Pattern formation in bubbles emerging periodically from a liquid free surface

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.

About collagen, a tribute to Yves Bouligand

Yves Bouligand's analysis of the organizations of biological materials in relation to those of liquid crystals enabled the development of the idea that physical forces exerting their actions under strong spatial constraints determine the structures and morphologies of these materials. The different levels of organization in collagen have preoccupied him for a long time. We present here our recent works in this domain that we were still discussing with him a few months before his death at the age of 76 on 21 January 2011. After recalling the hierarchical set of structures built by collagen molecules, we analyse them, exploiting the properties of the curved space of the hypersphere and of the algorithm of phyllotaxis. Those two geometrical concepts can be proposed as structural archetypes founding the polymorphism of this complex material of biological origin.

Noise and robustness in phyllotaxis

A striking feature of vascular plants is the regular arrangement of lateral organs on the stem, known as phyllotaxis. The most common phyllotactic patterns can be described using spirals, numbers from the Fibonacci sequence and the golden angle. This rich mathematical structure, along with the experimental reproduction of phyllotactic spirals in physical systems, has led to a view of phyllotaxis focusing on regularity. However all organisms are affected by natural stochastic variability, raising questions about the effect of this variability on phyllotaxis and the achievement of such regular patterns. Here we address these questions theoretically using a dynamical system of interacting sources of inhibitory field. Previous work has shown that phyllotaxis can emerge deterministically from the self-organization of such sources and that inhibition is primarily mediated by the depletion of the plant hormone auxin through polarized transport. We incorporated stochasticity in the model and found three main classes of defects in spiral phyllotaxis--the reversal of the handedness of spirals, the concomitant initiation of organs and the occurrence of distichous angles--and we investigated whether a secondary inhibitory field filters out defects. Our results are consistent with available experimental data and yield a prediction of the main source of stochasticity during organogenesis. Our model can be related to cellular parameters and thus provides a framework for the analysis of phyllotactic mutants at both cellular and tissular levels. We propose that secondary fields associated with organogenesis, such as other biochemical signals or mechanical forces, are important for the robustness of phyllotaxis. More generally, our work sheds light on how a target pattern can be achieved within a noisy background.

A PHYLLOTACTIC APPROACH TO THE STRUCTURE OF COLLAGEN FIBRILS

Collagen fibrils, cable-like assemblies of long biological molecules, the so-called triple helices, are dominant components of connective tissues. Their determinant morphological and functional roles motivated a large number of studies concerning their formation and structure. However, these two points are still open questions and, particularly, that of the lateral assembly of the triple helices which is certainly dense but not strictly that of a well-ordered molecular crystal. We examine here the geometrical template provided by the algorithm of phyllotaxis which gives to each element of an assembly of points or parallel rods the most homogeneous and isotropic dense environment in a situation of cylindrical symmetry. The scattered intensity obtained from a phyllotactic distribution of triple helices in collagen fibrils presents features which could contribute to the scattering observed along the equatorial direction of their X-ray patterns. Following this approach, the aggregation of triple helices in fibrils should be considered within the frame of soft condensed matter studies rather than that of molecular crystal studies.