Role of Mechanical Cues in Cell Differentiation and Proliferation: A 3D Numerical Model

Abstract

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.

Fig 1. Cell mechanosensing.

a- Spherical configuration of the cell in which sensing forces are exerted at each membrane node towards the cell centroid (mechano-sensing process). b- Calculation of the cell internal deformation due to cell mechano-sensing. Deformed cell due to mechano-sensing. e _pol stands for polarisation direction of the reoriented cell while ${\mathbf{\text{F}}}_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, F _prot and F _drag represent the net traction force, protrusion force and drag force, respectively.

Fig 2. Interaction of two cells in contact.

For the assumed cell configuration, two cells can have four common nodes (n ₁:n ₄). x _i and x _j are position vectors of the ith and the jth cells, respectively, while x _ij is a vector passing by the centroids of the ith and jth cells. The distance between their centroids (O_i and O_j) is equal to or greater than the proposed cell diameter, ‖x _ij‖ ≥ 2r.

Fig 3. Computational algorithm of cell mechano-sensing and consequent cell fate due to mechanotaxis.

Fig 4. MSC proliferation and differentiation within a substrate of 45 kPa stiffness after 5.5 days.

MSC proliferation (see also S1 Video).

Fig 5. MSC proliferation and differentiation within a substrate of 45 kPa stiffness after 32 days.

The first commitment of a mature mother MSC to osteogenic lineage specification (see also S1 Video)

Fig 6. MSC proliferation and differentiation within a substrate of 45 kPa stiffness after 50 days.

Continuing differentiation and proliferation of MSCs and osteoblasts (see also S1 Video).

Fig 7. MI of MSCs within substrates of different uniform stiffnesses.

E represents substrate elasticity modulus.

Fig 8. Average cell traction force within a hard substrate of 45 kPa stiffness.

Average cell traction force, $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, versus time within a hard substrate of 45 kPa stiffness during MSC proliferation and differentiation. Point A represents the instant of MSC proliferation which causes a considerable jump in the average net traction force while point B is the initial instant of MSC differentiation to osteoblast leading to an enhancement of the average net traction force.

Fig 9. Osteoblast proliferation in hard substrates.

a- 30 kPa and b- 45 kPa stiffness (see also S2 Video).

Fig 10. Average cell traction force within a hard substrate of 30 kPa and 45 kPa stiffness.

Average cell traction force, $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, versus time within hard substrates during MSC differentiation and osteoblast proliferation. Points A represent the instant of MSC differentiation to osteoblast which instantly causes a traction force increase while points B are the initial instant of osteoblast proliferation causing a jump in the average net traction force.

Fig 11. Normalized density of each cell phenotype.

Normalized density of each cell phenotype versus substrate stiffness during identical times as a consequence of MSC differentiation and proliferation of each cell phenotype. The error bars represent mean standard deviation of different runs.

Fig 12. Chondrocyte proliferation in intermediate substrates.

a- 20 kPa and b- 25 kPa stiffness (see also S3 Video).

Fig 13. Average cell traction force within intermediate substrates of 20 kPa and 25 kPa stiffness.

Average cell traction force, $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, versus time within intermediate substrates during MSC differentiation and chondrocyte proliferation. Point A represents the moment of MSC differentiation to chondrocyte which instantly causes the traction force to increase while point B is the initial moment of chondrocyte proliferation causing a jump in the average net traction force.

Fig 14. Neuroblast proliferation in soft substrates.

a- 0.1 kPa and b- 1 kPa stiffness (see also S4 Video).

Fig 15. Average cell traction force within soft substrates of 0.1 kPa and 1 kPa stiffness.

Average cell traction force, $F_{\mathrm{\text{net}}}^{\mathrm{\text{trac}}}$, versus time within soft substrates during MSC differentiation and neuroblast proliferation. Points A represent the instant of MSC differentiation to neuroblast which instantly causes the traction force to decrease while points B are the initial instant of neuroblast proliferation causing a jump in the average net traction force.