Phyllotaxis: a framework for foam topological evolution

Aperiodic crystals in biology

Biological systems display a broad palette of hierarchically ordered designs spanning over many orders of magnitude in size. Remarkably enough, periodic order, which profusely shows up in non-living ordered compounds, plays a quite subsidiary role in most biological structures, which can be appropriately described in terms of the more general aperiodic crystal notion instead. In this topical review I shall illustrate this issue by considering several representative examples, including botanical phyllotaxis, the geometry of cell patterns in tissues, the morphology of sea urchins, or the symmetry principles underlying virus architectures. In doing so, we will realize that albeit the currently adopted quasicrystal notion is not general enough to properly account for the rich structural features one usually finds in biological arrangements of matter, several mathematical tools and fundamental notions belonging to the aperiodic crystals science toolkit can provide a useful modeling framework to this end.

Transitions between phyllotactic lattice states in curved geometries

Phyllotaxis, the regular arrangement of leaves or other lateral organs in plants including pineapples, sunflowers and some cacti, has attracted scientific interest for centuries. More recently there has been interest in phyllotaxis within physical systems, especially for cylindrical geometry. In this letter, we expand from a cylindrical geometry and investigate transitions between phyllotactic states of soft vortex matter confined to a conical frustum. We show that the ground states of this system are consistent with previous results for cylindrical confinement and discuss the resulting defect structures at the transitions. We then eliminate these defects from the system by introducing a density gradient to create a configuration in a single state. The nature of the density gradient limits this approach to a small parameter range on the conical system. We therefore seek a new surface, the horn, for which a defect-free state can be maintained for a larger range of parameters.

Ellipse packing in two-dimensional cell tessellation: a theoretical explanation for Lewis's law and Aboav-Weaire's law

Background

Lewis’s law and Aboav-Weaire’s law are two fundamental laws used to describe the topology of two-dimensional (2D) structures; however, their theoretical bases remain unclear.

Methods

We used R software with the Conicfit package to fit ellipses based on the geometric parameters of polygonal cells of ten different kinds of natural and artificial 2D structures.

Results

Our results indicated that the cells could be classified as an ellipse’s inscribed polygon (EIP) and that they tended to form the ellipse’s maximal inscribed polygon (EMIP). This phenomenon was named as ellipse packing. On the basis of the number of cell edges, cell area, and semi-axes of fitted ellipses, we derived and verified new relations of Lewis’s law and Aboav-Weaire’s law.

Conclusions

Ellipse packing is a short-range order that places restrictions on the cell topology and growth pattern. Lewis’s law and Aboav-Weaire’s law mainly reflect the effect of deformation from circle to ellipse on cell area and the edge number of neighboring cells, respectively. The results of this study could be used to simulate the dynamics of cell topology during growth.

Type-I collagen fibrils: From growth morphology to local order

The length of type-I collagen fibrils in solution increases through the development and progress of pointed tips appearing successively at the two ends of an axis-symmetric shaft with constant diameter. Those tips, respectively fine ([Formula: see text]) or coarse ([Formula: see text]) have opposite molecular orientations. The [Formula: see text]-pointed tips, the first to appear, are particularly remarkable as they all show, on most of their length, a common parabolic profile which stays constant during the growth. Assuming that the latter occurs by lateral accretion of individual molecules in staggered configuration, we propose to give account of this prominent morphological feature along a purely geometrical argument, the profile of a tip being linked to the shape of the trajectories followed all along the accretion process. Among several possible trajectories, Fermat spirals lead to a parabolic profile in perfect agreement with the one observed for [Formula: see text]-pointed tips. This is to be put in relation with the presence of such spirals in phyllotactic patterns which ensure the best packing efficiency in cases of axis-symmetry, which is indeed that of dense collagen fibrils. Moreover, those patterns are structured by concentric circles of dislocations, constitutive of the structure itself, whose behaviour might contribute to the mechanical properties of the fibrils.

Curvature-driven spatial patterns in growing 3D domains: A mechanochemical model for phyllotaxis

Here we discuss the formation of phyllotactic patterns in the shoot apical meristem (SAM) of plants, where the spatial distribution of the phytohormone auxin determines phyllotaxis in a domain that is growing and changing in time. We assume that the concentration of auxin modifies the mechanical properties of the domain and that the mechanical stress field in the SAM orients the flux of auxin. To study this problem we propose a mechanism for pattern formation in growing domains with variable curvature. The dynamics of chemicals is modeled by a reaction-diffusion system that produces a three dimensional pattern of chemical concentrations that changes the stress field in the domain while growing. The growth process is modeled by a phase-field order parameter which determines the location of the boundaries of the domain. This field is coupled to the chemical concentration through a curvature term that affects the local mechanical stress in the domain. The local stress changes in turn modify the chemical patterns. Our model constitutes a useful and novel approach in theoretical biology, as many developmental processes in organisms seem to be affected by the changes of curvature, size, mechanical stress and other physical aspects. Several patterns seen in many plants are reproduced under certain conditions by our model.

Phyllotaxis, disk packing, and Fibonacci numbers

We consider the evolution of the packing of disks (representing the position of buds) that are introduced at the top of a surface which has the form of a growing stem. They migrate downwards, while conforming to three principles, applied locally: dense packing, homogeneity, and continuity. We show that spiral structures characterized by the widely observed Fibonacci sequence (1, 1, 2, 3, 5, 8, 13, ...), as well as related structures, occur naturally under such rules. Typical results are presented in an animation.