Instrinsic oscillations of treadmilling microtubules in a motor bath

We analyse the dynamics of overlapping antiparallel treadmilling microtubules in the presence of crosslinking processive motor proteins that counterbalance an external force. We show that coupling the force-dependent velocity of motors and the kinetics of motor exchange with a bath in the presence of treadmilling leads generically to oscillatory behavior. In addition we show that coupling the polymerization kinetics to the external force through the kinetics of the crosslinking motors can stabilize the oscillatory instability into finite-amplitude nonlinear oscillations and may lead to other scenarios, including bistability.

💡 Research Summary

The authors present a theoretical study of two antiparallel, treadmilling microtubules (MTs) that overlap over a region of length ℓ and are cross‑linked by processive tetrameric kinesin motors (e.g., Eg5). An external constant force F pushes the MTs toward their minus ends, while the motors generate a counter‑force that balances F. The model reduces the motor population in the overlap to two variables: nc, the number of motors simultaneously bound to both MTs, and nb, the number bound to only one filament. Motor binding occurs with rate kb, while unbinding occurs with a force‑dependent rate ku = k0u exp(fm b/kBT), where fm = F/nc is the load per motor and b is a nanometer‑scale length. This exponential dependence creates a strong nonlinear feedback: as nc decreases, the load per motor rises, accelerating unbinding.

The overlap length ℓ evolves according to dℓ/dt = 2(Vp – Vs). The sliding velocity Vs is assumed to decrease linearly with load: Vs = V0(1 – F/(nc fs)), where V0 is the unloaded motor speed and fs the stall force. The polymerization speed Vp may be constant (first part of the analysis) or depend on nc (later part). After nondimensionalisation (˜F, ˜n_c, ˜ℓ, etc.) the steady state is given by Eq. (4). Linear stability leads to a cubic eigenvalue equation (5). The key dimensionless parameters are

• g = ˜f(1 – Vp/V0), which measures the effective motor‑MT coupling,
• Δ, a ratio of motor influx to outflux proportional to the bulk motor concentration, and
• γ = kb/k0u, the asymmetry between binding and unbinding at zero load.

For g < 1 the fixed point is always stable. When 1 < g < 2, the sign of the real part λ0 of a complex conjugate pair of eigenvalues determines stability. The condition λ0 = 0 defines a Hopf bifurcation curve (Eq. 8). On the stable side (λ0 < 0) the system exhibits damped oscillations; on the unstable side (λ0 > 0) the oscillations grow exponentially. The oscillation frequency θ = Im(λ) is given by θ = √(2g²Δ − G) with G = (g − 1) e^g, and depends on both kinetic parameters and the motor bath through Δ.

When the time scales of the fast mode (λ1) and the slow oscillatory mode (λ0) are well separated (|λ1| ≫ |λ0|), a center‑manifold reduction yields a second‑order equation for nc: ¨nc − λ0 · ṅc + θ² nc = 0. Thus the system behaves as a harmonic oscillator with growth or decay set by λ0. The other variables (nb, ℓ) follow linearly from nc, with ℓ acting as a weighted motor density that keeps the combination G nc + γ nb roughly constant.

The authors then introduce a feedback of motor density on polymerization: Vp(nc) =