Statistics and fluctuations of biological and nano-scale non-Euclidean sheets

Non uniform growth, or the assembly of incompatible building blocks can lead to the formation of solid sheets with intrinsic non-Euclidean geometry. Such sheets often form elaborate three-dimensional configurations and have unusual mechanical properties. I will address two cases, in which the mechanics and geometry of non-Euclidean sheets are coupled to stochastic processes: 

1. A growing leaf can be viewed as a thin sheet of an active solid. Its effective rheology is nontrivial, allowing the leaf to increase its area by orders of magnitude, keeping its "proper" geometry. In attempting to uncover the this rheology, we find that leaf growth is a highly fluctuating process in both time and space, and that it is affected by mechanical stress.


2. On the nano-scale, we study self assembled ribbons, made of lipids and peptides with chiral head groups. We find that the average ribbon’s configurations are well described by the elastic theory of non-Euclidean sheets. We then combine elasticity with statistical mechanics to derive predictions for various statistical properties of ribbon shapes at finite temperature. Some of the predictions are quantitatively verified by analyzing experimental data.