Twist-to-bend ratio: an important selective factor for many rod-shaped biological structures

Mechanical optimisation plays a key role in living beings either as an immediate response of individuals or as an evolutionary adaptation of populations to changing environmental conditions. Since biological structures are the result of multifunctional evolutionary constraints, the dimensionless twist-to-bend ratio is particularly meaningful because it provides information about the ratio of flexural rigidity to torsional rigidity determined by both material properties (bending and shear modulus) and morphometric parameters (axial and polar second moment of area). The determination of the mutual contributions of material properties and structural arrangements (geometry) or their ontogenetic alteration to the overall mechanical functionality of biological structures is difficult. Numerical methods in the form of gradient flows of phase field functionals offer a means of addressing this question and of analysing the influence of the cross-sectional shape of the main load-bearing structures on the mechanical functionality. Three phase field simulations were carried out showing good agreement with the cross-sections found in selected plants: (i) U-shaped cross-sections comparable with those of Musa sp. petioles, (ii) star-shaped cross-sections with deep grooves as can be found in the lianoid wood of Condylocarpon guianense stems, and (iii) flat elliptic cross-sections with one deep groove comparable with the cross-sections of the climbing ribbon-shaped stems of Bauhinia guianensis.


Appendix A: L 2 -Gradient Flow
Introducing an artificial time variable t (hereafter called pseudo-time) and using the L 2 scalar product (·, ·) as well as the first variation d I d u [v] of a functional I in u in the direction of v the choice of a gradient flow dynamic leads to the pseudo-time stepping approach given by for all v 2 H 1 0 (W). The variation of the perimeter term yields the well-known Allen-Cahn equation. To solve equation (A1) by a finite element method, we thus still need to compute the first variations of D min/max (u) and D z (u) in u. Concerning D min/max (u) these are given by simple calculations and with For regularisation near RM(u) = 0, we introduce a parameter 0 < q 1 ⌧ 1 and approximate d RM Finally, using D min/max = D mean ⌥ RM, the variation of the flexural rigidities can be computed.
Concerning the first variation of D z , we consider the weak formulation of equation (4) which is given by for all test functions v 2 H 1 0 (W) and f = 0 on ∂ W. Remark. We note that for every admissible phase field variable u 2 A the Riesz representation theorem ensures that equation (A2) is uniquely solvable. Since the variation of D z (u) depends on the solution f (u) of equation (A2), we apply a Lagrangian approach, see, e.g., Hinze et al. 1 . Introducing the adjoint variable p : W ! R we can formulate the Lagrangian as Seeking stationary states (u, f , p) of L, we find that, if the first variation for (u, f , p) vanishes, both f and p solve equation (A2). Since equation (A2) is uniquely solvable we conclude that f = p and obtain with the solution f (u) of equation (A2). This finally gives us the gradient flow

Appendix B: Finite Element Approximation
To solve equation (A1) by the finite element method we choose the discrete subspace S 1,0 (T h ) ⇢ H 1 0 (W), which is given by Further we choose a finite difference quotient in order to discretise the time derivative. Using an explicit treatment of the appearing nonlinear terms in equation (A1), and an implicit treatment of the Laplacian, we obtain the semi-implicit time stepping with inner nodal basis points x i 2 N h and j = 1,...,|N h |. The nodal interpolant I 1,h (v) of a function v 2 C 0 (W) is given by The mass constraint is imposed by the additional condition In every time step t we thus solve the problem where S = M + tgK 2 R n⇥n and B = L. Here the mass matrix M, stiffness matrix K and lumped mass matrix L 2 R n (represented only by its diagonal) are given by The right hand sideF is given byF j = t e F j + (M ·U n ) j for j = 1,...,N, andF N+1 = m, where we have In order to achieve an energy stability such that for all n we have (g Per e +s 1 D z + s 2 D min + s 3 D max )(u n )  (g Per e +s 1 D z + s 2 D min + s 3 D max )(u n 1 ), it is necessary to demand t 2 O(e 3 ).
For the experiments we set the interfacial parameter e = 3 · 10 3 and choose the time step as t = 5.5 · 10 6 . The parameters for the material elastic moduli are chosen to be E = 71, G = 26. For the domain W we use a triangulation T of the square W = [ 1, 1] 2 with mesh size h T ⇠ 10 3 . The parameter q 0 from equation (4) is set to q 0 = e.

3/9 Appendix C: Model plants
Different cross-sections of the three model plants (Musa acuminata, Condylocarpon guianense, Bauhinia guianensis) with labelling of the mechanically important tissues. The three cross-sections of Musa acuminata represent different parts of a leaf petiole ( Figure C1). For Condylocarpon guianense ( Figure C2) and Bauhinia guianensis ( Figure C3) cross-sections of the stems originate from different ontogenetic stages of each plant.   Cross-section of an adult stem in which a remnant from the young stage can be found in the centre. In addition, a huge amount of less-dense wood comprising wide-diameter vessels and broad wood rays (ww) only forms on two opposite sides of the young circular stem resulting in an elliptical shape with a wide groove.

Appendix D: Pseudo-time evolution of the different phase field simulations
We show the different shapes of the phase field at selected time steps. The evolution with respect to pseudo-time t of the gradient flow for minimising torsional rigidity and maximising minimum flexural rigidity D min is displayed in Figure D2. The evolution of the gradient flow for minimising merely torsional rigidity D z is displayed in Figure D3. The evolution of the gradient flow for minimising torsional and minimal flexural rigidity is shown in Figure D4. The same evolution along the graph of the twist-to-bend ratio is depicted in Figure D1. Figure D1. Evolution of the shape of the phase field in terms of minimising torsional rigidity and minimal flexural rigidity (s 1 = s 2 = 1, s 3 = 0) with respect to the twist-to-bend ratio D max /D z and pseudo-time t.