The question of how ribonucleic acid (RNA) molecules self-assemble into unique three-dimensional folds has direct implications for the biology of modern cells, for the physics of structure formation, and for the origin of life. The work presented in this thesis focuses on two universal but enigmatic features of RNA chains that distinguish them from proteins, the better-studied structured biopolymer. First, all nucleic acids are surrounded by a diffusely bound "atmosphere" of counterions that help neutralize the backbone's massive negative charge and strongly influence folding. The shape and energetic consequences of this usually hidden atmosphere have been made visible through solution x-ray scattering, chemical structure mapping, and precise ion-counting methods on DNA model systems and on more complex RNA folds, including the P4--P6 domain of the Tetrahymena ribozyme. The observed experimental results disfavor the existence of significant attractive forces between RNA helices and exhibit agreement with the simplest theoretical picture of the ion atmosphere, the nonlinear Poisson-Boltzmann mean-field theory. Second, the native and many alternative folds of the vast majority of known RNA molecules have not been amenable to standard structural techniques like x-ray crystallography. A new method, based on Multiplexed •OH Cleavage Analysis (MOHCA) read out by two-dimensional gel electrophoresis has been developed to rapidly determine RNA backbone folds under any solution condition at nucleotide resolution. These efforts to visualize the counterion atmosphere and the solution folds of structured RNA are fundamental first steps towards a fully predictive theory of how RNA self-assembles.
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