Spontaneous Polaron Transport in Biopolymers

Polarons, introduced by Davydov to explain energy transport in $\alpha$-helices, correspond to electrons localised on a few lattice sites because of their interaction with phonons. While the static polaron field configurations have been extensively studied, their displacement is more difficult to explain. In this paper we show that, when the next to nearest neighbour interactions are included, for physical values of the parameters, polarons can spontaneously move, at T=0, on bent chains that exhibit a positive gradient in their curvature. At room temperature polarons perform a random walk but a curvature gradient can induce a non-zero average speed similar to the one observed at zero temperature. We also show that at zero temperature a polaron bounces on sharply kinked junctions. We interpret these results in light of the energy transport by transmembrane proteins.

💡 Research Summary

The authors investigate the dynamics of Davydov‑type polarons on flexible polymer chains when next‑nearest‑neighbour (NNN) electronic couplings are included. Starting from a semi‑classical Holstein‑Mingaleev Hamiltonian that couples an electron wavefunction φₙ to lattice displacements Rₙ, they add an exponentially decaying transfer integral Jₙₘ that extends beyond adjacent sites. This long‑range term is controlled by a decay parameter α, allowing the electron to feel the geometry of the chain over several lattice spacings.

Physical parameters are chosen to represent an α‑helix: the excitation transfer energy W≈2×10⁻²² J, mass M̂≈2×10⁻²⁵ kg, elastic constant σ̂≈19.5 N m⁻¹, and a bending stiffness derived from the persistence length of helices. After nondimensionalisation, the key dimensionless quantities become g≈2.5 (electron‑phonon coupling), σ≈5.1, k≈777 (bending rigidity), and Γ≈814 (viscous damping). The decay constant is set to α=2, following previous work.

Numerical experiments are performed on a chain of N=60 sites. The central segment (sites 25–45) is pre‑bent by imposing a linearly increasing bond angle ϕₙ, producing a curvature gradient dϕ. A stationary polaron is first relaxed on a straight chain, then the bending is switched on and the full coupled equations of motion (including a Langevin noise term for finite temperature) are integrated with a fourth‑order Runge‑Kutta scheme.

At zero temperature the results show that the polaron is attracted toward regions of higher curvature. The physical origin is that the long‑range electronic coupling lowers the energy when neighboring lattice points are closer together; a bent region naturally brings non‑adjacent sites nearer, creating an effective potential well. Consequently, the polaron accelerates along the curvature gradient, reaches the end of the bent segment, and is reflected by the straight portion, which acts as a potential barrier. The acceleration depends strongly on α: smaller α (longer‑range coupling) yields larger speeds, while larger α suppresses motion.

The dependence on the curvature gradient dϕ exhibits a depinning transition. Below a critical gradient dϕ_c≈5.6×10⁻³ the polaron remains pinned; above it, the average velocity follows a power law ⟨V⟩∝(dϕ−dϕ_c)^ν with ν≈0.5, reminiscent of elastic interface depinning. In physical units the maximal speed is about 16 nm ns⁻¹, comparable to the speed of energy transport reported in some protein experiments.

At physiological temperature (T≈300 K) thermal fluctuations are introduced via white noise satisfying the fluctuation‑dissipation theorem. The polaron now performs a random walk, but the curvature gradient biases the walk, producing a small net drift toward the more curved region. Because the chain ends act as reflecting walls, a polaron initially placed nearer one end can undergo multiple reflections, each time gaining a slight bias from the bend, leading to a measurable average drift over long times.

Biologically, the authors argue that many transmembrane proteins and α‑helices are intrinsically curved, so the curvature‑induced attraction of polarons could provide a passive, structure‑driven mechanism for directed energy transport without external forcing. This contrasts with earlier models that required an explicit “kick” or external field to set the polaron in motion.

The paper acknowledges several limitations: the model remains one‑dimensional, neglecting the full three‑dimensional geometry of real proteins; the electron‑phonon coupling constant g is chosen larger than experimentally inferred values, which may overestimate mobility; and the hard‑core repulsion term is largely ignored in equilibrium configurations. Future work is suggested to extend the model to multi‑strand helices, incorporate realistic electronic structure calculations, and seek experimental signatures of curvature‑driven polaron motion (e.g., ultrafast spectroscopy on engineered peptide chains).

Overall, the study provides a clear theoretical demonstration that curvature gradients in flexible biopolymers can spontaneously drive Davydov‑type polarons, offering a plausible route for directed energy flow in biological macromolecules.