We have simulated polymer translocation across the a α-hemolysin nano-pore via a coarse grained computational model for both the polymer and the pore. We simulate the translocation process by allowing the protein cross a free-energy barrier from a metastable state, in the presence of thermal fluctuations. The deformation in the channel, which we model by making the radius of pore change from large to small size, can be originated by the random and non-random (systematic) cellular environment, drive out the polymer out of equilibrium during the transport dynamics. We expect that in more realistic conditions, effects originating on the translocation phenomena due to the deformability of the nano-pore can either decrease or increase the transport time of biomolecule passing through the channel. Deformation in channel can occurred because the structure of αhemolysin channel is not completely immobile, hence a small pore deformation can be occurred during translocation process. We also discuss the effects of polymer deformation on the translocation process, which we achieve by varying the value of the empirical and dihedral potential constants. We investigate the dynamic and thermodynamical properties of the translocation process by revealing the statistics of translocation time as a function of the pulling inward force acting along the axis of the pore under the influence of small and large pore. We observed that a pore with small size can speed down the polymer translocation process, especially at the limit of small pulling force. A drastic increase in translocation time at the limit of low force for small pore clearly illustrate the strong interaction between the transport polymer and pore. Our results can be of fundamental importance for those experiments on DNA-RNA sorting and sequencing and drug delivery mechanism for anti-cancer therapy.