A first principle (3+1) dimensional model for microtubule polymerization

In this paper we propose a microscopic model to study the polymerization of microtubules (MTs). Starting from fundamental reactions during MT’s assembly and disassembly processes, we systematically derive a nonlinear system of equations that determines the dynamics of microtubules in 3D. %coexistence with tubulin dimers in a solution. We found that the dynamics of a MT is mathematically expressed via a cubic-quintic nonlinear Schrodinger (NLS) equation. Interestingly, the generic 3D solution of the NLS equation exhibits linear growing and shortening in time as well as temporal fluctuations about a mean value which are qualitatively similar to the dynamic instability of MTs observed experimentally. By solving equations numerically, we have found spatio-temporal patterns consistent with experimental observations.

💡 Research Summary

In this work the authors present a first‑principles, three‑dimensional (3+1 D) model of microtubule (MT) polymerization and depolymerization that is rooted in the underlying biochemical reactions. Starting from a solution containing GTP‑tubulin, GDP‑tubulin and free GTP, they identify four elementary processes: (i) GTP hydrolysis, (ii) conversion of tubulin from the GTP‑bound to the GDP‑bound state, (iii) addition of a tubulin layer to the MT end (growth), and (iv) removal of a layer (shrinkage). Experimental free‑energy changes for these steps (Δ₁≈220 meV, Δ₂≈160 meV, Δ₃≈Δ₄≈420 meV) are taken from the literature and used to justify a quantum‑mechanical treatment of each reaction.

The MT is represented as a quantum state |N⟩ where N counts the number of tubulin layers. Creation and annihilation operators a†, a raise or lower N, while b†, b act on tubulin dimers and d†, d on the energy quanta associated with GTP hydrolysis. The two fundamental reactions are compactly written as operator products a† b d (growth) and d† b† a (shrinkage). A Hamiltonian is constructed that contains a free part (linear in number operators with coefficients ω, σ, Γ) and an interaction part consisting of all possible powers of the composite operators, the exponent n representing the number of successive growth or shrinkage events. Momentum conservation imposes constraints on the indices of the operators.

Applying the Heisenberg equation of motion i∂ₜq = –