<p>We investigate the nonlinear partial differential equation of special relevance in nano-biosciences, which describes microtubules (MT) model as nonlinear transmission lines. Within the nano and sub-nano scales for either space and time, the resulting solutions of the ionic current and the corresponding voltage are estimated via the emergence of the Tan-function, Cole-Hopf and factorization methods, and through the illustrating graphs, behave along the microtubules as shock and solitary waves. The application of the phase portrait method reveals the system's stability characters. The irreversible thermodynamics of open systems is examined through the evaluation of the entropy production rate together with the Lagrangian density, which describe the energy evolution in the MT, and prove their oscillatory character between negative and positive signs insuring its instability. The obtained results show fair agreements with the biophysical situation of the problem.</p>