Ion-RNA interactions thermodynamic analysis of the effects of mono- and divalent ions on RNA conformational equilibria

Abstract

RNA secondary and tertiary structures are strongly stabilized by added salts, and a quantitative thermodynamic analysis of the relevant ion-RNA interactions is an important aspect of the RNA folding problem. Because of long-range electrostatic forces, an RNA perturbs the distribution of both cations and anions throughout a large volume. Binding formalisms that require a distinction between "bound" and "free" ions become problematic in such situations. A more fundamental thermodynamic framework is developed here, based on preferential interaction coefficients; linkage equations derived from this framework provide a model-free description of the "uptake" or "release" of cations and anions that accompany an RNA conformational transition. Formulas appropriate for analyzing the dependence of RNA stability on either mono- or divalent salt concentration are presented and their application to experimental data is illustrated. Two example datasets are analyzed with respect to the monovalent salt dependence of tertiary structure formation in different RNAs, and three different experimental methods for quantitating the "uptake" of Mg(2+) ions are applied to the folding of a riboswitch RNA. Advantages and limitations of each method are discussed.

Figure 1

Ion - RNA interaction coefficients defined with respect to equilibrium dialysis. A, Cartoon of a dialysis experiment. An RNA with 16 negative charges (red), together with 16 K⁺ ions, is added (gray box) to a solution of KCl previously in equilibrium across a dialysis membrane (blue, K⁺; yellow, Cl⁻; different shades of blue are used to distinguish K⁺ contributed as a counterion to RNA (dark blue) or KCl (light blue).) In the approach to equilibrium (arrow), two K⁺ and two Cl⁻ ions migrate from left to right across the dialysis membrane. (The right side dialysis chamber is presumed to be very large compared to the left side, such that the ion migration does not appreciably change the right side salt concentration.) Calculations of the single ion interaction coefficients (Γ₊, Γ₋) are shown as the differences between the left and right side ion concentrations (Eq. 3). B, Histogram illustrating charge balance in the equilibrium dialysis experiment cartooned in panel A. The vertical heights of the colored bars represent the total concentration of the various ions. Color coding of mobile ions and RNA phosphate is as in panel A. The left-most pair of bars (L_init) represents the addition of the potassium salt of an RNA with Z negative charges (K_Z·RNA) to the left dialysis chamber. The middle pair of bars (marked L_eq) represents the ion concentrations in the left chamber after dialysis equilibrium has been achieved with the concentrations of salts indicated in the right dialysis chamber (marked R_eq). The relations between the single ion coefficients (Γ₊, Γ₋) and histogram heights are marked, and the relationships between the interaction coefficients and the number of RNA charges (Z) are listed.

Figure 2

Monovalent salt dependences for the formation of RNA tertiary structures. For both RNAs shown, log(K_obs) was calculated at 20 °C from the melting temperature (T_m) and enthalpy (ΔH°) of the tertiary folding transition observed in melting curves by the formula ln(K_obs) = −(ΔH°/R)(1/T_m − 1/T₀), where R is the gas constant and T₀ is 293 K. A, log(K_obs) for folding of the A-riboswitch (buffer: 4 μM 2,6-diaminopurine, 20 mM MOPS adjusted to pH 6.8 with KOH, 0.1 mM EDTA, and various KCl concentrations). Data are taken from (Lambert, et al., 2009). Each point is the average of three experiments; error bars are smaller than the data points. The salt molality and mean ionic activity are calculated from the total K⁺ concentration contributed by both KCl and MOPS buffer. B, log(K_obs) for formation of the complex between tar and tar* hairpins (buffer: 5 mM cacodylic acid adjusted to pH 6.4 with KOH, 0.1 mM EDTA, and various KCl concentrations). Data are taken from (Lambert, et al., 2009). Error bars are standard deviations from the average of three experiments.

Figure 3

Definition of ion - RNA interaction coefficients in mixed monovalent-divalent salt solutions. K⁺ and Cl⁻ ions are identified by the same colors as in Figure 1; green circles represent Mg²⁺. A, migration of ions taking place after addition of an RNA and its neutralizing K⁺ ions (in gray rectangle) to a mixture of KCl and MgCl₂ previously in equilibrium across a dialysis membrane. As equilibrium is re-established (arrow), there is a net migration of K⁺ into the right chamber and a net migration of Mg²⁺ into the left chamber. The movement of each cation type is accompanied by a neutralizing number of anions; in this example, there is a net flow of two Cl⁻ out of the left chamber. (As in Figure 1, the right side dialysis chamber is presumed to be very large compared to the left side, such that the ion migration does not appreciably change the right side salt concentration.) Calculations of all three single ion interaction coefficients (Γ₊, Γ₋, Γ₂₊) are shown as the differences between the left and right side ion concentrations (Eq. 3 and 24). B, histogram illustrating charge balance in the equilibrium dialysis experiment cartooned in panel A. The vertical heights of the colored bars represent the total concentration of charge contributed by various ions, with the same color coding as in panel A. (The Mg²⁺ ion concentration is half the charge concentration.) The pair of bars labeled L_init represents the combination of salts added to the “left” chamber of a dialysis apparatus: the potassium salt of an RNA with Z negative charges (K_Z·RNA), KCl, and MgCl₂. The next pair of bars (marked L_eq) represents the ion charge concentrations in the left chamber after dialysis equilibrium has been achieved with the concentrations of salts indicated in the right dialysis chamber (marked R_eq). The L_eq and R_eq concentrations of the three ions are individually compared on the right side of the diagram, and relationships between the single ion coefficients, neutral salt coefficients, and the number of RNA charges are listed.

Figure 4

Three different measurements of ΔΓ₂₊ for the A-riboswitch RNA, all reported under the same temperature and ionic conditions (50 mM KCl, 20 mM K-MOPS pH 6.8, 20 °C). A, isothermal titration of A-riboswitch RNA (3 μM) with buffer containing MgCl₂. 11 μM adenine was also present. Absorbances at 260 and 295 nm were collected after each addition. After correction of the absorbance for dilution, the data were normalized to the initial reading before MgCl₂ addition. The curves are least squares fits of Eqs. (31) and (34), with the assumption that m_U = m_F. B, ln(K_obs) was calculated from the T_ms and ΔH° (average value, 56.3 ±2.4 kcal/mol) obtained from melting experiments with A-riboswitch RNA in buffer containing 11 μM 2,6-diaminopurine and various concentrations of MgCl₂. The curve is a least squares fit of Eq. (37). Errors in ln(K_obs) are principally the uncertainty in ΔH°. C, direct measurement of Γ₂₊ by titration with MgCl₂ in the presence of a fluorescent dye that senses Mg²⁺ activity. The ‘+DAP’ titration contained 250 μM 2,6-diaminopurine. The points and error bars are the averages and standard deviations of five different titrations for each data set. D, comparison of ΔΓ₂₊ obtained by three different methods. Data points with black or gray centers are from isothermal titrations observed at 260 and 295 nm, respectively, in which different ligands (purine, adenine, 2-aminopurine, or 2,6-diaminopurine) were used to vary the midpoint of the folding transition. The fitted midpoint of the folding transition, [Mg²⁺]₀ in Eq. (34), is plotted on the x-axis. Errors on both axes are standard deviations from three independent experiments. The solid curve is the calculated slope of the fitted line in panel B; the single error bar is the uncertainty in the slope at the inflection point. Solid gray data points are the difference between the two curves in panel C, with the corresponding cumulative error. Data in panels A, C, and D were taken from (Leipply and Draper, 2009).

Figure 5

Potential errors introduced by the assumption that ΔΓ₂₊ is constant in the derivation of the linkage Eq. (30). The dependence of ΔΓ₂₊ on [Mg²⁺] was modeled by a polynomial fit to the solid gray data points in Figure 4D. The polynomial was used in the integration of Eq. (30) to give expressions for lnK_obs and θ with [Mg²⁺]-dependent ΔΓ₂₊ (in contrast to Eqs. 33 and 34, which assume ΔΓ₂₊ is constant). θ is plotted for the calculated titration curve when the midpoint of the titration, [Mg²⁺]₀, is 10 μM (circles), 30 μM (squares),or 100 μM (diamonds). ΔΓ₂₊ (as used to calculate the displayed curves) at the titration midpoints (θ = 0.5) is 1.45, 2.30, and 2.73, respectively. The simulated data points have been fit to either a modified version of Eq. (34) that assumes the y-intercept of the curve has the value θ = 0, $$\theta ={\theta }_0+(1-{\theta }_0){([{Mg}^{2+}]/{[{Mg}^{2+}]}_0)}^{\text{n}}/[1+{([{Mg}^{2+}]/{[{Mg}^{2+}]}_0)}^{\text{n}}],$$, or to an equation that allows a non-zero y-intercept, Eq. (35). The residuals of the fits are shown in the lower three panels; closed symbols correspond to Eq. (34) and open symbols to Eq. (35). For the curve with a midpoint of [Mg²⁺]₀ = 100 μM, Eq. (35) could not be fit to the data because the value of C₀ became vanishingly small. The values of ΔΓ₂₊ at the titration curve midpoints obtained from the modified Eq. (34) are 1.99, 2.39, and 2.73, in order of increasing [Mg²⁺]₀. ΔΓ₂₊ obtained by fitting of Eq. (35) and application of Eq. (36) are 1.43 and 2.29 (10 and 30 μM transition midpoints, respectively).