For many years, researchers have studied the striking dependence of RNA tertiary structure stability on Mg2+ ions (1-3). The Schimmel laboratory was one of the first to consider the effects of Mg2+ on RNA folding reactions using equations originally derived to describe ligands binding to a fixed number of specific sites on a multisubunit protein (i.e., the Hill equation) (4). Since then, it has become customary to think about ion-RNA interactions as binding events characterized by equilibrium constants and fixed stoichiometries—an approach we refer to as the ‘binding formalism’. In particular, the Hill equation and Hill coefficient (5) have become a standard means for characterizing Mg2+-induced RNA folding reactions and are often interpreted in terms of the binding formalism. But Schimmel himself recognized that “these equations are, essentially, semiempirical and as such provide little insight into the actual mechanism of the binding equilibria. Nevertheless, in order to catalog the information and to compare results of various investigations, they provide a useful common framework” (6). The question remains as to whether the adjustable variables of the Hill equation have meaningful molecular interpretations or should be considered simply empirical parameters.
A thorough consideration of ion effects on RNA folding must take into account the fact that ions interacting with an RNA can experience a variety of different environments, ranging from partially dehydrated ions essentially buried within the RNA to fully hydrated ions some distance from the RNA surface (7). The binding formalism presupposes a model that excludes the possibility of ions interacting via long-range electrostatic interactions, and therefore does not provide a complete description of Mg2+-RNA interactions. Consequently it has been useful to develop a more general formalism for addressing the effects of ions on RNA stability—one that does not specify any particular model of ion-RNA interactions.
A general approach for describing interactions between ions and macromolecules is based on parameters known as preferential interaction coefficients (8). We previously extended this formalism to address the effect of Mg2+ on RNA folding reactions, and derived an equation that simplifies to the form of the Hill equation when two approximations are made (9, 10). Where the approximations are valid, the Hill coefficient n quantifies the ‘uptake’ of Mg2+ ions that accompany a folding reaction, but has a different molecular interpretation than that attributed to n by the binding formalism.
In the present paper, we experimentally test the two approximations necessary for interpretation of the Hill coefficient. In the first approximation, Mg2+ concentrations are substituted for thermodynamic activities. Because of the strong interactions taking place between ions (e.g. Mg2+ and Cl−), the effective concentration of an ion—known as its activity—is usually very different from its concentration. We present experiments showing how the inclusion of a monovalent salt (such as KCl) can suppress the errors that can arise when Mg2+ concentrations are used in the Hill equation. The second approximation is that the Hill coefficient is a constant, independent of the concentration of Mg2+ present in solution. Using two independent methods for measuring the ion uptake that accompanies folding of a riboswitch RNA, we find that it varies from nearly zero to a maximum of ~2.6 ions per RNA as the Mg2+ concentration required to fold the RNA increases. This strong Mg2+-dependence of the Hill coefficient restricts use of the Hill equation to analysis of folding data over a narrow Mg2+ concentration range, and prevents extrapolation of the derived coefficient to other Mg2+ concentrations.