Hyphal network whole field imaging allows for accurate estimation of anastomosis rates and branching dynamics of the filamentous fungus Podospora anserina.

The success of filamentous fungi in colonizing most natural environments can be largely attributed to their ability to form an expanding interconnected network, the mycelium, or thallus, constituted by a collection of hyphal apexes in motion producing hyphae and subject to branching and fusion. In this work, we characterize the hyphal network expansion and the structure of the fungus Podospora anserina under controlled culture conditions. To this end, temporal series of pictures of the network dynamics are produced, starting from germinating ascospores and ending when the network reaches a few centimeters width, with a typical image resolution of several micrometers. The completely automated image reconstruction steps allow an easy post-processing and a quantitative analysis of the dynamics. The main features of the evolution of the hyphal network, such as the total length L of the mycelium, the number of "nodes" (or crossing points) N and the number of apexes A, can then be precisely quantified. Beyond these main features, the determination of the distribution of the intra-thallus surfaces (Si) and the statistical analysis of some local measures of N, A and L give new insights on the dynamics of expanding fungal networks. Based on these results, we now aim at developing robust and versatile discrete/continuous mathematical models to further understand the key mechanisms driving the development of the fungus thallus.

Curvature, metric and parametrization of origami tessellations: theory and application to the eggbox pattern.

Origami tessellations are particular textured morphing shell structures. Their unique folding and unfolding mechanisms on a local scale aggregate and bring on large changes in shape, curvature and elongation on a global scale. The existence of these global deformation modes allows for origami tessellations to fit non-trivial surfaces thus inspiring applications across a wide range of domains including structural engineering, architectural design and aerospace engineering. The present paper suggests a homogenization-type two-scale asymptotic method which, combined with standard tools from differential geometry of surfaces, yields a macroscopic continuous characterization of the global deformation modes of origami tessellations and other similar periodic pin-jointed trusses. The outcome of the method is a set of nonlinear differential equations governing the parametrization, metric and curvature of surfaces that the initially discrete structure can fit. The theory is presented through a case study of a fairly generic example: the eggbox pattern. The proposed continuous model predicts correctly the existence of various fittings that are subsequently constructed and illustrated.