Cell differentiation, proliferation and migration are essential processes in tissue regeneration. Experimental evidence confirms that cell differentiation or proliferation can be regulated according to the extracellular matrix stiffness. For instance, mesenchymal stem cells (MSCs) can differentiate to neuroblast, chondrocyte or osteoblast within matrices mimicking the stiffness of their native substrate. However, the precise mechanisms by which the substrate stiffness governs cell differentiation or proliferation are not well known. Therefore, a mechano-sensing computational model is here developed to elucidate how substrate stiffness regulates cell differentiation and/or proliferation during cell migration. In agreement with experimental observations, it is assumed that internal deformation of the cell (a mechanical signal) together with the cell maturation state directly coordinates cell differentiation and/or proliferation. Our findings indicate that MSC differentiation to neurogenic, chondrogenic or osteogenic lineage specifications occurs within soft (0.1-1 kPa), intermediate (20-25 kPa) or hard (30-45 kPa) substrates, respectively. These results are consistent with well-known experimental observations. Remarkably, when a MSC differentiate to a compatible phenotype, the average net traction force depends on the substrate stiffness in such a way that it might increase in intermediate and hard substrates but it would reduce in a soft matrix. However, in all cases the average net traction force considerably increases at the instant of cell proliferation because of cell-cell interaction. Moreover cell differentiation and proliferation accelerate with increasing substrate stiffness due to the decrease in the cell maturation time. Thus, the model provides insights to explain the hypothesis that substrate stiffness plays a key role in regulating cell fate during mechanotaxis.