Dependence of RNA Tertiary Structural Stability on Mg²⁺ Concentration: Interpretation of the Hill Equation and Coefficient

The Mg(2+)-induced folding of RNA tertiary structures is readily observed via titrations of RNA with MgCl(2). Such titrations are commonly analyzed using a site binding formalism that includes a parameter, the Hill coefficient n, which is sometimes deemed the number of Mg(2+) ions bound by the native RNA at specific sites. However, the long-range nature of electrostatic interactions allows ions some distance from the RNA to stabilize an RNA structure. A complete description of all interactions taking place between Mg(2+) and an RNA uses a preferential interaction coefficient, Gamma(2+), which represents the "excess" Mg(2+) neutralizing the RNA charge. The difference between Gamma(2+) for the native and unfolded RNA forms (DeltaGamma(2+)) is the number of Mg(2+) ions "taken up" by an RNA upon folding. Here we determine the conditions under which the Hill coefficient n can be equated to the ion uptake DeltaGamma(2+) and find that two approximations are necessary: (i) the Mg(2+) activity coefficient is independent of concentration during a titration, and (ii) the dependence of DeltaGamma(2+) on Mg(2+) concentration is weak. Titration experiments with a Mg(2+)-binding dye and an adenine-binding riboswitch were designed to test these approximations. Inclusion of a 30-fold excess of KCl over MgCl(2) was sufficient to maintain a constant Mg(2+) activity coefficient. We also observed that Mg(2+) uptake by the RNA varied from near zero to approximately 2.6 as the Mg(2+) concentration increases over an approximately 100-fold range. It is possible to determine DeltaGamma(2+) from Mg(2+)-RNA titrations, but the values are only applicable to a limited range of solution conditions.