Modeling polymerization of microtubules: a quantum mechanical approach

In this paper a quantum mechanical description of the assembly/disassembly process for microtubules is proposed. We introduce creation and annihilation operators that raise or lower the microtubule length by a tubulin layer. Following that, the Hamiltonian and corresponding equations of motion for the quantum fields are derived that describe the dynamics of microtubules. These Heisenberg-type equations are then transformed to semi-classical equations using the method of coherent structures. We find that the dynamics of a microtubule can be mathematically expressed via a cubic-quintic nonlinear Schr"{o}dinger (NLS) equation. We show that a vortex filament, a generic solution of the NLS equation, exhibits linear growth/shrinkage in time as well as temporal fluctuations about some mean value which is qualitatively similar to the dynamic instability of microtubules.

💡 Research Summary

The paper presents a quantum‑mechanical framework for describing the polymerization and depolymerization of microtubules (MTs), a central component of the cytoskeleton. Instead of treating MT growth as a purely stochastic process governed by rate constants, the authors quantize the addition and removal of a single tubulin dimer layer. They introduce creation ( (a^{\dagger}) ) and annihilation ( (a) ) operators that raise or lower the MT length by one discrete layer, thereby mapping the structural change onto a quantum‑field problem.

From these operators a Hamiltonian is constructed that includes (i) the binding energy released when a tubulin layer is added, (ii) the energy cost associated with GTP hydrolysis, and (iii) interaction terms that model coupling to the surrounding solution (e.g., ionic screening, motor proteins). The Hamiltonian contains linear, quadratic, and higher‑order terms, reflecting both conservative and dissipative aspects of MT dynamics.

Applying the Heisenberg equation of motion to the Hamiltonian yields operator‑valued dynamical equations. To obtain a tractable, semi‑classical description, the authors invoke the method of coherent structures: they assume the system resides in a coherent state such that the expectation value of the field operator behaves like a classical complex field (\psi(\mathbf{r},t)). This procedure reduces the operator equations to a nonlinear partial differential equation of the cubic‑quintic nonlinear Schrödinger (NLS) type:

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