A model for hierarchical patterns under mechanical stresses

Abstract

We present a model for mechanically-induced pattern formation in growing biological tissues and discuss its application to the development of leaf venation networks. Drawing an analogy with phase transitions in solids, we use a phase field method to describe the transition between two states of the tissue, e.g. the differentiation of leaf veins, and consider a layered system where mechanical stresses are generated by differential growth. We present analytical and numerical results for one-dimensional systems, showing that a combination of growth and irreversibility gives rise to hierarchical patterns. Two-dimensional simulations suggest that such a mechanism could account for the hierarchical, reticulate structure of leaf venation networks, yet point to the need for a more detailed treatment of the coupling between growth and mechanical stresses.

Keywords:

• mechanical instabilities
• phase field
• numerical modeling
• pattern dynamics

Acknowledgements

This work was supported by EC NEST project MechPlant. The authors would like to thank one anonymous referee for detailed remarks and helpful suggestions on a previous version of this manuscript.

Notes

Notes

1. Conversely, localized deformation in non-linear elastic materials can be interpreted as a first-order phase transition.

2. The parameters are chosen such that the critical strains defined below satisfy ε _c1 < ε _c0.

3. Similar calculations could be carried out for the states of equilibrium in which the two phases coexist if the width of the interfaces is neglected.

4. If the two layers were instead assumed to grow at different rates, then the mismatch would become arbitrarily large with time.

5. The equations of mechanical equilibrium (a set of linear equations) are solved at each step of the relaxation. The relaxation is stopped when the rate of change of the phase field falls below a certain threshold.

6. We are thus assuming that there are three well separated time scales τ_mech ≪ τ_diff ≪ τ_growth, where τ_mech, τ_diff, and τ_growth are the characteristic time scales corresponding to mechanical relaxation, cell differentiation, and growth, respectively.

7. Such an ad hoc rule would clearly be unsatisfactory to describe a physical system, but in the case of biological systems, it can be seen as a necessary simplification of the complex phenomena involved.

8. Note that the potential defined by Equation (17) cannot be written in this form.

9. A typical example in the case of an irreversible evolution can be seen in Figure 8a.

10. A typical example in the case of an irreversible evolution can be seen in Figure 8c.

11. To this end, we define a ‘maximum deformation tensor’ ε _m that keeps track of the maximum past compression in any direction, i.e. the tensor ε _r (t) − ε _m (t′) is positive for all times t < t′. From this maximum deformation tensor, we define an ‘irreversible deformation tensor’ ε _i . In a basis that diagonalizes ε _m , ε _i = diag(f(λ₁), f(λ₂)), where ε _m = diag(λ₁, λ₂). The function f is defined by f(x) = 0.9x if x < −0.5 and f(x) = ∞ if x > −0.5, so that compressions in excess of 0.5 are stored in ε _i . Irreversibility is implemented by imposing that the tensor ε _r − ε _i is negative. The factor 0.9 in the definition of f was introduced to prevent numerical instability.