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Cytoskeletal filaments propelled by surface-bound motor proteins can be viewed as active polymers, a class of active matter. When constraints are imposed on their movements, the propelled cytoskeletal filaments show dynamic patterns distinct from equilibrium conformations. Pinned at their leading ends, propelled microtubules or actin filaments form rotating spirals, whose shape is determined by the interplay between motor forces and the mechanics of the cytoskeletal filaments. We simulated the spiral formations of microtubules propelled by kinesin motors in an overdamped dynamics framework, which in addition to the mechanics of the spiralling microtubule explicitly includes the mechanics of kinesin motors. The simulation revealed that spiral formation was initiated by localized buckling of microtubules near the pinned ends, and the conditions for occurrence of spiral formation were summarized in a phase diagram. The radius of the formed spirals scaled with the surface motor density with an exponent of approximately − 1/4, distinct from previous theoretical and simulation studies based on implicit modelling of motor proteins. This result can be understood as a consequence of the contributions of kinesin motors to the total elastic deformation energy, highlighting the importance of mechanics of motor proteins in the behaviour of the active polymers. These findings can be useful in accurate modelling of active polymers and in designing active polymer-based applications such as molecular shuttles driven by motor proteins.