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The ionic composition of a solution strongly influences the folding of an RNA into its native structure; of particular importance, the stabilities of RNA tertiary structures are sharply dependent on the concentration of Mg2+. Most measurements of the extent of Mg2+ interaction with an RNA have relied on equilibrium dialysis or indirect measurements. Here we describe an approach, based on titrations in the presence of a fluorescent indicator dye, that accurately measures the excess Mg2+ ion neutralizing the charge of an RNA (the interaction or Donnan coefficient, Γ2+) and the total free energy of Mg2+ - RNA interactions (ΔGRNA-2+). Automated data collection with computer-controlled titrators enables the collection of much larger data sets in a short time, compared to equilibrium dialysis. Γ2+ and ΔGRNA-2+ are thermodynamically rigorous quantities that are directly comparable with the results of theoretical calculations and simulations. In the event that RNA folding is coupled to the addition of MgCl2, the method directly monitors the uptake of Mg2+ associated with the folding transition.
Early RNA folding studies demonstrated the unusual sensitivity of tRNA tertiary structure to Mg2+ ions (Stein and Crothers, 1976a); it has subsequently been recognized as a general principle that strong stabilization of an RNA structure by Mg2+ is diagnostic of the formation of tertiary contacts (Tinoco and Bustamante, 1999). RNA folding studies have generally measured the net uptake of Mg2+ ions upon folding, or used spectroscopic or chemical probes to detect direct ion - RNA contacts (Draper, et al., 2005). To develop a comprehensive framework for understanding the ways in which Mg2+ stabilizes tertiary structures, it is necessary to quantitate the ion interactions taking place with both folded and unfolded states of the RNA. The interactions can be described both in terms of the number of excess Mg2+ ions neutralizing the RNA charge and the overall free energy of Mg2+ interactions. Both experimental parameters are useful benchmarks with which to compare results of theoretical calculations and simulations (Grilley, et al., 2007, Misra and Draper, 2002, Soto, et al., 2007).
Several studies in the 1970s used equilibrium dialysis to follow the excess Mg2+ accumulated by transfer RNA (Bina-Stein and Stein, 1976, Stein and Crothers, 1976b). Dialysis is a rigorous way to measure thermodynamic quantities, but its application has been limited by requirements for long-term stability of the RNA solutions and the time needed to collect an extensive data set. As an alternative to equilibrium dialysis, chelating dyes have been used to sense the change in Mg2+ activity caused by the presence of an RNA. Eriochrome black T, which changes its extinction coefficient upon binding to Mg2+, was used in an set of studies examining Mg2+ association with homopolymer RNAs (Krakauer, 1971). The fluorescence of a quinoline derivative has also been used to follow Mg2+ interactions with RNA (Römer and Hach, 1975, Serebrov, et al., 2001) and proteins (Pickett, et al., 2003). Here we present methods to measure Mg2+ – nucleic acid interactions using a fluorescent dye. The procedure can be largely automated by the use of computer-controlled titrators, enabling the collection of large data sets (>100 points) in less than a few hours. With appropriate controls, the results are as accurate and as thermodynamcally rigorous as those from equilibrium dialysis. A single titration data set can be analyzed to find the excess Mg2+ associated with an RNA, the free energy of Mg2+ interaction with an RNA, and/or the uptake of Mg2+ ions accompanying a folding reaction or RNA conformational change.
The high negative charge of a nucleic acid is neutralized by both an accumulation of mobile cations and an exclusion of mobile anions from the surrounding solution (Record, et al., 1998). The accumulated cations may interact with the nucleic acid in a variety of ways: fully hydrated ions interact strongly with the RNA electrostatic potential at a distance from the RNA surface (the “diffuse ion atmosphere”), cations may be bound to specific sites within the RNA, or cations may contact the RNA surface via an intermediate layer of water (Draper, 2004, Misra, et al., 2003). The depletion of anions near the RNA is caused primarily by repulsive interactions between the ions and RNA negative charges. To describe the accumulation or exclusion of ions in a model-independent way, we use the formalism of interaction coefficients. For Mg2+, the coefficient is defined as
where C2+, CRNA, and C± are the molar concentrations of Mg2+, RNA, and 1:1 salt, respectively; μ2+ is the chemical potential of the Mg2+ ion (Grilley, et al., 2006, Record, et al., 1998). Γ2+ is the number of Mg2+ ions that must accompany every RNA molecule added to a solution, in order to maintain a constant chemical potential of the ion. These Γ2+ “excess” ions neutralize 2Γ2+ RNA charges. The remainder of the RNA charge is neutralized by an excess of monovalent ions, Γ+, and a deficiency of anions, Γ− (a negative quantity).
The physical meaning of an interaction coefficient can be illustrated by an equilibrium dialysis experiment, which measures the Donnan coefficient
is the molar Mg2+ ion concentrations present “inside” the dialysis membrane with RNA (at concentration CRNA). , also known as the “bulk” concentration of ions, is the molar Mg2+ ion concentration “outside” the dialysis membrane. ( ) is the excess number of Mg2+ ions accumulated in response to the presence of the RNA. In a typical set of measurements, a series of RNA solutions would be equilibrated with solutions of increasing Mg2+ concentration (and thus increasing μ2+) to obtain as a function of μ2+. Formally, differs from Γ2+ (equation 1) in that the chemical potential of water is constant across the membrane in a dialysis experiment, while Γ2+ is defined for constant pressure. For the concentrations of salt and RNA considered here, the two coefficients are essentially identical and will not be distinguished in the following discussion (Anderson, et al., 2002).
For applications in analytical chemistry, chelators have been synthesized which exhibit large changes in their absorption or fluorescence emission spectra upon binding specific metal ions. In principle, low concentrations of such a Mg2+-binding dye could be used to sense the chemical potential of Mg2+ in a solution of RNA, yielding measurements of Γ2+ without the lengthy equilibration times needed for dialysis measurements. The experimental approach is sketched in Figure 1A. Sample and reference solutions are prepared with identical concentrations of dye and buffer; the sample solution also contains RNA. The two solutions are titrated with MgCl2 in parallel, and the fraction of dye molecules complexed with Mg2+, ν, is monitored spectroscopically. When ν is plotted vs. the total concentration of added MgCl2, C2+, a lag is observed in the curve for the sample solution (Figure 1A) because favorable interactions between Mg2+ and RNA reduce the effective concentration (or activity) of the Mg2+. The premise of this method, justified below, is that sample and reference solutions with identical values of ν also have the same Mg2+ ion chemical potential (μ2+) and would therefore be in thermodynamic equilibrium if separated by a dialysis membrane- they would be equivalent to the “inside” and “outside” solutions, respectively. The application of this premise is illustrated by the horizontal arrows in Figure 1A, which compare C2+ of the two titration curves at points with the same ν value. The Mg2+ interaction coefficient is calculated from these concentrations as (cf. equation 2)
To justify the assumption made above, that sample and reference solutions with the same value of ν would be in thermodynamic equilibrium across a dialysis membrane, imagine a Mg2+ - RNA solution that is in dialysis equilibrium with buffer. Samples of the “inside” and “outside” solutions are then spectroscopically assayed with the Mg2+ - binding dye and a value for ν is obtained after suitable normalization of the absorption or fluorescence signal. If the dye-ion complex has a 1:1 stoichiometry, the fraction of dye molecules complexed with Mg2+ is
where CC and CD are the molar concentrations of the complex and the unbound dye, respectively. Using the relationships between activity, concentration, and activity coefficient aC γCCC and aD γDCD, equation (4) becomes
The question is whether the same value of ν will be found in both solutions. Because the two solutions are in dialysis equilibrium, aC and aD must be identical between them. However, the activity coefficient γD or γC could differ between the solutions if either the free or complexed dye interacts with the RNA. For the dye used here, the Mg2+-dye complex is electroneutral and thus has no electrostatic interaction with the RNA. As long as there is no direct binding of the complex to the RNA, γC is insensitive to the presence or absence of RNA. The dye itself has a negative charge, and thus should be excluded from the RNA-containing solution to approximately the same extent as the chloride ion that is also present. If these experiments are carried out with a large excess of monovalent salt over Mg2+, the overall fraction of negative ions excluded by the RNA is so small as to introduce an error in the measurements of Γ2+ that is within the reproducibility of the experiment (see VI below for a calculation of typical error). Thus, under appropriate conditions, νin ≈ νout is true to an acceptable level of approximation and the dye can be used to monitor Mg2+ activity.
An advantage of using a Mg2+-sensing dye is that one pair of sample and reference titrations yields measurements of Γ2+ over a wide range of bulk Mg2+ concentrations, as graphed in Figure 1B. This Γ2+ curve can then be used to find the Mg2+-RNA interaction free energy by integration,
ΔGRNA-2+ is the net free energy change that takes place when MgCl2 is added to an RNA solution until the bulk Mg2+ concentration reaches the upper limit of integration. A detailed derivation of this equation is available elsewhere (Grilley, et al., 2006).
Several approximations have been made in the derivation of equations (3) and (6). The limitations these approximations impose on the measurement of Γ2+ and ΔGRNA–2+, as well as practical considerations in carrying out these measurements, are discussed in section VI below.
Equilibrium dialysis experiments are most often interpreted in terms of a binding density ν, the average number of ligands bound per macromolecule. A plot of ν vs. the “free” or bulk ligand concentration yields a binding curve, which is then fit with parameters describing the stoichiometry and affinities of different classes of binding sites (Wyman and Gill, 1990). Though this approach requires assumptions that are difficult to reconcile with long-range electrostatic interactions of ions, it is nevertheless widely used in the RNA folding literature. Here we contrast the binding density and interaction coefficient approaches, and note why the latter is preferable for the analysis of Mg2+-RNA titrations.
A fundamental conceptual difference between the two approaches is the way in which interaction free energies are described. In a binding formalism, each ligand-bound state of the macromolecule is considered a distinct specie with its own standard chemical potential (μ°). The interaction free energy is then the difference in standard state chemical potentials between the ligand and molecule in free and bound states, e. g.,
This approach works well when there are a small number of well-defined binding interactions. It breaks down in several ways when applied to ion - nucleic acid interactions. First, the repulsive interactions between anions and nucleic acids are not readily described in terms of specific “complexes”, and are therefore usually ignored in the binding density formalism. Second, long range electrostatic interactions create concentration gradients of ions near the RNA surface (the so-called “ion atmosphere”); closer cations experience stronger interactions. Because the binding formalism allows any one ion in solution to exist in only one of two states, “bound” in a complex or “free” in solution, these concentration gradients are also ignored. Lastly, the binding formalism relies on formulation of a model that describes all the possible “bound” states of the macromolecule. The usual models for Mg2+ - RNA interactions have postulated two or three classes of independent binding sites with “strong” or “weak” affinity (Stein and Crothers, 1976b); some models have included anti-cooperative interactions between ions (Laing, et al., 1994, Leroy and Guéron, 1977). Given enough classes of binding sites, the binding polynomials generated by these models invariably provide a good fit to a data set, but in most cases the equilibrium constants and stoichiometries do not have readily interpretable physical significance, and in any case are not unique descriptions- the data can always be fit by multiple models (Leroy and Guéron, 1977).
When using interaction coefficients, standard state chemical potentials are defined only for each electroneutral component added to the solution. For the cases considered here, there are formally four components: water, an RNA salt, monovalent (1:1) salt, and MgCl2. (Single ion chemical potentials may also be defined, subject to the constraint of electroneutrality, e.g. μ°KCl = μ°K+ + μ°Cl−). Interactions between any two components appear as mutual changes in their thermodynamic activities, which precludes any need to distinguish between bound and free fractions of a component. Thus, in an equilibrium dialysis experiment with Mg2+ and RNA,
because of the requirement that the Mg2+ activity (a2+ γ2+C2+) be the same on both sides of the membrane. Upon rearrangement, this equation shows how Mg2+ - RNA interactions are related to changes in the Mg2+ activity coefficient,
Favorable interactions between Mg2+ ions and RNA decrease (relative to ) and thus cause an accumulation of ions by the RNA ( ). The left hand side of equation (9) is the numerator of the Donnan coefficient, (equation 2). Thus, the Mg2+ activity coefficient (γ2+) is mathematically related to the interaction coefficient Γ2+, and Γ2+ is in turn related to the Mg2+ - RNA interaction free energy (equation 6). In contrast to the parameters obtained from a binding density analysis, these three related thermodynamic quantities are model-independent, As such, they may be directly compared with values of Γ2+ and ΔGRNA-2+ that have been extracted from theoretical calculations and simulations of model systems (Ni, et al., 1999, Soto, et al., 2007).
8-hydroxyquinoline-5-sulfonic acid (HQS) was first described as a soluble quinoline derivative showing substantial changes in its absorption and fluorescence spectra upon chelation of a variety of metals with a valency of +2 or larger (Bishop, 1963a, Bishop, 1963b, Liu and Bailar, 1951). The structure of HQS and its complex with Mg2+ are shown in Figure 2A. The sulfonic acid group serves to increase the quinoline solubility in water; with a pK ≈ 4.0 (Smith and Martell, 1975), it is fully ionized in the pH range of interest here. The complex with Mg2+ has a net neutral charge, because ion chelation is coupled to deprotonation of the quinoline hydroxyl. Free HQS, with a protonated hydroxyl at pH 6–7, has an absorption maximum at 306 nm which is shifted to longer wavelengths (355–357 nm) in the presence of either high pH or saturating concentrations of Mg2+ (Tables 1 and and2).2). Superimposed spectra taken over the pH range 6.0 – 9.5, or with Mg2+ concentrations up to 110 mM, show clear isosbestic points (Figure 2B, C). The latter behavior is consistent with the formation of a single, 1:1 Mg2+ - HQS chelation complex.
Fluorescence is observed at ~500 nm upon excitation of the 355 nm absorption of the Mg2+ - HQS chelate. A ~100 fold increase in fluorescence is obtained upon titration of HQS with MgCl2 (Figure 3A). Fluorescence intensity data as a function of the total concentration of Mg2+ ion, C2+, is fit very well by a single site binding isotherm:
where Imax is the intensity of the Mg2+-HQS complex, Imin is the intensity of the free HQS, and KHQS is the apparent equilibrium constant for formation of Mg2+-HQS. At the concentration of HQS typically used in these titrations, the fraction of added ions that are bound to HQS is very small and accuracy is not compromised if the total ion concentration (C2+) is used in place of the free (unbound) ion concentration.
KHQS is pH dependent, because chelation of Mg2+ by HQS promotes deprotonation of the quinoline hydroxyl group. Log(KHQS) increases linearly with pH over the range 6 – 8 (Figure 3B); the calculated pKa is 8.43 and the intrinsic K°HQS is 11.3 × 103 M−1, in agreement with literature values (Smith and Martell, 1975). The large fluorescence enhancement and range of Mg2+ binding affinities around neutral pH make HQS a suitable indicator for experiments measuring Mg2+-RNA interactions.
Starting from Mg2+ and going down the periodic table column of group II ions, HQS-metal ion binding constants become weaker and the fluorescence enhancements become smaller. At pH 8.0 the apparent equilibrium constant for Ca2+-HQS formation is 182 M−1 and the fluorescence enhancement is ~75 fold (Figure 3B). The Ba2+ complex is even weaker, with an apparent binding constant of 36 M−1 and a fluorescence enhancement of ~10 fold. HQS may be a good sensor for Ca2+- as well as Mg2+- RNA interactions, but its affinity for Sr2+ or Ba2+ is probably too weak for this purpose.
Measurement of metal ion-RNA association is critically dependent on the purity of the components that are used. All solutions should be made from pure water with at least 18 MΩ resistivity. Purchased buffers and salts should be at least 99.5% pure, but because EDTA is included in all the buffers to scavenge transition metals, salts of higher purity (and expense) are not necessary.
HQS can be purchased from Sigma Chemicals in a relatively pure acid form. The small amount of metal contaminating the HQS can be removed by recrystallization at acid pH. First saturate ~50 mL of pure water with the dye (~5 g). To this solution add concentrated (12 N) HCl until no further color change or precipitation is perceptible. Heat the solution to boiling while stirring to fully re-dissolve the HQS. Cool the solution on ice and drain the excess water leaving the crystalline HQS as very fine needles. Add fresh pure water and HCl to the HQS crystals, then heat with stirring until the HQS is all dissolved. Cool, drain and repeat the process seven times, adding less water and HCl at each step to account for the slight loss of HQS that remains dissolved at the end of the cooling step. After the final step, add 50 mL water and raise the pH to approximately 7 with KOH. Continue to add water and maintain pH 7 until all crystals are dissolved. When HQS treated this way is diluted into freshly made buffered solution, no change in its absorbance spectra is seen upon addition of 100 μM EDTA. The concentration of the stock solution can be determined using an extinction coefficient at the 326 nm isosbestic point (Table I) of 2600 M−1 cm−1. The purified HQS can be divided into aliquots (~100 mM stock is typical) and stored in acid rinsed glassware for years.
The 99.99% purity MgCl2•hexahydrate available from Aldrich can be used as supplied to make ~2 M MgCl2 stock solutions. MgCl2 solutions for titrations are prepared by diluting the stock 2 M MgCl2 solution into the appropriate titration buffer (see IV.1 below).
Titration buffers should be made from weighed salts and the acid form of the buffer; titrate with the appropriate 99.5% pure hydroxide salt to adjust the pH. Some convenient buffer concentrations and pHs for different monovalent cation concentrations are as follows:
(Both buffer and EDTA stock solutions were prepared from the respective free acids and adjusted to the desired pH with KOH. The K+ concentration is the sum of the weighed KCl and the KOH added to adjust the pH.) Lower pH, which weakens the affinity of HQS for Mg2+, is paired with higher salt concentration to match the weakened Mg2+ -RNA interactions. These pairings have given reproducible measurements of Γ2+ with duplex DNA restriction fragments, RNAs that have Mg2+ -dependent or Mg2+ -independent tetiary structures, and RNAs with only secondary structure. Titrations at pH 8.0 (and thus larger KHQS) extend the range of useful data to concentrations as low as ~1 μM (D. Leipply, personal communication).
Glassware, plastic Eppendorf-style micro-centrifuge tubes and other preparative labware leach metals into solution. For that reason, 20 μM EDTA was included in all titration buffers. It is important to note that this level of EDTA serves only to scavenge transition metals, which bind EDTA much more tightly than does Mg2+; an order of magnitude higher EDTA concentration is needed before a significant amount of Mg2+ is bound.
The hygroscopic properties of MgCl2 affect the accuracy with which the salt can be weighed. Therefore either of the following two protocols is used to standardize MgCl2 solutions. Both are based on EDTA-Mg2+ complex formation but use different methods to detect the stoichiometric break point.
The first protocol uses HQS as a reporter of free Mg2+. An accurately-known concentration of EDTA (1 – 5 mM) in 100 mM EPPS (pH 8.0) and 100 μM HQS is titrated with MgCl2 solution. Either fluorescence or absorbance of the HQS is plotted vs. added MgCl2 to find the amount of Mg2+ needed to titrate the EDTA to a stoichiometric endpoint. pH 8.0 provides an appropriate ratio of the EDTA and HQS binding affinities for accurate determination of the stoichiometry. A high buffer concentration is necessary because of the pH dependence of EDTA-Mg2+ binding (Martell and Smith, 1974).
The second protocol depends on the hypochromicity of EDTA in the low UV region (200–250 nm) upon metal ion binding. Titrations are conducted in 10 mM MOPS (pH 7) using semimicro quartz cuvettes and recording an absorbance spectrum between 200 and 350 nm. In a typical experiment, 800 μL of a ~1 mM solution of the metal ion of interest is titrated with 1 to 3 μL aliquots of a 40 mM EDTA solution. Plots of the absorbance at 230 nm versus the concentration of EDTA yield stoichiometric titrations with sharp break points that are used to calculate the concentration of the metal ion.
Metal-free DNA or RNA fragments can be prepared using a DEAE matrix made by Qiagen, which comes pre-packaged in disposable columns appropriate for small step gradients. A Qiatip-2500 column holds 2.5 mg of double stranded nucleic acid (5 mg single strand), and has a column volume of approximately 30 mLs. The resin can be removed and repacked into a Biorad Econo-column for use with pumps or gradient makers. To use the column first make the following two buffers:
All the solutions that follow are adjusted to a final pH of 7.0 with NaOH after mixing buffer A or B with the other components. The resin is first washed with 0.15% Triton X-100, 25% water, and 75% buffer A, and then equilibrated with 25% water, 75% buffer A prior to loading the sample. Samples are diluted with 3 volumes Buffer A and loaded onto the column. If an RNA transcription mixture is being loaded, add sufficient EDTA to dissolve any magnesium pyrophosphate precipitate. After loading, rinse with 25% water, 75% buffer A until a stable baseline (monitored at ~260 nm) is achieved. The column is then eluted in steps consisting of mixtures of buffers A and B in different ratios and 25% (v/v) urea (e.g., 100 mL of a 600 mM NaCl solution is made by mixing 67 mL buffer A, 8 mL buffer B and 36.2 g urea). Since addition of urea slightly shifts the pH, mix buffers A and B and dissolve urea before bringing the solution to pH 7.0 with NaOH. The Qiagen columns are particularly sensitive to pH. Double-stranded DNA tends to elute between 0.9 and 2 M NaCl, RNA between 0.4 and 1 M NaCl; larger nucleic acids elute at higher salt concentrations. Very large (greater than approximately 400 basepairs) nucleic acids may require higher pH for elution (e.g., 2 M salt and pH 8.0). Nucleic acids should be precipitated out of column fractions by addition of 50% isopropanol, which minimizes precipitation of the salt and urea.
RNA transcripts for titrations can also be purified by denaturing gel electrophoresis followed by electroelution from an Elutrap (Schleicher & Schuell). The high concentration of EDTA and urea used in standard electrophoresis buffers are as efficient in removing Mg2+ and transition metal contamination as the above column protocol.
Precipitated, purified nucleic acids should be resuspended in 50 mM EDTA, pH 7.0, 1 M ammonium acetate to help remove any remaining metal contaminants. The nucleic acid can then be exchanged into titration buffer. Amicon centrifugal filtration devices (Millipore) are large enough (4 or 15 mL) to provide a reasonable dilution factor without concentrating the sample too much. Extremely high concentrations of nucleic acid (greater than 40 mM nucleotide) equilibrate slowly and should be avoided. A good final stock concentration is between 10 and 30 mM in nucleotide. Typical equilibrations involve at least eight fivefold dilutions. Nucleic acids approach equilibrium with low salt buffers very slowly by either dialysis or repeated rounds of concentration and dilution. We therefore first attempt to equilibrate samples with buffers containing half of the desired final salt concentration, and then bring the salt concentration up to the final desired concentration in the final dilution/concentration cycles.
The approach of the sample to the final salt concentration can be tracked by UV-monitored melting curves. Sample RNA concentrations are adjusted with equilibration buffer to an RNA appropriate for a titration (1 – 4 mM nucleotide), and melting is monitored by absorbance at an appropriate wavelength in a 1 mm path length cuvette. The same experiment repeated on a sample diluted 10 fold (0.1 mM nucleotide) in 1 cm cuvette should have the same melting profile if equilibration has been achieved. Because of the high concentration of the sample, formation of RNA dimers is a potential artifact that could also cause differences between the melting profiles at two different RNA concentrations.
All HQS titrations were performed on an Aviv ATF-105 differential/ratio spectrofluorometer designed for computer-controlled titrations into reference and sample cells. The instrument includes a pair of automatic dispensers (Hamilton) and “J”-shaped tubes for titrant delivery into standard 1 cm2 cuvettes. This setup works well for a high density solution (such as concentrated urea), but the buffered, 12 mM MgCl2 titrant solutions used here easily mix with the cuvette solution during set-up. The problem can be overcome by using a glass capillary tube that has been pulled into a very fine tip. The glass tips can be held in fittings with M6 threads by wrapping the tube with Teflon tape (Figure 4). The assembly is held by a teflon cap machined for the purpose (E in Figure 4). This arrangement enhances “wicking” of sample up the cuvette walls and increases evaporation, but these problems are minimized by applying a hydrophobic coating to the walls. (Soak inside of cuvettes with 15 M nitric acid for at least one hour. Rinse well with pure water and dry. Add Sigmacote (Sigma-Aldrich) to the cuvette and let sit for at least an hour. Rinse well with pure water.)
The Hamilton dispensers can be used with syringe sizes from 25 μL to 25 mL. With 1000 steps per stroke, the 25 μL syringe could have a resolution in volume delivery of as little as 0.025 μL. However, the 25 μl syringes tend to wear out more quickly than the larger volume syringes. Thus we use 50 μl syringes as a good compromise between resolution and long-term stability. Regardless, the valves and syringe plungers wear down over time, leading to inconsistent results. With daily use, syringe plungers should be replaced about once a month, the entire syringe about every six months, and the valves every three to four months. The syringes and valves on the dispensers also leach metals and need to be rinsed with 5 mM EDTA solution prior to each day of use.
In designing an HQS titration, the total time of the experiment and the number and volume of injections are important and related considerations. In general, we have found that the shorter the experiment, the more reproducible the data. For nucleic acid systems in which Mg2+ does not induce any folding reaction, the only limitation on the interval between additions is the time taken for mixing, about 15 seconds at moderate stirring speeds. The folding kinetics of some RNAs can require equilibration times of five or more minutes for each Mg2+ addition. (A necessary control is to obtain time courses of RNA folding after Mg2+ addition in the UV spectrophotometer under similar solution conditions as desired for the HQS-monitored titration; based on such kinetic data, an appropriate interval can be designed for the HQS-monitored titration. Apparent folding rates are usually not uniform as one titrates across a folding transition.) The longest a titration should be is about five hours; longer titrations require corrections for drift in instrument signal (which is exacerbated by fluctuations in room temperature) and titrant and sample evaporation (to allow titrant addition, the cuvettes cannot be completely sealed). It is easier to repeat a titration several times with different nucleic acid concentrations and schedules of Mg2+ additions than to try to obtain a complete range of data in a single long titration.
A single titration experiment requires data from two cuvettes run in parallel: a reference cuvette with HQS and titration buffer and a sample cuvette with RNA. The same batch of titration buffer with which the RNA has been equilibrated is used in both curvettes. When preparing the sample and reference cuvettes, absorption spectra should be taken to check the RNA concentration and test for metal ion contamination. First, scan buffer alone in both cuvettes (550 – 200 nm). Add the nucleic acid to the sample cuvette (typically a five fold dilution) and perform a second scan. The absorbance at 260 will be far beyond the linear range of any spectrophotometer; use the absorbance values between 0.2 and 1.2 OD in the range 295 – 310 nm to check the RNA concentration. Add HQS, typically 1–2 μL of stock to a final concentration of 20 – 50 μM, and perform a final scan. (HQS concentrations as low as 10 μM may be used, if accurate data at bulk Mg2+ concentrations in the μM range are to be collected.) Metal ion contamination of the RNA stock solution is evident in the shape of the HQS absorbance curve between 350 and 400 nm (Figure 2). A final diagnosis of contamination may be made by adding EDTA to 100 μM and looking for changes in the HQS spectrum; however, the sample cannot then be used without re-purification.
A typical titration protocol with the automatic titrators uses starting volumes of 2.00 mL in sample and reference cuvettes. The two titrant solutions are:
The same buffer used to equilibrate samples (see IV.2 above) is also used for the titration buffer in each solution; unbuffered stock MgCl2 solutions are as described in IV.1. Because titration buffer is slightly diluted when making the sample and reference titrants, there will be a ~0.2% change in salt and buffer concentration over the course of the entire titration; the magnitude is not large enough to be of concern.
A typical titration schedule is as follows (concentrations listed are for the reference cell, but the same volume additions are used for sample cell):
|Final [MgCl2]||12 mM||20 mM||40 mM|
|Step size||200 μM||400 μM||1 mM|
|Number of Points||60||20||20|
The Aviv fluorometer software calculates the injection volumes needed to achieve constant increments in MgCl2 concentration; constant volume additions could be used as well. The standard stirring time between additions is 30 sec; nucleic acid samples that fold may require adjustment in the stirring times programmed for each section.
To reduce problems with self-quenching, data are collected on the edge of the HQS absorption maximum by exciting at 405 nm with a 2 nm bandwidth; emission is monitored at 500 nm with an 8 nm bandwidth.
After the automatic titration is finished, manual additions are made to help determine the fluorescence intensity at saturating Mg2+ (Imax; see section VI. below). Unbuffered 2 M MgCl2 is added in the following volumes: 5, 5, 10, 20, 20, 20, and 20 μl.
HQS titrations can also be performed by manual additions of Mg2+ with standard pipetters. Manual titrations can be carried out in small volume cuvettes, which is useful when the RNA of interest cannot be obtained in large amounts. In a typical titration, a stoppered microcuvette containing 400 μL of an HQS solution is titrated in parallel to a cuvette containing 400 μL of HQS and the RNA of interest. We recommend using a 10 μL pipetter during the entire experiment for consistency. Use three MgCl2 stock solutions (made in titration buffer): 7, 35 and 1000 mM. Titrate the RNA containing cuvette initially with 7 mM MgCl2 (45 additions of 1–10 μL), then with 35 mM MgCl2 (three 10 μL additions) and finally with 1000 mM MgCl2 (five 10 μL additions). Titrate the cuvette containing only HQS with 35 mM MgCl2 (48 additions of 1–10 uL) and 1000 mM MgCl2 (five 10 μL additions). Add the same volume of titrant to both the RNA and HQS cuvettes. Monitor the fluorescence intensity until the HQS signal stabilizes, and take six additional measurements of the intensity to average for a single data point.
An example titration data set is shown in Figure 5. The most reliable and informative data are obtained near the beginning of the titration, where the difference between the sample and reference curves is most pronounced. In the example shown, this region corresponds to HQS normalized fluorescence (saturation) values between 0 and 0.15 (cf. the inset in Figure 5).
To begin data analysis, measured intensities for the sample cuvette are first normalized:
Inorm is the fraction of Mg2+-bound HQS, which will be compared with the reference titration (Figure 1A). Imin is taken as the initial data point (before MgCl2 addition) of the sample cuvette. Imax is obtained from the fit of a single site binding isotherm (equation 10) to only the manual titration points of the sample cuvette (Figure 5). More complicated schemes for obtaining Imax were tried; all gave identical results within error.
The data from the reference titration are fit to a single site binding isotherm (equation 10), allowing KHQS, Imax, and Imin values to float. KHQS is then used to calculate the “bulk” Mg2+ concentration, , for each data point in the sample curve:
Finally, the preferential interaction coefficient is computed by calculating the excess Mg2+ present in the sample cuvette over the calculated bulk Mg2+ concentration:
Errors in four experimental measurements can affect the accuracy of the calculated Γ2+ values. These quantities and their estimated typical uncertainties are Imax (<1%), KHQS (<1%), nucleic acid concentration (3%), and stock MgCl2 concentration (1%). Figure 6 shows the effect on the calculated value of Γ2+ when each of these quantities is perturbed by 3 – 10%. Errors in Imax result in small changes in Γ2+ for the initial titration points, and much larger errors for the final titration points. Any error in determining the stock MgCl2 concentration will be partly compensated in the calculation of by the fact that an overestimation of the total added MgCl2 decreases the calculated KHQS (and vice versa). Errors in the RNA concentration result in the same percentage change in Γ2+ at all . Combining all of the typical estimated magnitudes of errors, their net effect on Γ2+ in the Figure 6 data set is similar to the variation observed between titrations repeated under identical conditions in the range from ~3 × 10−6 to nearly 10−3 M bulk Mg2+.
To average data from separate titrations, we choose a set of values, calculate the corresponding Γ2+ for each data set by linear interpolation between neighboring data points, and average the Γ2+ values so obtained. Titrations carried out at different pH values can be averaged to extend the range of values for which errors in Γ2+ are acceptable. Three to five independent titrations are typically averaged to obtain a reliable data set; the standard deviations of the averaged values give an estimate of the associated error. Integration of the averaged Γ2+ curve to obtain interaction free energies (Figure 1B) is done either by using the trapezoidal rule of numerical integration or by fitting a polynomial that asymptotically approaches the x-axis, y = b(x−a)2 + c(x−a)3 + d(x−a)4, and integrating the polynomial. The two methods generally agree within error. In either case, free energies calculated at low are potentially subject to systematic error from the way the curve is extrapolated to the x-axis.
Here we discuss a number of factors that should be taken into account in designing a titration experiment and analyzing the results, either because of practical limitations in the experimental set-up or assumptions that are inherent in equations (6) and (12).
This work was supported by NIH grant GM58545 (D.E.D.) and a Burroughs Wellcome Fellowship (A.M.S.).