Supporting Text Z = 2Γ 2+ + Γ + Γ [1]

Transcription

1 Supporting Text RNA folding experiments are typically carried out in a solution containing a mixture of monovalent and divalent ions, usually MgCl 2 and NaCl or KCl. All three species of ions, Mg, M + (Na + or K + ), and Cl, interact with the RNA. Cations are distributed among different environments on and near an RNA molecule in ways that depend on energetic factors, such as coulombic interactions of the ions with the RNA electrostatic field and the costs of partially dehydrating the ions and RNA (1). Regardless of the details of the equilibrium ion distribution, the favorable interactions result in a net accumulation of cations by the RNA in excess of the bulk concentrations of the ions.* In a similar manner, unfavorable interactions of Cl ions with the negative RNA electrostatic field cause a depletion of Cl ions near an RNA, relative to the bulk Cl solution concentration. The net accumulation of a cation or exclusion of an anion by a polynucleotide is sometimes called the ion interaction coefficient (2, 3). The principle of electroneutrality requires that cation accumulation and anion exclusion balance the total RNA charge, Z = 2Γ + Γ + Γ [1] where Z is the total (negative) charge carried by RNA phosphates and Γ is the interaction coefficient for the individual ions. (Note that anion exclusion, Γ, is negative.) Values of the three Γ parameters depend on the bulk concentrations of MgCl 2 and MCl and on the molecular details of the RNA. Virtually any RNA conformational change affects the electrostatic potential near the RNA, the positioning of electronegative groups, and other factors contributing to the energetics of ion interactions; all of these factors have the potential to alter the magnitudes of the three ion interaction terms of Eq. 1. The question of interest here is the problem of Mg -induced RNA folding, which asks to what degree the addition of MgCl 2 to a solution containing a constant amount of MCl shifts an RNA folding equilibrium,

2 I K obs n G obs = RT ln m N. [2] m I (I is partially folded RNA, typically containing only secondary structure, and N is the RNA native state conformation. m is the molality of the indicated specie.) G obs is calculated from concentrations rather than thermodynamic activities and, thus, depends on solution conditions. For the problem of Mg -induced RNA folding, we define the RNA standard state as a dilute RNA in a solution of specified ph and NaCl or KCl concentration. An intrinsic free energy of folding, independent of the concentration of MgCl 2, is then G fold = µ N µ I = RT ln γ m N N = RT ln γ N + G obs. [3] γ I m I γ I γ is the activity coefficient of the I or N species. In the absence of Mg and at RNA concentrations dilute enough to minimize RNA RNA interactions (generally the case at the 1 µm RNA concentrations used to measure G obs ), γ N and γ I are unity; as MgCl 2 is added, G obs, γ N, and γ I change in such a way that G fold remains constant. From the perspective of Eq. 3, MgCl 2 affects the observed RNA-folding free energy by altering the activity coefficients [or, from µ = µ + RTln(γm), the chemical potentials] of the I and N RNA conformations. This formulation of RNA folding thus distinguishes an intrinsic folding free energy, G fold, from the free energies of MgCl 2 RNA interactions as reflected in RNA chemical potentials. The ion interaction coefficients of Eq. 1 and G obs are all affected by the addition of MgCl 2 to an RNA solution and are therefore related quantities. In this supporting information, we derive the relationship between RNA chemical potentials or G obs (Eq. 3) and the Mg interaction coefficient. We further show how titrations in which an indicator dye is used to monitor the effective concentration of Mg in the presence of

3 RNA yield the required ion interaction coefficient. The derivation closely follows the work of Anderson and Record (), who considered the effects of monovalent salts on polynucleotide equilibria. In essence, we have extended their analysis to include MgCl 2 as a fourth solution component. Relation Between Free Energies of Ion RNA Interactions and Γ. We first define the three electroneutral components present in solution and their molalities as RNA M + m 2 MCl m 3 MgCl 2 m. (Following a standard nomenclature for solutions, component 1 is water, present in constant concentration when molal units are used.) We wish to find changes in µ 2 upon addition of m while keeping the other components constant: dµ 2 = µ 2 dm. [] m m 2,m 3 Temperature and pressure are assumed constant in this and all following partial derivatives. To put the partial derivative in Eq. in terms of the chemical potential of MgCl 2, µ, rather than that of the RNA, µ 2, the Euler reciprocity relation µ 2 = µ [5] m m m 2,m 3 2 m 3,m is obtained from the fundamental equation for dg at constant temperature and pressure, dg =µ 2 dm 2 +µ 3 dm 3 +µ dm [6] with m 3 held constant. Application of the Euler chain relation to the right hand side of Eq. 5 gives µ 2 = µ m [7] m m m 2,m 3 m m 2,m 3 2 µ,m 3

4 The rightmost partial derivative is the definition of the interaction coefficient for MgCl 2. In the nomenclature introduced by Record s group (3), it would be called Γ µ. It refers to the electroneutral species MgCl 2, in contrast to the single ion interaction terms defined in Eq. 1. The relation between Γ µ and Γ is discussed below. The other derivative on the right-hand side tells how the activity of MgCl 2 varies with the total concentration of the salt. It can be evaluated by first considering a solution with only MCl and MgCl 2. The activity of MgCl 2 in such a solution is defined as a = ( a )a () 2 = γ m ( 2m + m 3 ) 2, [8] where a is the activity of Mg ion and a the activity of Cl. Taking the logarithm of both sides of Eq. 8 and differentiating with respect to ln(m ) gives ln a = lnγ m +1+ = m µ. [9] ln m lnm m 3 2m m 3 + m 3 RT m m 3 The term containing m 3 and m varies between 2 and 0, depending on the ratio of the two salt concentrations. Under the conditions explored in this work, m 3 exceeds m by a factor of at least 60 and by factors of 270 or 600 for the free-energy changes summarized in Figs. 1 and 5, respectively. Thus the term is neglected here. The dependence of γ on m is also small. As derived from a theory of mixed electrolytes, lnγ 3 ln m m B 23, [10] subject to the condition that the contribution of m to the solution ionic strength is small compared to that of m 3 (compare equation from ref. 5). B 23 reflects the strength of the interactions between Mg and Cl ions and is calculated from the experimental activity coefficient of a solution of MgCl 2 having the same ionic strength as the solution

5 of interest. For m 3 = 0.3 m, much larger than considered here or in most literature experiments examining Mg -induced RNA folding, we calculate ( lnγ / lnm ) m (0.685) from literature data (6). Because m is usually <0.01 m in such experiments, this derivative is negligible. Thus, in excess MCl over MgCl 2, the approximation µ m m 3 RT m [11] is applicable. (The excellent fits of Mg HQS titration data to single site-binding isotherms when MgCl 2 concentrations are used without activity corrections, as in Fig. 2, is confirmation that the Mg activity coefficient does not change with MgCl 2 concentration under the conditions used here.) Lastly, we note that the concentration of Mg ion (m ) is identical to the concentration of the electroneutral specie MgCl 2, m, for the four-component system defined here. Therefore Γ µ is identical to the single-ion coefficient Γ used in Eq. 1, because m may be substituted for m in the partial derivative defining Γ µ (Eq. 7). This identity does not hold for three component systems in which the same cation is added with both the nucleic acid and the electroneutral salt; in those cases, Γ µ3 is directly related to the anion coefficient, Γ (compare equation 13 and its discussion in ref. 7). Combining this last identity with Eqs., 7, and 11, we obtain the desired expression for the free energy of Mg interaction with RNA in a solution with excess MCl: m G RNA,Mg = µ 2,Mg µ 2,0 RT Γ d ln m. [12] 0 RNA chemical potentials in the presence (µ 2,Mg ) or absence (µ 2,0 ) of added MgCl 2 are specified. The ion interaction coefficient, Γ, is used here as a measure of all interactions taking place between added Mg and an RNA. Because additional complicating interactions, for instance between RNAs, can take place at high RNA concentrations, it is

6 necessary that Γ be evaluated in the limit of infinite RNA dilution (7). Under such conditions, m takes on the value of the bulk Mg concentration. The changing interactions taking place between three small ions (Mg, M +, and Cl ) and an RNA when two of the species are varied (i.e., titration by MgCl 2 ) are so complex that it would not be possible to evaluate G RNA-Mg in any straightforward way if not for the approximations made possible by the presence of a large excess of monovalent salt (MCl) over added MgCl 2. The large amount of Cl contributed by MCl means that (i) the Cl concentration remains approximately constant when MgCl 2 is added, and (ii) the Mg ion activity coefficient remains approximately constant during the titration. In effect, the Mg concentration becomes isolated as the only significantly changing variable during titration. It is sometimes assumed that RNA folding experiments carried out at extremely low monovalent salt concentrations in some way simplify the system so that only the effects of Mg on the RNA are being observed. To the contrary, the preceding analysis shows that it is only in the presence of excess MCl that the effects of Mg on RNA folding may be quantitatively evaluated. Experimental Measurement of Γ. We next show how Γ and m of Eq. 12 can be related to experimental measurements discussed in the accompanying paper. In the experiment, two identical solutions (reference and sample) of buffered MCl are prepared, RNA equilibrated with the same buffered MCl is added to the sample solution to a molar concentration C RNA, and then the solutions are titrated with MgCl 2 while keeping the M + concentration constant. In effect, the molar amounts of MgCl 2 added to the solutions, C Mg,sample and C Mg,ref, are adjusted so that the same Mg activity, as sensed by a fluorescent chelator, is maintained in reference and sample solutions during the titration. The sample solution will have an excess of added MgCl 2, from which the quantity C RNA Mg = C Mg,sample C Mg,ref C RNA = C Mg [13] C RNA is calculated. This equation is similar to the definition of Γ as

7 lim m 2 0 m Mg m 2 = µ,m 3 m Mg m 2 Γ, [1] µ,m 3 where m Mg is the difference between the Mg concentrations in two solutions having the same MgCl 2 activity but only one containing RNA (2, 7). Several approximations are necessary to equate the right-hand side of Eq. 13 with the left-hand side of Eq. 1. First, molar and molal concentrations are equivalent for the relatively dilute solutions used here; thus, C RNA m 2, etc. Second, the imposition of constant MgCl 2 chemical potential (µ ) in Eq. 1 implies that µ must be identical between the sample and reference cuvettes in the experiment. This is not strictly true; the two solutions are adjusted to the same Mg ion activity (a ), as detected by the fluorescent dye, but the Cl accompanying the excess Mg in the sample cuvette may cause the MgCl 2 activities (a ) to differ (see Eq. 8). But because there is a large excess of MCl over Mg (m 3 >> m ) in these experiments, the fractional difference in Cl activities between the solutions will be small. (The same consideration as to the necessary excess of m 3 over m for acceptable error levels in Eq. 9 also applies here.) Lastly, the experiment must be conducted at low enough RNA concentration for the limit in Eq. 1 to apply. Potential errors at high RNA concentrations include unwanted contributions to C RNA Mg from RNA RNA interactions and large differences in Cl concentration between sample and reference solutions at constant a. We have not detected any systematic dependence of C RNA Mg measured over RNA concentration ranges as large as mm in nucleotides and, therefore, have not introduced any corrections for concentration dependence. Experiments reported in the accompanying paper used a maximum of.2 mm RNA nucleotides. Comparing Eqs. 13 and 1, it follows that C Mg,ref is equivalent to the bulk Mg concentration when the dependence of Γ on RNA concentration is negligible. Thus, substitutions of experimental quantities into Eq. 12 yields

8 C G RNA Mg RT Mg RNA,ref C Mg d lnc. [15] Mg,ref 0 In the main text, the simplified notation C is used for the experimentally determined bulk Mg concentration. Observed Free Energies of RNA Stabilization by MgCl 2. In the formulation of the equilibrium between the N and I forms of an RNA in Eqs. 2 and 3, the Mg -induced stabilization of the N state relative to the I state is defined as G Mg = G N Mg G I Mg = (µ N,Mg µ N,0 ) (µ I,Mg µ I,0 ) = RT ln γ γ N,0 I,Mg γ I,0 γ N,Mg [16] In this equation, subscripts on the various chemical potentials (µ) and activity coefficients (γ) specify N or I state RNAs in the presence or absence of Mg. (The different RNA chemical potentials correspond to the four RNA energy levels in Figs. 1 and 5.) Eq. 15 provides a route to this Mg RNA stabilization free energy via measurements of the Mg RNA interaction free energies. The same stabilization free energy also can be obtained from the observed stability of the RNA in the presence ( G obs,mg ) and absence ( G obs,0 ) of added Mg. From Eq. 3, noting that G fold is independent of Mg concentration, RT ln γ N,0 + G obs,0 = RT ln γ N,Mg γ I,0 γ + G [17] obs,mg I,Mg

9 Rearrangement of Eq. 17 and comparison with Eq. 16 gives G Mg = G obs,mg G obs,0. Mg -Induced RNA Stabilization and the Wyman Linkage Relation. Some of the approximations made in the derivation of Eq. 15 also apply to the use of the Wyman linkage relation to analyze the Mg -dependence of RNA folding reactions. In its most general formulation, the linkage relation describes the effect of changing ligand activity on the observed free energy of a macromolecular conformational transition in terms of the difference between the preferential ligand interaction coefficients for the two conformers. The derivation outlined in section 7 of ref. 8 proceeds as in Eqs. 7 above and, for the folding reaction of interest in this work, results in 1 G obs,mg RT ln a m 2,m 3 = Γ µn Γ µ I = Γ µ [18] Using the approximations discussed above in reference to Eq. 8, ln(c ) (measured when the RNA is dilute enough that Mg is in large excess over nucleotides) can be substituted for lna in the above equation if m 3 >> m by a factor of at least 30. Γ may also be substituted for Γ µ, as in Eq. 12. Concluding Remarks. The problem of Mg -induced RNA folding has usually been posed in terms of the stoichiometric binding of Mg to the folded RNA (9-11) or of Mg binding to both unfolded and folded RNAs with different affinities and stoichiometries (12). This kind of approach follows a well established formalism for describing allosteric changes in proteins as induced by stoichiometric binding of ligands at specific sites (8) but introduces substantial approximations when applied to ion nucleic acid interactions. In particular, it is not possible to define a Mg RNA binding stoichiometry, because there is no way to distinguish free and bound Mg ions in the presence of RNA; long-range electrostatic interactions take place between an RNA and all ions in solution. Thus, it has been problematic to relate free energy changes calculated using a stoichiometric Mg RNA binding formalism to quantities that can be calculated

10 from theoretical models of folding (13). Ion interaction coefficients (Eq. 1), which describe the fractional RNA charge neutralization by ions, provide a general approach for obtaining ion RNA interaction free energies without the necessity of distinguishing sitebound, diffuse, or other classes of ions (1). Although it has been necessary to introduce approximations in relating RNA folding energetics (Eq. 2), experimentally accessible quantities (Eqs. 13 and 15), and Γ, it is possible to evaluate the potential error for a given set of experimental conditions. An additional advantage of this approach is that the measured ion interaction coefficients can be directly compared with the results of theoretical calculations and simulations (, 13). *The bulk concentration of an ion is the ion concentration in an RNA solution dilute enough that the excess number of ions associated with the RNA is small compared with the total number of ions. It can also be thought of as the ion concentration that would be found an infinite distance away from any RNA molecule in a solution at equilibrium. 1. Draper DE, Grilley D, Soto AM (2005) Annu Rev Biophys Biomol Struct 3: Record MT, Jr, Richey B (1988) in ACS Sourcebook for Physical Chemistry Instructors, ed Lippincott ET (Am Chem Soc, Washington, DC), pp Anderson CF, Felitsky DJ, Hong J, Record MT (2002) Biophys Chem : Anderson CF, Record MT, Jr (1993) J Phys Chem 97: Harned HS, Robinson RA (1968) Multicomponent Electrolyte Solutions (Pergamon, Oxford, UK). 6. Robinson RA, Stokes RH (1959) Electrolyte Solutions (Butterworths, London), 2nd Ed. 7. Record MT, Jr, Zhang W, Anderson CF (1998) Adv Protein Chem 51: Wyman, J, Jr (196) Adv Protein Chem 19: Lynch DC, Schimmel PR (197) Biochemistry 13: Silverman SK, Cech TR (1999) Biochemistry 38: Fang X, Littrell K, Yang XJ, Henderson SJ, Siefert S, Thiyagarajan P, Pan T, Sosnick TR (2000) Biochemistry 39:

11 12. Laing LG, Gluick TC, Draper DE (199) J Mol Biol 237: Misra VK, Draper DE (2002) J Mol Biol 317: