In this article a brief review of the theory of one-dimensional nonlinear lattice is presented. Special attension is paid for the lattice of particles with exponential interaction between nearest neighbors (the Toda lattice). The historical exposition of findings of the model system, basic equations of motion, special solutions, and the general method of solutions are given as chronologically as possible. Some reference to the Korteweg-de Vries equation is also given. The article consists of three parts. Firstly, the idea of dual system is presented. It is shown that the roles of masses and springs of a harmonic linear chain can be exchanged under certain condition without changing the eigenfrequencies. Secondly, the idea is applied to the anharmonic lattice and an integrable lattice with exponential interaction force between adjacent particles is obtained. Special solutions to the equations of motion and general method of solution are shown. In the last part, some studies on the Yang-Yang's thermodynamic formalism is given.
Particles formed by a feast/famine regulatory protein (FFRP), pot0434017 (FL11), in solution in the absence of DNA were analyzed using electron microscopy (EM). By applying conventional (i.e. dry) EM to the protein negatively stained with uranyl acetate, top views of tetrameric assemblies of dimers were obtained, where four pairs each of N-domains were extending from C-domains assembled around the centers. In cryo-EM images of the protein embedded in 3D amorphous ice, sets of four densities were arranged around ellipsoids having similar lengths for their long axes but of different lengths for their short axes. These images were interpreted as projections with different tilts of four pairs of N-domains arranged inside flat assemblies: the positively charged N-domains only were stained with ammonium molybdate, but the negatively charged C-domains were unstained and thus unobservable. Using seventeen such cryo-images, in combination with a crystal structure equivalent to an assembly of C-domains, a disk-like 3D structure was reconstructed.