Mechanical explanation for simultaneous formation of different tissues in an organ morphogenesis

An interesting recent finding is that, in a medium of nearly linear elastic materials, the stem cell differentiation responds to the Young’s modulus of elasticity. It is discussed here that the more general mechanical identifier of differentiation is indeed the stiffness (rather than the Young’s modulus). Specifically, for nonlinear (and thus more realistic/physiologically mimetic) materials the stiffness experienced by the cells is affected by the intracellular distance and hence by the local density of the cells. It is proposed here that the stiffness-directed differentiation is affected not only by the substrate elasticity but also by the cell density and diffusion effects (relocations). Thus, it is suggested that nonlinear effects actually augment fascinating more versatile capabilities to the stiffness-directed differentiation compared to what was supposed by the traditional linear elastic materials. In particular, it is argued how such modified stiffness-directed differentiation criteria can explain simultaneous differentiation paths leading to the formation of an organ. Formation of organs like a limb requires simultaneous formation of different tissues from the hardest (central cartilage and bone) to softest ones (vascular networks of blood vessels and neurons) at correct locations. This means that while the pattern of initial stem cells is changing, they should follow different differentiation paths at the same time on the same substrate (a primary limb cross section, for instance). Here, it is shown that the distribution and relocation of the stem cells affect their sensing of the surrounding mechanics and hence their differentiation. This means various kinds of tissues automatically form, while the stem cells are redistributing and/or relocating.

Authors and Affiliations

1. Department of Materials Science and Engineering, Shiraz University, Shiraz, Iran

   Amir Hossein Haji

Corresponding author

Correspondence to Amir Hossein Haji.

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Appendix A

In order to assign the stiffness to a structural point, the common method is to set an arbitrary force on the desired location and find the resulting displacement, noting the fact that the equilibrium condition for the whole system is guaranteed by the system supports responsible to afford the reaction forces. This is what we can do for the stiffness experienced by an inserted bead in a matrix. Far from the cells, an inserted bead can be modeled as a small force distribution which stretches and contracts two half-spaces. Figure 7a shows such a bead at 2D case for better illustration. The experienced stiffness by such a bead [equal to \(K_{0} \) in Eq. (2)] would be obtained by adding the stiffness due to the semi-infinite spaces in tension to that in compression. So the stiffness measured by the bead reads:

where \(K_{S} \) is the semi-infinite space stiffness. In vicinity of the cell, another contribution to the equivalent-spring model should be considered due to the connection of the bead and the cell (Fig. 7b):

where \(K_{\mathrm{I}} =a/d^{2}\) as given by Eq. (2) is the spring constant for either of the connecting springs stretched to left or right with respect to the bead–cell connection centerline. (Note that as these springs are in series the stiffness of the whole connecting segment equals \(K_{\mathrm{I}} /2)\).

We may do the same for a singular contractile cell. The equivalent-spring model for a singular cell is shown in Fig. 7c. Compared to the bead at far location, there is another contribution to the cell’s experienced stiffness. This is due to the under-cellular spring in parallel to the semi-infinite springs. Let \(K_{U} /2\) be the stiffness of the equivalent spring for the whole under-cellular segment AB. The equivalent spring model then gives the stiffness experienced by the cell at its left and right sides (points A and B) as:

For the cells in assembly (as shown in Fig. 7d), the stiffness should include the intracellular stiffness \(K_{\mathrm{I}} \):

Equivalent spring models for an inserted bead (a), a singular cell–bead interaction (b), a singular cell (c), and a cellular network (d)

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As \(K_{U} \) is of the same order as \(K_{S} \) and much smaller than \(K_{\mathrm{I}} \), there is no fundamental difference between this stiffness and the stiffness measured by the bead (given by Eq. A.2).

It should be noted that the cells in the above situation are assumed to be in-place (no relocation) where the symmetry imposes the static equilibrium (equal forces on opposite sides of the cells). For such a case, the deflections at left and right sides are equal, and hence the total deflection that the cell measures along AB is two times the displacement at each side. Therefore, the stiffness that the cell could assign to the whole under-cellular segment would be half as much as the value it assigns to its left or right side (\(K_{AB} =0.5K_{C} =0.5K_{L} =0.5K_{R} )\).

An important notion is due to the relocation of the cell when the traction at one side is zero and the mechanical balance is provided by diffusion force (see “Appendix B”). In contrast to the in-place condition, the assignment of any stiffness to the whole segment AB is obviously not applicable in such case. It also follows that in such case the under-cellular portion is omitted from the spring model: \(K_{U} =0\). However, the cell can still have measurements for one-side (half-length) stiffness. For instance, consider the cell is relocating to the right, which means that the stiffness at right equals the intracellular stiffness at right side \(K_{R} =\left. {K_{\mathrm{I}} } \right| _{R} =a/d_{R}^{2} \), while at left side it would be measured \(K_{L} =0\). The average stiffness then reads \(0.5(K_{L} +K_{R} )=0.5K_{R} \), and the cell feels the matrix significantly softer compared to the in-place case [the stiffness \(K_{C} \) given by Eq. (A.4)].

Appendix B

An energy-based description of the diffusion process assumes that the flux of diffusing cells is directly related to their configuration. In isothermal condition, the internal energy (corresponding to the cells configuration) can be calculated over the entire matrix volume as:

where e(C) is the energy density. To extract some expression as force, one should try variation of this energy by the extended Hamilton’s principle:

which in the case the kinetic energy T is negligible reads:

where \({W}'\) denotes the work done by non-conservative forces and \(U(=E)\) the potential energy. Using (B.1):

where without loss of generality we have reduced the model to some specified direction. Now arguing some force term in the integrand calls for some term as displacement in turn. The relevant candidate present in the model is the concentration change which eventually depends on the cells displacement. From mass conservation, we have:

which takes the following form for C expressed as a spatial field (Eulerian view):

So:

and upon substituting into (B.4):

As any change in the potential energy negatively equals the work done by the conservative force, we get:

where \(\mathrm{d}m=C\mathrm{d}x\) and

The simplest option for the energy density (considering the even symmetry of the energy function) is \(e=C^{2}/2\), which yields:

Diffusion is usually taken as some steady-flux process which calls for some resistant force R against the potential force. Actually, this is the force we can assign the work done by the non-conservative forces \({W}'\). In the simplest case, R can be taken proportional to the flux:

Now:

gives:

upon (B.11) and (B.12). Finally, via the continuity equation:

we reach the classical Fickian diffusion within our force–displacement framework. Of special interest, however, is the interpretation of the resistant force R in such a mechanical–chemical framework. The best candidate for this is actually the resultant traction force applied by the cells by which the interplay of the mechanical and diffusion mechanisms is trivially achieved. In other words, the two mechanisms are conjugate as the resistant to each one’s flux is provided by the flux of the other.

Appendix C

By substituting the perturbations \(n=1+\tilde{{n}}\), \(\rho =1+\tilde{{\rho }}\) and \(u=\tilde{{u}}\) in (12)–(14), one gets the following up to linear order:

where \(\tau _{1} =\tau /(1+\lambda )\) and \(\tau _{2} =\tau (1-\lambda )/(1+\lambda )^{2}\).

Substituting \(\tilde{{n}}=\tilde{{n}}_{0} \exp (\sigma t+i\vec {{k}}\cdot \vec {{r}})\), \(\tilde{{\rho }}=\tilde{{\rho }}_{0} \exp (\sigma t+i\vec {{k}}\cdot \vec {{r}})\), and \(\tilde{{u}}=\tilde{{u}}_{0} \exp (\sigma t+i\vec {{k}}\cdot \vec {{r}})\)in (C.1)–(C.3):

which yields:

where \(\mu =\mu _{1} +\mu _{2} \). Dividing by \((1+{\nu }')\)and letting \(\mu /(1+{\nu }')\rightarrow \mu \), \(\tau _{1} /(1+{\nu }')\rightarrow \tau _{1} \), \(\tau _{2} /(1+{\nu }')\rightarrow \tau _{2} \) and \(s/(1+{\nu }')\rightarrow s\), we reach Eq. (17).

Appendix D

For the pure mechanical case (I), the model reads:

The cell conservation equation in this case is just comprised of the convective term emphasizing that the cells just passively relocate due to the ECM deformations. On the other hand, such ECM deformations are due to the in-place (with respect to the ECM) contractions applied by the cells [described by Eq. (D.2)].

For the diffusion–mechanical case (II), we have:

The cell conservation equation in this case is composed of not only the convective term but also the diffusion and mechanotaxis terms. However, another important difference compared to the case (I) is that we have also set \(\gamma =0\) in the cells traction term emphasizing that it does not have any long-range dependence in this case. This is based on the assumption that the cells in this case do not need to reach a uniform static balance but rather move in a continuous dynamic manner (relocating via locomotion as stated in Sect. 2.1), and hence no long-range modifications in tractions are applied.

Another important notion regarding these two cases is that due to their different wavelengths (as discussed in Sect. 3.1), they actually evolve together but at different scales: case (I) at large and case (II) at small scales.

Appendix E

1.1 Discussion on limb bud attachment to the substrate

In order to clarify the attachment condition considered in the mechanical model, we need to consider the formation of another important part of the musculoskeletal system, i.e., the joints. The joints are the articulated regions of two adjacent skeletal bones. The limb joints commonly hinge the limb to the body (like the shoulder and pelvis joints) or two articulated parts alongside the limb (like the knee and elbow joints). The limb joints form from initial cartilage, and actually, they are those parts of the cartilage which have not been replaced by bones.

From a mechanical-model point of view, the joint formations are mainly due to large-scale instabilities and/or symmetry breaking. The longitudinal growth of the limb gives rise to large aspect ratios responsible for axial instabilities. On the other hand, lateral symmetry break (for instance from a circular to an elliptic cross section) causes the lateral instabilities [21]. These effects may work individually or collectively. Formation of the phalanges, for instance, is mainly due to the axial instabilities, while for knees and elbows both the axial and lateral instabilities take effect.

Figure 8 shows different kinds of joints and corresponding instabilities and/or symmetry breaks leading to their formation. For the limb bud, the main effect is the translational symmetry break as two different parts with different geometry are attached together (the shoulder and pelvis joints (Fig. 8a). The formation of phalanges is dominated by the axial instabilities (Fig. 8b), while for knees and elbows the axial instabilities are combined with rotational symmetry break (Fig. 8c). Symmetry breaks actually imply the mechanical attachment condition (\(s\ne 0\)) to be a reasonable assumption for the limb bud and hence the initial cartilage/vascular formation.

Instabilities and symmetry breaks in joint formation

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