Invertebrate

Mammalian Organs

Smith, et al. (2015)

Bobtail squid

Trichoplax

Nawroth, et al. (2017)

Trachea Lining

Stentor

Susumu Nishinaga (2013)

Merismo (2013)

Ciliated Channel

\(P_{in}\)

\(P_{out}\)

\(h_{free}\)

Ciliated Channel

\(P_{in}\)

\(P_{out}\)

\(h_{free}\)

 What we want to know:​

   Effect of pressure head (\(P_{out}-P_{in}>0\)) on flow rate

   Effect of cilia overgrowth (\(h_{free}\rightarrow0\)) on flow rate

Ciliated Surfaces: What do we know?

Taylor's swimming sheet (1951)

Blake's solutions (1971)

On Shun Pak and Lauga... (2010s)

 

Simulations of many interacting cilia are also starting to be commonplace

 

But there is still a search for the most flexible and
  efficient framework

Osterman, N., and Vilfan A. (2011)

Not claiming what follows will be the answer...

Channel Model

Assumptions

  • Creeping Flow i.e.,\(\text{Re}\approx0\)
    (instantaneity \(\Rightarrow\) no time stepping)
  • Rectangular Geometry
  • Left/Right Periodic Boundary
    (Pressure Gradient)
  • Top/Bottom Symmetry
    (Cilia in Full Coordination)
  • Implicit Fluid-Structure Interaction

Stokes' Flow Model

Treat cilia as boundary conditions:

In \(\Omega\in[0,1]\times[0,1],\)

 

\(u_x(x,0)=u_x(x,1)=f(x)\cos(2\pi x-t)\)

\(u_y(x,0)=u_y(x,1)=g(x)~~~\;\,\,~\qquad\qquad\)

\(u_x(0,y)=u_x(1,y)\qquad\qquad\,\quad\)

\(u_y(0,y)=u_y(1,y)\,\quad\qquad\qquad\)

\(-\nabla^2 \mathbf{u} + \nabla p = 0\)

\(\nabla \cdot \mathbf{u} = 0\)

Analytical Solutions

Using the stream function:

\(u_x=\frac{\partial\Psi}{\partial y},\quad u_y=\,-\frac{\partial\Psi}{\partial x}\)

\(\nabla^4\Psi=0~~~~~~~~~\)

For the simplest case: \(f(x)=1,\,g(x)=0\)

 we can solve by separation of variables

\(\Psi(x,y)=X(x)Y(y)\)

\(\Psi(x,y)\;\)\(=\frac{1}{\sinh (2 \pi )-2 \pi }((y-1) \sinh (2 \pi y)+y \sinh (2 \pi -2 \pi y))\cos (2 \pi x-t)\)

Derivation

\(\Psi(x,y)=X(x)Y(y)\) + Boundary Conditions

\(\Psi(x,y)=\frac{1}{\sinh (2 \pi )-2 \pi }((y-1) \sinh (2 \pi y)+y \sinh (2 \pi -2 \pi y))\cos (2 \pi x-t)\)

\(\partial_y\Psi|_{y=0,1}=XY'|_{y=0,1}=\cos(2\pi x-t)\)

\(\partial_x\Psi|_{y=0,1}=X'Y|_{y=0,1}=0\).

\(Y'(0)=Y'(1), Y(0)=Y(1)=0\)

\(X(x)=c\cos(2\pi x-t)\).

\(Y(y)=c_1e^{2\pi y}+c_2e^{-2\pi y}+c_3ye^{2\pi y}+c_4ye^{-2\pi y}\)

\[Y''''-2(2\pi)^2 Y''+(2\pi)^4Y=0,\quad Y'(0)=Y'(1)=1, \quad Y(0)=Y(1)=0,\]

\(\nabla^4\Psi(x,y)=(\partial_x^2+\partial_y^2)^2\Psi(x,y)=(\partial_x^4+2\partial_x^2\partial_y^2+\partial_y^4)\Psi(x,y)=0\)

\(Y(y)=\frac{1}{\sinh (2 \pi )-2 \pi }((y-1) \sinh (2 \pi y)+y \sinh (2 \pi -2 \pi y))\)

Derivation

\(\Psi(x,y)=X(x)Y(y)\) + Boundary Conditions

\(\Psi(x,y)=\frac{1}{\sinh (2 \pi )-2 \pi }((y-1) \sinh (2 \pi y)+y \sinh (2 \pi -2 \pi y))\cos (2 \pi x-t)\)

\(\partial_y\Psi|_{y=0,1}=XY'|_{y=0,1}=\cos(2\pi x-t)\)

\(\partial_x\Psi|_{y=0,1}=X'Y|_{y=0,1}=0\).

\(Y'(0)=Y'(1), Y(0)=Y(1)=0\)

\(X(x)=c\cos(2\pi x-t)\).

\(Y(y)=c_1e^{2\pi y}+c_2e^{-2\pi y}+c_3ye^{2\pi y}+c_4ye^{-2\pi y}\)

\[Y''''-2(2\pi)^2 Y''+(2\pi)^4Y=0,\quad Y'(0)=Y'(1)=1, \quad Y(0)=Y(1)=0,\]

\(\nabla^4\Psi(x,y)=(\partial_x^2+\partial_y^2)^2\Psi(x,y)=(\partial_x^4+2\partial_x^2\partial_y^2+\partial_y^4)\Psi(x,y)=0\)

\(Y(y)=\frac{1}{\sinh (2 \pi )-2 \pi }((y-1) \sinh (2 \pi y)+y \sinh (2 \pi -2 \pi y))\)

NO NET FLOW!

Asymmetric Solution

To produce realistic flow, we need nontrivial \(f(x,y)\) and \(g(x,y)\)

Previous solution generalizes with \(\partial_yf(x,0)=\partial_yf(x,1),f(x,y)=f(x+1,y)\) and \(g=0\)

But also need \(\partial_x f(x,0)=\lambda f(x,0)\)

It is more straightforward to pursue numerical solutions (first)

Numerical Method

FEM on a 'staggered mesh'

 

\(\begin{bmatrix} -1 & \text{Curl} & {0} \\ \text{Curl} & {0} & \text{Grad}\\ {0}&\text{Div}&{0}\end{bmatrix}\begin{bmatrix}{\omega}\\\mathbf{u}\\p\end{bmatrix}=\begin{bmatrix}0\\0\\0\end{bmatrix}\)

Weak form equations need cascading basis

Guarantees divergence-free velocity field

Tangential BCs are enforced weakly

Normal/Flux BCs can be enforced strongly

'Mixed' Finite Element Scheme

\(-{\omega}+\nabla\times\mathbf{u}=0\)

\(-\nabla\times {\omega}+ \nabla p = 0\)

\(\nabla \cdot \mathbf{u}= 0\)

Validation: Lid-driven Cavity Flow

Standard test case

Constant tangential velocity

  at one boundary, solid no-slip        boundaries otherwise

Additional symmetry due to

  lack of convective nonlinearity

Validation: Lid-driven Cavity Flow

Error Convergence

error in \(u_x\)

error in \(u_y\)

error in \(p\)

Boundary Driven Flow

Cilia as Porous Media

Ciliary region is in fact permeable

  and Move with velocity \(\mathbf{v}\)

To leading order we add 'drag force' \(\propto \mathbf{u}-\mathbf{v}\)

 

 

 

 

 

Note: \(k=k(x,y)\)!

\(\Rightarrow\) Brinkman Equations:

 

\(\begin{bmatrix} -1 & \text{Curl} & {0} \\ \text{Curl} & k & \text{Grad}\\ {0}&\text{Div}&{0}\end{bmatrix}\begin{bmatrix}{\omega}\\\mathbf{u}\\p\end{bmatrix}=\begin{bmatrix}0\\k\mathbf{v}\\0\end{bmatrix}\)

Boundary Driven Flow \(+\;k\neq0\)

Boundary Driven Flow

Velocity at Interface

2D Velocity Field

Traveling Wave of Rigid Rods

Envelope model of inextensible rigid rods based at

\[x_{base} = (i-1)\Delta/L\]

with a traveling wave motion at the tip

\[x_{tip}=x_{base}+\epsilon\cos(2\pi x_{base}+\omega t),\]

\[y_{tip}=~~~~~~\,~~~~~~\epsilon\sin(2\pi x_{base}+\omega t).\]

Then \(k\) can be interpolated from \(x_{tip}\) and \(\mathbf{v}\) can be interpolated from \(\mathbf{v}_{tip}\)

\(\Delta\)

\(L\)

Traveling Wave of Rigid Rods

Effective and Recovery Strokes

Realistic \(k,\mathbf{v}\) are more asymmetric

Symplectic Metachronal Waves

 

 

 

Antiplectic Metachronal Waves

Guirao and Joanny (2007)

Velocity Profile in Experiment

Sears, Patrick R., et al. (2012)

Faster effective stroke

 

Slower recovery stroke

Effective and Recovery Strokes

Realistic \(k,\mathbf{v}\) are more asymmetric

Symplectic Metachronal Waves

 

 

 

Antiplectic Metachronal Waves

Guirao and Joanny (2007)

Effective and Recovery Strokes

Realistic \(k,\mathbf{v}\) are more asymmetric

Symplectic Metachronal Waves

 

 

 

Antiplectic Metachronal Waves

Guirao and Joanny (2007)

Symplectic Waves

Antiplectic Waves

In the Near Future

A Project in Progress:

  • Pressure head and Cilia Overgrowth
  • Optimization of \(f(x,y), g(x,y)\) and/or \(k,\mathbf{v}\)
    • Better interpolation scheme from real data
    • Directly from filament computations
  • Cross plane 3-dimensional effects
    • Active surfaces
  • Ambient fluids and structures
  • ...

Conclusions and Outlook

Accomplishments:

  • instability driven dynamics of axially actuated elastic microfilament
  • formulation of cilia pumping problem

 

Plans:

  • continue to pursue established problems
  • more geometry (and topology?)
  • 'active matter' type questions?

Thanks!

References:

  • F. Ling, H. Guo, E. Kanso, “Instability-triggered Oscillations of Active Microfilament” In preparation

  • Bayly, P. V., & Dutcher, S. K. (2016). Steady dynein forces induce flutter instability and propagating waves in mathematical models of flagella. Journal of The Royal Society Interface, 13(123), 20160523.

  • De Canio, G., Lauga, E., & Goldstein, R. E. (2017). Spontaneous oscillations of elastic filaments induced by molecular motors. Journal of The Royal Society Interface, 14(136), 20170491.

  • Gazzola, M., Dudte, L. H., McCormick, A. G., & Mahadevan, L. (2016). Dynamics of soft filaments that can stretch, shear, bend and twist. arXiv preprint arXiv:1607.00430.

  • Bergou, M., Wardetzky, M., Robinson, S., Audoly, B., & Grinspun, E. (2008). Discrete elastic rods. ACM transactions on graphics (TOG), 27(3), 63.

  • Osterman, N., & Vilfan, A. (2011). Finding the ciliary beating pattern with optimal efficiency. Proceedings of the National Academy of Sciences, 108(38), 15727-15732.

  • Schouteden, C., Serwas, D., Palfy, M., & Dammermann, A. (2015). The ciliary transition zone functions in cell adhesion but is dispensable for axoneme assembly in C. elegans. J Cell Biol, 210(1), 35-44.

  • Lin, J., Yin, W., Smith, M. C., Song, K., Leigh, M. W., Zariwala, M. A., ... & Nicastro, D. (2014). Cryo-electron tomography reveals ciliary defects underlying human RSPH1 primary ciliary dyskinesia. Nature communications, 5, 5727.

  • Wan, K. Y., & Goldstein, R. E. (2016). Coordinated beating of algal flagella is mediated by basal coupling. Proceedings of the National Academy of Sciences, 113(20), E2784-E2793.

  • Machemer, H. (1998). Motor control of cilia. In Paramecium (pp. 216-235). Springer, Berlin, Heidelberg.

  • Sears, P. R., Thompson, K., Knowles, M. R., & Davis, C. W. (2012). Human airway ciliary dynamics. American Journal of Physiology-Lung Cellular and Molecular Physiology, 304(3), L170-L183.

  • Smith, C. L., Pivovarova, N., & Reese, T. S. (2015). Coordinated feeding behavior in Trichoplax, an animal without synapses. PLoS One, 10(9), e0136098.

  • Guirao, B., & Joanny, J. F. (2007). Spontaneous creation of macroscopic flow and metachronal waves in an array of cilia. Biophysical journal, 92(6), 1900-1917.

  • Lodish, H., Berk, A., Zipursky, S. L., Matsudaira, P., Baltimore, D., & Darnell, J. (2000). Molecular cell biology 4th edition. National Center for Biotechnology Information, Bookshelf.

Dynamics of Active Microfilaments

By Feng Ling

Dynamics of Active Microfilaments

Mechanics of Flagella and Cilia in Motion (qualifying exam)

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