Localized modulated waves in microtubules

Abstract

In the present paper, we study nonlinear dynamics of microtubules (MTs). As an analytical method, we use semi-discrete approximation and show that localized modulated solitonic waves move along MT. This is supported by numerical analysis. Both cases with and without viscosity effects are studied.

Mechanical Models of Microtubules

“…(38) even though the terms including φ n 2 and φ n 4 appear as a result of a series expansion of the cosine function. Instead of viscosity force introduced in the previous section, we introduce viscosity momentum M v ¼ÀΓ _ φ, where Γ is the viscosity coefficient [51][52][53]. Following the procedure explained earlier, we obtain…”

“…Due to viscosity momentum M v ¼ÀΓ _ Φ n, the final result φ n t ðÞincludes the expected exponential term e Àβ t , where β ¼ Γ=2I [51].…”

“…¼ÀQdE cos φ n would be better choice for the potential energy, where d is the distance between the centers of positive and negative charges within the dipole. This potential indicates the angle as a coordinate instead of the projection u and the Hamiltonian for the radial model, which we call as φ-model, is [50,51] …”

“…Finally, viscosity should be introduced in the semi-discrete approximation [51]. Due to viscosity momentum M v ¼ÀΓ _ Φ n, the final result φ n t ðÞincludes the expected exponential term e Àβ t , where β ¼ Γ=2I [51].…”

“…We will return to this issue in the next section. Now, we switch to the semi-discrete approximation within the φ-model to solve the dynamical equation of motion, which is [51] εI…”