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pictorum (Figure 66a) first deposits a prismatic structure, made of relatively large elongated aragonitic prisms glued together by an organic phase, followed by a highly regular nacreous structure, made of platelets (up to 10 µm in diameter) that are assembled to form layers, separated by 40-nm organic membranes [1]. The structure of early nacre exhibits numerous topological faults that resemble structural dislocations in a periodic lattice: spiral steps arising from one nacre layer to the next (inset, Figure 66a). Each defect has a specific topological sign: a right- handed and a left-handed spiral. Convolutional Neural Networks (CNNs) were trained to analyse the extremely fine nacreous ultrastructure from 3D data obtained by synchrotron-based X-ray holographic nano-tomographic imaging at beamline ID16A. The segmented data provided unprecedented information on the 3D structure of nacre and on the behaviour of these topological defects during nacre deposition (Figure 66b) [2].

It became evident that during growth, structural dislocations in nacre move and interact. Defects with opposite screw behaviour are attracted to each other and eventually mutually annihilate, leaving perfectly periodic

nacre behind. The interaction between dislocations can be described by a comparison to the motion of dislocations in atomic lattices or topological defects in liquid-crystals, where each defect generates an elastic deformation field around its core. This results in an attractive force between two defects with opposite screw directions located on the same gliding plane.

Furthermore, the periodic layered structure of nacre was described by an oscillatory function, which represents the local modulating concentration of the organic and the mineral phases with a period that corresponds to the thickness of nacre platelets. The early disordered nacre can be represented by an array of analogous out-of-phase oscillators, which are coupled by the elastic field described before. In this case, the observed structural dislocations act as dissipative topological defects and, like many other oscillatory systems, the process of synchronisation of this array can be modelled by the classical Kuramoto model [3].

Experimentally obtained coordinates and the screw directions of dislocations at the prism-to-nacre transition

Fig. 66: a) Scanning electron microscopy image of a fractured cross-section of the U. pictorum shell, showing the nacreous (top) and prismatic (bottom) ultrastructures. Inset: Segmented X-ray holographic nanotomography view of a spiral defect connecting adjacent nacre layers. b) A representative 3D segment of the prismatic-to-nacre transition in U. pictorum obtained by X-ray holographic nanotomography. Reprinted from J. Struct. Biol. 209, M. Beliaev et al., Quantification of sheet nacre morphogenesis using X-ray nanotomography and deep learning, 107432 (2019) with

permission from Elsevier.

Fig. 67: Simulated snapshots of the topography, the phase field and the elastic energy field at the front of the growing nacreous structure. Adapted by permission from Springer Nature: Nat. Phys. 17, Dynamics of topological defects and structural

synchronization in a forming periodic tissue, M. Beliaev et al., 410-415 (2021).