Wood grain patterns at branch junctions: modeling and implications

Abstract

We apply a recent model of wood grain pattern formation to the junction between two tree branches. In model simulations, the export of indole-3-acetic acid (IAA) from a branch into the junction is necessary to maintain the continuity between the grain pattern of the branch and the grain of the subjacent stem. Increased IAA export corresponds to a larger effect on the overall grain pattern. Conversely, if IAA export stops, the grain pattern diverts around the branch. These results do not depend on specific values for the model parameters and appear to be quite general. Since long-range water transport is largely parallel to the wood grain, greater IAA export from a branch is expected to give improved access to the water resources of the tree. IAA export thus emerges as a likely regulator of branch vigor. In this way, the basipetal flux of IAA through a branch, and its morphological consequences at branch junctions, may play an important role in several aspects of tree form.

Appendices

Appendix 1. Model equations

In this appendix we discuss the differential equations used in our model of wood grain at a branch junction. We begin with Eqs. 1 and 2 from Kramer and Groves (2003) and use their notation. The model is a pair of coupled, nonlinear partial differential equations in the variables m and u, where m is the local mass of IAA per unit area of cambial surface and u is the local grain direction. The equations as cited have been transformed to the dimensionless variables ${\mathbf{{r}'}} = {\mathbf{r}}v/D_{\parallel }$, ${t}' = tK(v/D_{\parallel } )^{2}$, and ${m}' = m/\langle m\rangle$, where 〈m〉 is a characteristic IAA concentration (we use the value m=m ₁ imposed at the top of the simulation domain). Since we are seeking time-independent solutions, we set ∂m/∂t=0 and ∂u/∂t=0. Dropping all primes gives:

$$\nabla\cdot{\left[{(\nabla_{{||}}m-m){\mathbf{\hat{u}}}+(r_{D}\nabla_{\bot}m){\mathbf{\hat{w}}}}\right]}=0$$

(1)

$$ \nabla ^{2} {\mathbf{u}} + \frac{2} {{L^{2} }}{\mathbf{u}}(1 - u^{2} ) + \frac{1} {\lambda }{\left[ { - (\nabla m){\left( {\frac{{3 - u^{2} }} {2}} \right)} + ({\mathbf{u}} \cdot \nabla m){\mathbf{u}}} \right]} = 0 $$

(2)

Following Kramer and Groves (2003), we set r _D =0.23 and L=0.5 for all the simulations reported in this paper. This leaves just one dimensionless parameter of interest, $\lambda = Kv/(\mu D_{\parallel } m_{1} )$.

Appendix 2. Boundary conditions

Separate from the model equations that dictate the solution in the interior of the simulation domain (see Appendix 1), one also needs to specify the shape of the domain and the boundary conditions on m and u. As shown in Fig. 4, we use a square simulation domain of side S, with an disk of radius R excised from the center.

Fluxes  There is a uniform flux of IAA into the domain at the top edge of the square and a second flux into the domain at the edge of the disk. These two fluxes represent the IAA supplied by the two distal branches. There is an IAA sink along the bottom edge that allows IAA to drain out of the domain via active transport. Side-to-side the domain has periodic boundary conditions, which is to say that IAA exiting the domain along the right edge immediately re-enters on the left and vice versa. This is not a necessity, but it is a useful means to enforce IAA conservation.

To set the apical flux, we impose the conditions m=m ₁ and ∂m/∂y=0. The net flux into the domain through the top edge is thus Φ ₁=m ₁ vS. Implementing the flux at the circular boundary was nontrivial. Rather than imposing a condition at the boundary per se, we include the disk as part of the simulation domain. At each time step, every lattice site in the disk is given a small additional mass of IAA, Δm. The direction of active transport at all lattice sites in the disk is chosen to be radially outward, so that all mass deposited in the disk eventually exits the disk. The value of Δm is chosen so the net flux out of the disk is Φ ₂.

Grain  The grain direction along the top and side edges of the square domain is chosen to point downward (u=−e _y). We impose the Neumann condition ∂u/∂y=0 at the bottom edge. The condition on u at the boundary of the disk is again problematic. Since we want the flux of IAA to point radially outward, we originally chose u=e _r on the disk boundary. While this works fine for λ<1, it causes problems when the grain stiffness is too large. In particular, it leads to an anomalously large region of approximately radial grain outside the disk. The fix for this problem was to recognize that the direction of IAA active transport in the disk could be uncoupled from the value for u as it appears in the calculation of the Laplacian (see Eq. 2, Appendix 1). We then set u=0 at the boundary of the disk.