Stability Analysis of Jacobi Elliptic Solutions of Microtubule

Abstract

A nonlinear dynamic model which elucidate the mechanism of the internal motion occuring during the assembly of microtubules (MTs) is considered. The nonlinear space-time dynamics, defined in terms of celebrated ‘solitonic’ equations, brings indispensable tools for understanding, prediction and control of complex behaviors in both physical and life sciences. MTs is assumed to be a single longitudinal degree of freedom per tubulin dimer, and the motion equation is reduced to the discrete nonlinear Schr ¨ odinger equation. Exact solutions are investigated using the Jacobian elliptic fonctions approach. It is shown that such a nonlinear model can lead to the existence of kink solitons, breather as well as Jacobi solutions which can propagate along the microtubule outer surface, and the tubulin tail soliton collisions could serve as elementary computational gates that control cytoskeletal processes. We examine the stability of these solutions and found that the kink solution is stable with respect to small-amplitude fluctuations.