What is flagellar wave reversal?

Quick turn around

Rapid braking

Journey to the Microcosmos (2020)

But why wave reversals?

• a common navigation strategy is to intersperse a "running/forward" gait with bursts of "tumble/reverse/stop" phase where sensing could occur

• this means mono-flagellates need flagellar wave reversals

• two model organisms exhibiting wave reversal traits are parasites that separately cause sleeping sickness and Leishmaniasis.

Tip-to-base (T2B) "forward" waves

Base-to-tip (B2T) "reversed" waves

Diversity for Complex Function

metachronal waves

Diversity from Similar Structures

Tug-of-War Model of Flagellum

• model a flagellum as two micro-filaments coupled elastically and via molecular motor dynamics*

• each filament ($\,+$ and $−$ ) has a population of motors that bend the centerline in opposite directions ($\,n_+$ and $n_-$)

• Motors attach under a fixed rate but detach due to both natural transitions & geometric feedback

Motor Dynamics & Geometric Feedback

• binding/unbinding (attach/detach) of dynein $\rightarrow$ inter-doublet sliding

• sliding force couple + elastic resistance  $\rightarrow$ active bending moment

• feedback could arise via sliding control, curvature control, or "geometric clutch"

Qiu et al., Nature Struct. Mol. Bio (2012)

Sartori et al., eLife (2016)

Dimensional & Dimensionless Parameters

System I/O

Ramping up the Motor Activity

fixing all parameters except $\text{Sp}$ and dimensionless activity $\mu_a$, we see that sliding control give rise to

• a Hopf bifurcation where instability induces sustained oscillations
• larger activity give rise to a 2nd transition from T2B to B2T waves
• the transition zone can show mixed wave states, especially under added thermal noise*

sperm number

active moment

Biasing the Attachment Rate

If we fix $\mu_a$ but vary the attachment duty ratio $\alpha$ linearly as a function of arclength $s$

• more activity near the base, or        [$\partial\alpha/\partial s>0$], leads to base-to-tip B2T waves
• more activity near the tip, or            [$\partial\alpha/\partial s<0$], leads to tip-to-base T2B waves

$\text{Sp}=10$

$\text{Sp}=10$

$\text{Sp}=10$

sperm number

motor asymm.

$n_+$

$n_-$

$n_+$

$n_-$

$n_+$

$n_-$

Bistable Zones

motor activity distributed in the  shaded region can support both B2T and T2B waves!

sperm number

motor asymm.

• increase in total activity magnitude introduces a T2B to B2T transition under sliding control*

• biasing activity at either end leads to wave initiations near that end in general, e.g., base heavy activity $\rightarrow$ B2T waves

• there exists a bi-stable region where same activity pattern can produce different wave propagation directions

How activity affect wave directions

Tanhks 4 Yo Attention

Funding Sources: