Hydrodynamic instabilities provide a generic route to spontaneous biomimetic oscillations in chemomechanically active filaments

Non-equilibrium processes which convert chemical energy into mechanical motion enable the motility of organisms. Bundles of inextensible filaments driven by energy transduction of molecular motors form essential components of micron-scale motility engines like cilia and flagella. The mimicry of cilia-like motion in recent experiments on synthetic active filaments supports the idea that generic physical mechanisms may be sufficient to generate such motion. Here we show, theoretically, that the competition between the destabilising effect of hydrodynamic interactions induced by force-free and torque-free chemomechanically active flows, and the stabilising effect of nonlinear elasticity, provides a generic route to spontaneous oscillations in active filaments. These oscillations, reminiscent of prokaryotic and eukaryotic flagellar motion, are obtained without having to invoke structural complexity or biochemical regulation. This minimality implies that biomimetic oscillations, previously observed only in complex bundles of active filaments, can be replicated in simple chains of generic chemomechanically active beads.

Following Ref. [1], we construct an active elastic filament by chaining active beads using potentials. We place N such beads at points r 1 , r 2 , . . . r N and define bond vectors b m = r m+1 − r m between adjoining beads. The potentials U S (b m ) = 1 2 k(|b m | − b 0 ) 2 and U B (b m , b m+1 ) = (κ/b 0 )(1 − cos φ m ) model inextensibility and semiflexibility respectively, penalizing departures of the filament from the equilibrium bead-bead separation of b 0 or the bond-bond angle φ m = 0. Here k is the spring constant and κ is the bending modulus. Self-avoidance is enforced through a purely repulsive Lennard-Jones potential U LJ which vanishes smoothly at a distance σ LJ . The total potential U (r 1 , . . . , r N ) is the sum of these three potentials. Stretching, bending and self-avoidance causes the total elastic force f n = −∂U (r 1 , . . . , r N )/∂r n to act on the n-th bead.
We model the activity using force-free and torque-free singularities [2][3][4][5][6], of which the second-rank symmetric stresslet tensor is the most dominant [7], and produces flows that decay inverse squarely with distance. With a tensorial strength σ n and an axis of uniaxial symmetry p n , the stresslet can be parametrised as σ n = σ 0 (p n p n − I/d) where d is the spatial dimension and σ 0 sets the scale of the activity. Extensile flows correspond to σ 0 > 0, while contractile flows correspond to σ 0 < 0. In the present case, we set p n =t n , the local unit tangent vector, to reflect the tangential stresses generated by the active particles.
The filament exerts forces on the surrounding fluid due to its elasticity and activity. The resultant force density at the point r due to N active beads is given by summing over the Stokeslet and stresslet singularities, Integrating this on a surface enclosing the filament gives (2) Since the elastic forces f m are internal to the filament and obey Newton's third law, they cancel exactly on summation over all the beads. The active terms are total divergences and thus vanish on integration over a bounding surface. The symmetry of the stresslet tensor ensures that no angular momentum is added to the fluid. Consequently, the force density considered here does not add any net linear or angular momentum to the fluid. Models in which this fundamental constraint is not imposed will fail to correctly reproduce HI due to active energy transduction.
The velocity of the n-th bead is obtained by summing the force and activity contributions from all beads, including itself, to the fluid velocity at its location. Thus, we obtain the equation of motion [1] An isolated spherical bead with a force f acquires a velocity µ f where µ is its mobility. By symmetry, an isolated spherical bead with a stresslet σ cannot acquire a velocity. Therefore, for m = n, O αβ = µδ αβ and D αβγ = 0. In the absence of activity, σ n = 0, and bending, κ = 0, the equation reduces to the Zimm model of hydrodynamic interactions of a polymer in a good solvent [10]. Dimensionally, the active and elastic forces are of the form σ 0 /L and κ/L 2 , where L = (N − 1)b 0 is the length of the filament. The balance of these forces gives the dimensionless quantity A = Lσ 0 /κ, which is also the ratio of the active and elastic rates of relaxation, respectively Γ σ = σ 0 /ηL d and Γ κ = κ/ηL d+1 [1]. The dynamics of the filament is completely captured by its length L and the relative active strength, given by this activity number A.
In the free-draining approximation to our model, we ignore HI. Thus the velocity of the n-th bead due to elastic forces is µf n , where µ is the mobility. For the active velocity we retain contributions from immediate neighbours of the n-th bead. This gives a local equation of motionṙ The parameter values used for the simulation and analysis are: fluid viscosity η = 1/6, radius of monomer a = Video S3 : Hopf bifurcation in clamped active filament. Description : Variation of real and imaginary parts of eigenvalues with activity number A for L = 188. The main panel shows the two largest eigenvalue pairs (red and blue pentagons) while the inset shows the entire spectrum. All eigenvalues are real and negative for A < 6, beyond which the first pair converge and become complex conjugates, indicating the transition from the stable node to the stable focus. This pair crosses the imaginary axis at A ≈ 12.5, that is, at A c1 , indicating the transition from the stable focus to a limit cycle through a Hopf bifurcation. The second pair replicates this entire behaviour at higher A.