Chirality is a widespread feature existing in nature and can be critical in the proper functions of some organisms. In our previous work, a rotational clutch-filament model for a radial fiber was built to reveal the critical role of α-actinin in the cellular chiral swirling. Here, we assume two mobility modes of α-actinin along actin filaments. In Mode A, where α-actinin concomitantly moves together with a growing filament, our model analysis suggests that cells cannot swirl clockwise; in Mode B, where α-actinin is fixed along the axial direction of the radial fiber instead, our model analysis suggests that both counter-clockwise and clockwise chiral swirling occur, consistent with experiments. Thus, our studies suggest that how α-actinin moves along growing filaments within a radial fiber would strongly affect cellular swirling. In addition, the previous rotational clutch-model has been improved by considering the elastic response of a radial fiber to a torque and distributed biomechanical properties of varied cell phenotype.