Flexural Rigidity and Shear Stiffness of Flagella Estimated from Induced Bends and Counterbends

Abstract

Motile cilia and flagella are whiplike cellular organelles that bend actively to propel cells or move fluid in passages such as airways, brain ventricles, and the oviduct. Efficient motile function of cilia and flagella depends on coordinated interactions between active forces from an array of motor proteins and passive mechanical resistance from the complex cytoskeletal structure (the axoneme). However, details of this coordination, including axonemal mechanics, remain unclear. We investigated two major mechanical parameters, flexural rigidity and interdoublet shear stiffness, of the flagellar axoneme in the unicellular alga Chlamydomonas reinhardtii. Combining experiment, theory, and finite element models, we demonstrate that the apparent flexural rigidity of the axoneme depends on both the intrinsic flexural rigidity (EI) and the elastic resistance to interdoublet sliding (shear stiffness, ks). We estimated the average intrinsic flexural rigidity and interdoublet shear stiffness of wild-type Chlamydomonas flagella in vivo, rendered immotile by vanadate, to be EI = 840 ± 280 pN⋅μm(2) and ks = 79.6 ± 10.5 pN/rad, respectively. The corresponding values for the pf3; cnk11-6 double mutant, which lacks the nexin-dynein regulatory complex (N-DRC), were EI = 1011 ± 183 pN·μm(2) and ks = 39.3 ± 6.0 pN/rad under the same conditions. Finally, in the pf13A mutant, which lacks outer dynein arms and inner dynein arm c, the estimates were EI = 777 ± 184 pN·μm(2) and ks = 43.3 ± 7.7 pN/rad. In the two mutant strains, the flexural rigidity is not significantly different from wild-type (p > 0.05), but the lack of N-DRC (in pf3; cnk11-6) or dynein arms (in pf13A) significantly reduces interdoublet shear stiffness. These differences may represent the contributions of the N-DRCs (∼40 pN/rad) and residual dynein interactions (∼35 pN/rad) to interdoublet sliding resistance in these immobilized Chlamydomonas flagella.

How does cilium length affect beating?

“…Global force, torque, and power generated by the cilium were estimated from cilia motion (21) and compared to estimates from body motion. In addition, internal forces attributable to dynein motor protein activity were estimated from the waveform, using previous measurements of viscous resistive force coefficients, flexural modulus, and shear stiffness (32).…”

“…The local viscous contribution to bending, , at any axial location on the cilium is calculated by integrating the viscous force per unit length from that location to the distal end of the cilium and taking the normal component, . Finally, elastic shear, ℎ , can be estimated from the product of the tangent angle of the cilium and shear stiffness ( , pN/rad).Bending rigidity ( ) and shear stiffness ( ) were measured in wild-type cilia by Xu, et al(32). As a first-order approximation for estimation of internal dynein force, all cilia were assumed to have the flexural rigidity and shear stiffness previously measured in normal length wild-type cilia, respectively = 840 pN-µm 2 (mean ± std.…”

How Does Cilium Length Affect Beating?

“…Global force, torque, and power generated by the cilium were estimated from cilia motion (20) and compared to estimates from body motion. In addition, internal forces attributable to dynein motor protein activity were estimated from the waveform, using previous measurements of viscous resistive force coefficients, flexural modulus, and shear stiffness (31).…”

The biomechanical role of extra-axonemal structures in shaping the flagellar beat of Euglena

“…with γ p⊥ (s) = − sin φ p γ 1 (s) + cos φ p γ 2 (s). From (8) and (37), leading order calculations give V 1 , V 3 ∼ ρ p , whereas V 2 ≈ γ p⊥ / 1 + γ 2 p⊥ . Linearizing in γ p⊥ we have V 2 ≈ γ p , from which follows (11).…”

“…In the dynamic simulation in Figure 7 we used the following numeric values for the physical parameters of the system. The bending modulus of the Ax is B a = 840 pN · µm 2 , taken from [37]. We set L = 28 µm, T = 25 ms, and µ ⊥ = 3.1 fN · s · µm −2 , which are all values estimated in [25].…”