Spontaneous oscillations, beating patterns, and hydrodynamics of active microfilaments

Cilia and flagella are ubiquitous in nature and are known to help in transport and swimming at the cellular scale by performing oscillations. Fundamental to these periodic waveforms is the core internal structure of the filaments known as the axoneme, consisting of an array of microtubule doublets, protein linkers, and dynein motors. In the presence of ATP, the collective action of the molecular motors drives internal sliding motions that are converted to spontaneous oscillations by a mechanism that still remains elusive. A sliding controlled axonemal feedback mechanism has recently been proposed and explored in the limit of small deformations, where it was shown to result in nonlinear amplitude selection through a mechanical regulation of dynein kinetics. Here, we build on that model to derive a more complete set of planar nonlinear governing equations that retains all the geometric nonlinearities, incorporates intrinsic biochemical noise and accounts for long-range, nonlocal hydrodynamic interactions. For a clamped filament, motor activity drives a Hopf bifurcation leading to traveling wave solutions that propagate from tip to base, in agreement with previous weakly nonlinear studies. Quite remarkably, our results demonstrate the existence of a second transition far from equilibrium, where nonlinearities cause a reversal in the direction of wave propagation and produce a variety of waveforms that resemble the beating patterns of swimming spermatozoa. We further extend the model to account for asymmetric ciliary beats and also allow for generalized dynein regulation mechanisms that can qualitatively reproduce Chlamydomonas reinhardtii flagellar dynamics. In the spirit of dimensional reduction, limit cycle representations are obtained for various waveforms and highlight the role of biochemical noise. We also analyze the velocity fields generated by the filaments and apply principal component analysis to derive low-order flow representations in terms of fundamental Stokes singularities that could be of use for constructing minimal models of swimming microorganisms.

Figure 1

(a) Thin section electron micrograph of axonemes of the green algae Chlamydomonas reinhardtii. Reproduced from Smith and Lefebvre [3]. (b) A simplified schematic of the cross-section of an axoneme showing the “9+2” structure.

Figure 2

Schematic of an elastic rod of length $L$ and arclength $s$ that is clamped at $s=0$. The rod is represented as a space curve in the $(x,y)$ plane, which we parametrize by the tangent angle $\varphi (s,t)$.

Figure 3

Schematic of the simplified two-dimensional projection of the axoneme. In this model, the axoneme is composed of two polar filaments ${\mathbf{x}}_{+}$ and ${\mathbf{x}}_{-}$ and has centerline $\mathbf{x}$. Nexin links are shown as elastic springs that resist sliding $\mathrm{\Delta }$. Mechanical loads exerted by bound dynein motors result in an internal shear force density $\pm f(s,t)$ that acts in the opposite direction on the two filaments, generating an active internal moment.

Figure 4

Phase chart highlighting the various dynamical transitions and beating patterns in the $\left(\mathrm{Sp},{\mu }_a\right)$ parameter space. Waveforms on both sides show characteristic filament conformations, where the arrows indicate the direction of wave propagation. Also see movie of the dynamics in the Supplemental Material [42]. Shown simulations are for $\mu =100,\zeta =0.3$, and $\eta =0.14$.

Figure 5

Dependence of (a) dimensionless amplitude, (b) dimensionless wavelength, and (c) frequency of the beating patterns on motor activity ${\mu }_a$ for two values of the sperm number $\mathrm{Sp}$. The dimensional frequency assumes the choice of ${\tau }_0=50\mathrm{ms}$ for the motor correlation time. Jumps in the data are associated with the transition from retrograde to anterograde wave propagation. Simulations were performed with $\mu =100,\zeta =0.3$, and $\eta =0.14$.

Figure 6

Overlaid characteristic waveforms at different time instants over the course of one beating period: (a) Symmetric spermlike beat. Parameters: $\mathrm{Sp}=12,{\mu }_a=12\times {10}^3$ and $\mu =100$. (b) Asymmetric cilialike beat obtained with biased dynein kinetics. Parameters: $\mathrm{Sp}=2.5,{\mu }_a=2\times {10}^3,\mu =90,({\pi }_0^{+},{\epsilon }_0^{+})=(0.17,0.73)$, and $({\pi }_0^{-},{\epsilon }_0^{-})=(0.25,0.75)$. (c) Asymmetric beating pattern resembling Chlamydomonas, obtained with biased kinetics in presence of both sliding and curvature control. Parameters: $\mathrm{Sp}=2.2,{\mu }_a=1.5\times {10}^3,\mu =80,f^{*}=1.9,({\pi }_0^{+},{\epsilon }_0^{+})=(0.24,0.76),({\pi }_0^{-},{\epsilon }_0^{-})=(0.39,0.61),{\kappa }^0=-2.0$, and ${\kappa }_c=12$. Also see movies in the Supplemental Material [42].

Figure 7

Power spectral density $\langle |\widehat{\varphi }{{|}^2\rangle }_s$ of the averaged tangent angle for: (a) a spermlike symmetric waveform and (b) Chlamydomonas-like breaststroke. Parameters are the same as in Figs. 6 and 6.

Figure 8

Tangent angle and motor population dynamics in the case of (a) spermlike, (b) cilialike, and (c) Chlamydomonas-like beating patterns. The top panel shows the evolution of the tangent angle $\varphi (s,t)$ measured at the midpoint of the filament $\left(s=1/2\right)$. The bottom panel shows the evolution of corresponding bound motor populations $n_{+}(s,t)$ and $n_{-}(s,t)$, also evaluated at the midpoint. Parameters are the same as described in the caption of Fig. 6.

Figure 9

The first row shows: (a) the kymograph, (b) the covariance function $C$, and (c) the two dominant mode shapes from PCA in the case of a symmetric spermlike beat. The bottom row (d, e, f) shows the same for the ciliary beating pattern. Parameters are identical to Figs. 6 and 6.

Figure 10

Limit cycle representation for (a) spermlike and (b) ciliary beating patterns. The red curves show the deterministic limit cycles. In panel (a), the gray points show deviations from the deterministic cycle due to biochemical noise.

Figure 11

Magnitude and streamlines of the instantaneous velocity field obtained using Eq. (39) for three representative cases: (a) spermlike beating pattern in free space, (b) ciliary beat in free space, and (c) ciliary beat against a no-slip wall at $y=0$. The three columns correspond to three time instants during one period of oscillation. Also see the movies of the instantaneous flow fields in the Supplemental Material [42].

Figure 12

(a) Streamlines and magnitude of the time-averaged velocity field ${\langle \mathbf{u}\rangle }_t$ for the three cases shown in Fig. 11. From left to right: sperm in free space, cilium in free space, and cilium against a no-slip wall. (b) Log-log plot of the time-averaged velocity magnitude vs. distance. The colors are associated with the lines along which the velocity are evaluated as shown in panel (a). In free space, the velocity fields show a decay of $1/r$ characteristic of a net Stokeslet. In the case of a cilium next to a wall, a faster decay of $1/r^3$ is found, which corresponds to a net force perpendicular to the no-slip wall.

Figure 13

First three PCA velocity modes ${\mathbf{V}}^{\left(i\right)}\left(\mathbf{x}\right)\left(i=1,2,3\right)$ for a clamped beating sperm. The first mode is dominated by a Stokeslet flow, whereas the next two modes have a more complex spatial structure with multiple velocity peaks that echo the structure of the deformation eigenmodes found in Fig. 9. All three velocity fields can be approximated in terms of regularized Stokeslets.

Figure 14

(a) Variation of the weighting coefficients ${\alpha }_i\left(t\right)$ of the three PCA velocity modes shown in Fig. 13 over one period of oscillation. (b) Comparison of the approximated instantaneous flow-field (right) to the exact numerical one (left) at $t/T=0.9$.