Probing RNA–Protein Interactions and RNA Compaction by Sedimentation Velocity Analytical Ultracentrifugation

Abstract

Recent advances in multi-wavelength analytical ultracentrifugation (MWL-AUC) combine the power of an exquisitely sensitive hydrodynamic-based separation technique with the added dimension of spectral separation. This added dimension has opened up new doors to much improved characterization of multiple, interacting species in solution. When applied to structural investigations of RNA, MWL-AUC can precisely report on the hydrodynamic radius and the overall shape of an RNA molecule by enabling precise measurements of its sedimentation and diffusion coefficients and identify the stoichiometry of interacting components based on spectral decomposition. Information provided in this chapter will allow an investigator to design experiments for probing ion and/or protein-induced global conformational changes of an RNA molecule and exploit spectral differences between proteins and RNA to characterize their interactions in a physiological solution environment.

1. Introduction

RNA molecules, although synthesized as extended polymeric structures, more often than not, adopt compact conformations to execute their biological functions [1]. Such compactions are almost always induced by specific and nonspecific interactions with counterions and sometimes by proteins [2]. Even compact conformations, by interacting with small metabolites and cellular proteins, can undergo distinct global topological changes which are manifested as alterations in the shape and hydrodynamic properties of a given RNA [3, 4]. One of the most sensitive techniques for detecting RNA compaction and global conformational changes is analytical ultracentrifugation (AUC) [5-7]. Although AUC has been more widely used for analyzing protein conformations and protein–protein interactions, the easy adaptability of analytical detection methods, offered by AUC, to biological and nonbiological samples has led to the adoption of this technique for structural characterization of RNA [5-11] and RNA–protein complexes [12-14], as well as investigations of RNA–protein interactions [15-18].

AUC is at an interesting crossroads where the development of custom-built open-source multi-wavelength (MWL) optics [19-21] has revolutionized the field by generating new information on macromolecular complexes in solution at unprecedented resolution [22]. Initially, the access to such technology was limited to a handful of laboratories that had the technical expertise to build the necessary hardware into existing ultracentrifuges. Excitingly, with the advent of the latest generation of Beckman AUC instruments, the Optima^™ series, which has appeared in the market in 2017, more laboratories now potentially have access to a commercially available instrument with MWL detection and analysis capabilities in each experiment. The number of publications from studies using these new instruments remains low, mostly centered around characterization of protein–protein interactions [23] and none, till date, on RNA–protein interactions. Another parallel exciting advancement in the field has been the development of the open-source data-analysis software, Ultrascan III [24], which can be used for analyzing single-wavelength (SWL) and MWL data from both the older Proteome Lab^™ and the latest Optima^™ AUC instruments. A recent series of articles appearing in a special issue of the European Biophysics Journal [25] highlight several major technological breakthroughs in AUC over the last decade, ranging from instrumentation and sample detection to data-analysis software and structural modeling. In this chapter, we elaborate on the application of MWL-AUC to studies of RNA and RNA–protein interactions and highlight the importance of MWL analysis in relevant areas. While additional optical systems exist (the Aviv fluorescence detector and Rayleigh interference optics), this chapter focuses on the applications of MWL-AUC because of the favorable spectral characteristics of proteins and RNA, which make these optics particularly useful for studying their interactions.

The basic experimental premise for both SWL and MWL AUC experiments is the same. As shown in Fig. 1a, a typical AUC experiment involves subjecting a solution, contained within sector-shaped chambers in an ultracentrifuge cell, to centrifugal forces that lead to sedimentation of the solute particles, away from the rotor center and toward the bottom of the cell (Fig. 1b). Consequently, a solute-depleted region is created toward the top of the cell, near the meniscus, and a solvent–solution boundary is generated. This emerging boundary gives rise to a concentration gradient which triggers diffusional transport in addition to sedimentation transport. The Optima^™ AUC’s MWL optical system is designed to measure the intensity of monochromatic light between 190 and 800 nm passing through the windows of the ultracentrifuge cells, which allows real-time monitoring of the movement of the solvent–solution boundary in each cell by measuring the amount of absorbed light by the solution. If the experiment is allowed to proceed indefinitely, an equilibrium state will be reached where the material collecting at the outward facing wall of the cell reaches very high concentration at the bottom of the cell. The large concentration gradient formed at the bottom generates strong diffusional transport in the opposite direction of the sedimentation transport. Eventually, diffusion and sedimentation transport cancel each other, an equilibrium state is reached, and net flow of solute is no longer observed.

Fig. 1.

Analytical ultracentrifugation: from the optical system to data output. (a) Overview of the optical system and rotor arrangement in the Beckman analytical ultracentrifuge (courtesy of Beckman Coulter, Inc. [Adapted from Greg Ralston; Introduction to Analytical Ultracentrifugation; http:/www.beckman.com/literature/Bioresearch/361847.pdf]). (b) One AUC cell is zoomed in, to show the schematic of a centerpiece inside an assembled cell. (c) Examples of SWL scans obtained from an AUC run, showing progressive movement of the boundary in a radial direction toward the bottom of the cell. Although the original scans are acquired in the intensity mode, the data, after processing, are presented as (pseudo)absorbance versus radius (cm) scans, as shown. (d) Data from MWL-AUC analysis of an RNA–protein interaction. A single time point of a four-dimensional data set as a function of time, radius, and wavelength is shown. The sample meniscus is at the left edge ([a, b adapted from Methods in Enzymology, Volume 469, Using analytical ultracentrifugation (AUC) to measure global conformational changes accompanying equilibrium tertiary folding of RNA molecules, Somdeb Mitra, Pages 209–236, Copyright 2009 [5], with permission from Elsevier; (d) adapted with permission from Analytical Chemistry 2017, 89 (1), Spectral and Hydrodynamic Analysis of West Nile Virus RNA—Protein Interactions by Multiwavelength Sedimentation Velocity in the Analytical Ultracentrifuge, Zhang, J. et al, Pages 862–870. Copyright (2017) American Chemical Society])

During the approach-to-equilibrium phase of the experiment, which is also called a sedimentation velocity (SV), the most useful data are obtained [26-28]. Molecules with more compact and spherical shapes, with a smaller hydrodynamic radius (Stokes radius or $R_{\mathrm{H}}$), travel faster compared with molecules that are less compact and elongated, with a larger $R_{\mathrm{H}}$ [10]. Denser molecules with the same hydrodynamic radius experience less buoyancy and also travel faster. The experimentally recorded data include the time-dependent changes of the concentration profile as a function of the distance from the center of the rotor (x), either monitored at a fixed wavelength for SWL experiments (Fig. 1c) or at a series of wavelengths for MWL experiments (Fig. 1d). The time-dependent changes in the position of the boundary of the concentration gradient ($\partial c∕\partial x$) are then fit to finite-element solutions of the Lamm equation, which describes the time-dependent evolution of the radial concentration gradient ($\partial c∕\partial t$):

$$\frac{\partial c}{\partial t}=D\left[\frac{{\partial }^{2}c}{\partial x^2}+\frac{1}{x}\frac{\partial c}{\partial x}\right]-s{\omega }^2\left[x\frac{\partial c}{\partial x}+2c\right]$$

where $\omega$ is the angular velocity, $c$ is the concentration of the solute, and $s$ and $D$ are two fundamental hydrodynamic properties of a molecule, its sedimentation, and diffusion coefficients, respectively [29-31].

The sedimentation coefficient, defined as the ratio of the linear velocity ($u$) to the angular acceleration (${\omega }^{2}x$) of the solute particle ($s=\frac{u}{{\omega }^{2}x}$), determines the speed of sedimentation under a given set of solution conditions. It is proportional to the molar mass ($M$) of the solute particle and inversely proportional to the frictional coefficient ($f$), which determines the viscous drag that a particle experiences in a direction opposite to its movement. Hence, the sedimentation coefficient, $s$, can be expressed as

$$s=\frac{m(1-\overline{\nu }\rho )}{Nf}$$ (1)

where $\overline{\nu }$ is the partial specific volume of the solute particle, $N$ is Avogadro’s number, and $\rho$ is the density of the solvent. According to Stokes law, the frictional coefficient of a particle ($f$) is proportional to the hydrated radius ($R_{\mathrm{H}}$) of the particle and the viscosity ($\eta$) of the solvent:

$$f=6\pi \eta R_{\mathrm{H}}$$ (2)

The diffusion coefficient ($D$) of a spherical solute particle is a function of the universal gas constant ($R$), the absolute temperature ($T$), and the frictional coefficient ($f$), according to the following relationship:

$$D=\frac{RT}{Nf}$$ (3)

Dividing Eq. 1 by Eq. 3 yields the following expression for the mass of a solute, provided that the partial specific volume and solvent density are known:

$$M=\frac{sRT}{D(1-\overline{\nu }\rho )}$$ (4)

Furthermore, we can define a hypothetical sphere with radius $R_0$ that has the same volume as the sedimenting particle:

$$R_0={\left(\frac{3M\overline{\nu }}{4\pi N}\right)}^{1∕3}$$ (5)

Using Stokes law, a frictional coefficient, $f_0$, can be derived for this particle:

$$f_0=6\pi \eta R_0$$ (6)

Dividing Eq. 2 by Eq. 6 gives rise to the frictional ratio or shape factor $F$(defined as the ratio, $f∕f_0$), which denotes the anisotropy of the particle. $F=1.0$ represents a perfect sphere, while higher anisotropies signify increasingly non-globular or asymmetric shapes [9, 32].

These relationships show how measurements of $s$ and $D$, by SV-AUC, report on the mass, the anisotropy, and the hydrodynamic radius of the sedimenting particle [26, 33]. Readers who are interested in more detailed theoretical treatments of hydrodynamic variables governing the bulk transport of solutes under centrifugal force are referred to [26, 33-36].

The premise of detecting counterion or protein-induced RNA tertiary compaction or global conformational change by SV-AUC is to monitor the change in the overall size or anisotropy of the RNA molecule. A series of measurements of $s$ and $D$ at varying counterion or protein concentrations reports on the values of $f$ and hence, $R_{\mathrm{H}}$ at each condition. If the investigator is monitoring global conformational changes of an already compact structure, such topological alterations can be detected as long as there is a change in the anisotropy [34].

The additional advantage offered by MWL AUC, in combination with new data analysis algorithms and high-performance computing, is the ability to spectrally resolve multiple interacting components in solution that differ in their molar absorptivity [19, 22, 23]. Therefore, SV-AUC experiments with MWL technology can directly report on molecular weights and stoichiometries of interacting partners [37]. In order to achieve clear spectral separation of the solutes, an investigator can either make use of the intrinsic ultraviolet–visible (UV–vis) absorptions of the macromolecules being studied [22, 23] or use macromolecules that have been tagged with a small fluorescent dye or fused to a fluorescent reporter protein [38].

Since the intrinsic UV absorption of RNA and proteins is significantly different, information on the sedimenting behavior ofeach ofthese components can be obtained from the same experiment by collecting the absorption signal over an entire accessible spectral range (depending on buffer composition, up to 215–300 nm) and then computationally deconvoluting the signals from each molecule. An excellent example of this approach has been described in detail in a recent publication [15]. To briefly summarize, this study investigated the interaction between a model RNA (a stem loop structure from the 3′ end of the negative-strand genomic RNA of the West Nile virus, WNV-RNA) and the human T-cell-restricted intracellular antigen-1-related protein (hTIAR). In each run of a MWL-AUC experiment, hTIAR was added at different molar ratios to a fixed molar concentration of WNV-RNA, which, for each titration point, yielded four-dimensional datasets in the form of intensity (converted to pseudo-absorbance) as a function of time, radius, and wavelength (Fig. 2a). Using the nonnegatively constrained least squares (NNLS) algorithms [39], the 4D data were then decomposed into three sets of individual absorption components: RNA, protein, and buffer (if the buffer has any absorbing component in the chosen spectral range) (Fig. 2b). Treating the buffer absorption as time and radius invariant signal, the time-dependent changes of the separated molar concentrations of the WNV-RNA and the hTIAR were then plotted as a function of the radial distance from the center of the rotor and separately stored as two three-dimensional datasets (Fig. 2c) (pseudoabsorbance vs. radius and time). Then each dataset was initially analyzed independently, using two-dimensional spectrum analysis (2DSA) [40], implemented in UltraScan [24], reporting the $s$, $D$, and $F$ values of the RNA and the protein separately. Finally, global fits of the RNA and protein datasets were performed, using genetic algorithm-Monte Carlo analysis [41, 42], to identify all distinct hydrodynamic species in solutions of RNA and proteins at varying molar ratios (Fig. 2d-g). Identical $s$ values for hydrodynamic species observed in both the RNA and protein signals indicate the presence of complexes. Comparing the molar concentrations of these complex species, one obtains the absolute amounts of RNA and protein present in the RNA–protein complexes which uniquely identify their ratios [15, 37]. These ratios are then used to determine the contributions of protein and RNA to the partial specific volume of the complex, which in turn permits a more precise prediction of the molar mass from the sedimentation and diffusion coefficients. Both the molar concentration contribution ratios and the molar mass, together, will then uniquely identify the complex stoichiometry.

Fig. 2.

MWL-AUC data and its analysis. Primary four-dimensional MWL-AUC data (a) (only one time point is shown; the meniscus is visible at the left edge of the radial range) are decomposed, by the nonnegatively constrained least square (NNLS) algorithms, into their spectral constituents (b), resulting in spectrally separated three-dimensional (3D) datasets (c). Each dataset only reflects the hydrodynamic contributions of one of the constituents, hTIAR and WNV-RNA, which have different chromophores in this case. The TCEP absorbance contribution must be considered in the spectral decomposition; however, since it does not sediment, it only contributes to the baseline, and a hydrodynamic analysis is not necessary. Panels d–g depict global 2DSA-Monte Carlo models obtained from decomposed RNA and hTIAR datasets, at different molar mixing ratios. Separate hTIAR and RNA controls (d), hTIAR:RNA 3:1 (e), hTIAR:RNA 6:1 (f), and hTIAR: RNA 10:1 (g). All plots show the protein data in red and the RNA data in blue. A clear shift to higher-molecular-weight complex formation is observed as the protein concentration is increased. At the highest ratio (10:1, g), the excess unbound protein appears as a peak at the same position as the free hTIAR control (d). Broad peaks indicate the presence of a reaction boundary, and absence of a free hTIAR peak, except when added in excess, indicates tight binding between hTIAR and WNV RNA. The control experiments and the 10:1 mixtures were performed at hTIAR and RNA loading concentrations of 11.6 μM and 1.16 μM, respectively. All plots are shown as g(s) distributions with dC/ds plotted on the Y-axis in the indicated units [Adapted with permission from Analytical Chemistry 2017, 89 (1), Spectral and Hydrodynamic Analysis of West Nile Virus RNA—Protein Interactions by Multiwavelength Sedimentation Velocity in the Analytical Ultracentrifuge, Zhang, J. et al, Pages 862–870. Copyright (2017) American Chemical Society]

In this chapter, we provide sufficient information for investigators to design, conduct, and analyze data from an SV-AUC experiment to measure counterion–protein-mediated tertiary compaction of RNA molecules, starting from an ensemble that possesses native secondary structure, as well as to detect global conformational changes in a folded RNA molecule upon ligand–protein binding. We will focus our discussions on the new Optima^™ instrument with its built-in MWL technology to highlight the additional insights into RNA–protein interactions that can be obtained with the new instrument. One of the authors (B.D.) offers recurring workshops on problem solving with AUC to help new users gain confidence in using this new technology effectively (for more information, please visit htp://ultrascan.aucsolutions.com).

2. Materials

To ensure the quality and integrity of the RNA samples, all buffers and solutions are prepared in ribonuclease (RNase)-free, deionized ultrapure water (resistivity 18.2 MΩ cm at 25 °C, TOC < 10 ppb) obtained from a Milli-Q^® water purifier (Millipore). Molecular biology grade, nuclease-free buffer components are purchased from Ambion^® (ThermoFisher Scientific^®), Fisher Scientific^®, or Sigma-Aldrich^®, and the final buffer stock solutions are filtered through 0.2 μm filters (Pall Corporation^®). Buffer exchanges, when needed, are performed using Amicon^® Ultra-15/50 (MilliporeSigma^®) Centrifugal Filter Units or EMD Millipore Ultra Centrifugal Filters, 3 kDa Ultracel, and 0.5 mL (Cole-Parmer Scientific Experts^®), of the appropriate molecular cutoffs. All equipment are routinely cleaned with RNaseZap (Ambion^®/ThermoFisher Scientific^®) and only autoclaved glasswares are used for storing reagents.

2.1. RNA Transcription

1. A bacterial plasmid which is constructed by cloning the DNA sequence corresponding to the RNA of interest, downstream of a T7 promoter sequence. Changing the first nucleotide after the T7 promoter (+1) to G greatly improves transcription yield. Usually, the 3′ side of the RNA sequence is flanked by an HδV ribozyme sequence to generate homogeneous 3′ ends [43] (see Note 1).

2. Polymerase chain reaction (PCR) machine (thermal cycler) and forward and reverse primers for amplification of the linear transcription template from the plasmid. The reverse primers are often designed to carry 2′-O methylation modification at the first two positions (see Note 2).

3. MEGAscript^® T7 Kit (Ambion^®) for in vitro transcription of RNA.

4. Denaturing gel solution containing 5% urea-acrylamide/bisacrylamide (19:1) and 7 M urea in 1× Tris–borate–EDTA (TBE) buffer.

5. Ammonium persulfate (Sigma-Aldrich^®) dissolved in water to make a 10% (wt/vol) solution.

6. N,N,N′,N′-tetramethylethylenediamine (TEMED >97%; Fisher Scientific).

7. Vertical gel running apparatus (Thermo Scientific, Owl Dual-Gel Vertical Electrophoresis Systems), glass plates (10 cm × 10 cm; one regular one notched), combs and spacers (1.5 mM thick), and a high-voltage programmable power supply (e.g., Thermo Scientific, Owl EC3000XL).

8. Hand-held UV torch and Fluor-coated TLC plate (Ambion^®) for UV shadowing.

9. Fluor-coated TLC plate (10 × 10 cm; Invitrogen^™ Ambion^™).

10. Elution buffer: 10 mM sodium cacodylate, 0.1 mM EDTA, 500 mM KCl (or NaCl), pH 7.3.

11. 100 and 70% ethanol.

12. Refrigerated tabletop centrifuge (e.g., Eppendorf 5402 Refrigerated Centrifuge Benchtop).

13. Lyophilizer/SpeedVac (e.g., Savant DNA120 SpeedVac Concentrator DNA–RNA Sample Dryer) for drying RNA pellets.

14. RNA stock solutions of 10–50 μM concentration in CE buffer (see Note 3).

2.2. Protein Expression and Purification

1. Proteins used in these experiments are almost always expressed in E. coli BL-21 (DE3) RIPL Codon Plus electro-competent cells (Agilent technologies), codon-optimized for eukaryotic gene expression.

2. All the proteins used in our studies are tagged with either a standard N-terminal 6X-His tag or a C/N-terminal Intein–Chitin binding domain (CBD) tag (IMPACT^™: Intein Mediated Purification with an Affinity Chitin-binding Tag; NEB^®). For N-His-tagged proteins, the corresponding genes are generally cloned in a suitable pET vector (e.g., pET-45b(+); Novagen^®) and for the C/N-Intein-CBD-tagged proteins, the corresponding genes are cloned in the pTYB1-2 (for C-tag) or pTYB11-12 (for N-tag) vectors (NEB^®).

3. HisPur^™ Ni-NTA Resin (Thermo Scientific^™) or Chitin Resin (NEB^®) for affinity purification of His-tagged and Intein-CBD-tagged proteins, respectively.

4. Luria-Bertani (LB) broth and agar powder from ThermoFisher Scientific^®.

5. Isopropyl-β-d-thiogalactoside (IPTG) (Sigma-Aldrich^®), stored as 10 mM stock at −20 °C.

6. Ampicillin (Sigma-Aldrich^®), stored as 100 mg/mL stock at −20 °C.

7. Any standard refrigerated floor-centrifuge that can accommodate a JA-10 (or similar) and a SS-34 (or similar) fixed-angle rotors.

8. Any standard model of Sonicator (e.g., Fisherbrand^™ Sonic Dismembrator) or French Press (e.g., French Press G-M^™; GlenMills^®) for bacterial cell lysis.

9. Amicon^® Ultra-15 or Amicon^® Ultra-50 Centrifugal Filters (MilliporeSigma^®), for protein concentration and buffer-exchange reactions of volumes up to 15 or 50 mL, respectively. Choose appropriate molecular weight cutoff, 2–3 times smaller than the molecular weight of the protein being used in the experiments.

10. EMD Millipore UFC500324 Amicon Ultra Centrifugal Filters, 3 kDa Ultracel, 0.5 mL (Cole-Parmer Scientific Experts^®) for buffer-exchange reactions of smaller volumes (up to 500 μL), immediately preceding RNA–protein MWL AUC experiments.

2.3. Buffers for RNA and Proteins

1. CE buffer: 10 mM sodium cacodylate trihydrate ((CH₃)₂As-O₂Na.3H₂O; Sigma-Aldrich^®), 0.1 mM EDTA, pH 7.3. A 10× stock solution is made and stored in a 4 °C refrigerator. The CE buffer is the starting point for experiments in which RNA folding is induced by monovalent ions, like sodium or potassium [10, 44] (see Note 3). It is also the buffer in which the RNA samples are routinely dissolved and stored.

2. CEK buffer: 10 mM sodium cacodylate, 0.1 mM EDTA, 100 mM potassium chloride, pH 7.3. This is the buffer routinely used while conducting Mg²⁺ (or other divalent ion) induced RNA folding experiments [45-47] (see Note 4). A 10× stock solution is made and stored in a 4 °C refrigerator.

3. We purchase ribonuclease-free solutions of MgCl₂ (1 M), EDTA (0.5 M, pH 8.0), KCl (2 M), and NaCl (5 M) from Ambion^®.

4. Standard protease inhibitors: 1 tablet Roche^® Complete EDTA-free protease inhibitor cocktail tablet per 50 mL buffer, 1 mg/mL leupeptin, 1 mg/mL aprotinin, 1 mg/mL pepstatin, 1 mM benzamidine.

5. SUPERase• In^™ RNase Inhibitor (20 U/μL; Invitrogen^™).

6. Protein lysis buffer (common to His-tagged and Intein–CBD-tagged proteins): 20 mM Hepes·KOH or potassium salt of 4-(2-hydroxyethyl)-1-piperazineethanesulfonic acid (C₈H₁₇KN₂O₄S; Sigma-Aldrich^®), pH 7.6, 0.5 M KCl, 0.1% Triton X100 (from 1% stock; Invitrogen^™), 1 mM EDTA, standard protease inhibitors.

7. Ni-NTA equilibration buffer: 20 mM Hepes·KOH, pH 7.4, 0.3 M KCl, 10 mM imidazole (from freshly made 1 M stock; purchased in crystalline form from Sigma-Aldrich^®).

8. Ni-NTA wash buffer: 20 mM Hepes·KOH, pH 7.4, 0.3 M KCl, 25 mM imidazole.

9. Ni-NTA elution buffer: 20 mM Hepes·KOH, pH 7.4, 0.3 M KCl, 250 mM imidazole.

10. Intein wash buffer: 20 mM Hepes·KOH, pH 7.4, 1 M KCl, 0.1% Triton X100, 1 mM EDTA.

11. Intein cleavage buffer (−DTT): 20 mM Hepes·KOH, pH 8.0, 0.5 M KCl, 1 mM EDTA (to initiate the Intein cleavage reaction, DTT is added later to a final concentration of 50 mM).

12. Protein storage buffer: 20 mM HEPES, pH 7.6,200 mM KCl, 10 mM beta-mercaptoethanol (2-mercaptoethanol, ≥99.0% stock; Sigma-Aldrich^®), and 10% glycerol (from ACS certified, ≥99.5% stock; Fisher Scientific^®). A 10× stock solution is usually made without the BME and stored in a 4 °C refrigerator. BME is added freshly, to a final concentration of 10 mM, right before diluting the buffer to 1× concentration.

13. AUC phosphate buffer: 10 mM sodium phosphate, pH 7.0, 50 mM NaCl, and 1 mM TCEP·HCl (Tris(2-carboxyethyl) phosphine hydrochloride, BioUltra, ≥98%; Sigma-Aldrich^®) (see Note 5). Stored at 4 °C as a 10× stock solution.

2.4. Analytical Ultracentrifugation

1. Beckman Optima^™ AUC by Beckman Coulter (see Note 6).

2. A four-slot An-60 Ti rotor with max. angular velocity of 60,000 rpm (alternatively, an eight-slot An-50 Ti rotor can also be used with a max. angular velocity of 50,000 rpm).

3. AUC cells (Beckman Coulter): Epon charcoal centerpieces with sector-shaped channels of 1.2 cm optical path length (Fig. 1b), quartz or sapphire windows, window liners and gaskets, upper and lower window holders, cell housing, screw ring, red polyethylene gasket sheet for sealing sample loading ports, brass screws for closing loading ports. It is recommended to have a full complement of 4 or 8 cells, one for each rotor hole, on hand so a maximum number of conditions (8 or 16, depending on rotor type) can be examined under identical run conditions.

4. A counterbalance (Beckman Coulter) for radial calibration (see Note 7).

5. 1 mL Disposable syringe with 30-gauge hypodermic needles.

6. Analytical Torque Stand and Wrench and screwdrivers (Beckman Coulter).

7. RNaseZap (Ambion^®) solution for cleaning the AUC cells.

8. Kimwipe^® (Kimberly-Clark^®) extra low-lint 2-ply paper wipers for delicate surfaces.

2.5. Software for AUC Run Setup, Data Acquisition, and Data Analysis

1. UltraScan-III (http://ultrascan.aucsolutions.com).

2. Origin^® (Origin Labs^®) or Prism (GraphPad^®) or similar software for data plotting and curve fitting.

3. Methods

3.1. RNA Sample Preparation

1. Using a PCR-generated linear DNA template containing the RNA sequence downstream of a T7 promoter, transcribe the RNA in vitro by run-off transcription, using components of the MEGAscript^® T7 Kit (Invitrogen^™ Ambion^®) or using a recombinant RNA polymerase, expressed and purified in our laboratory, with appropriate buffer conditions [43] at 37 °C for 2 h (see Note 8).

2. If a ribozyme is not used to obtain homogenous ends, the transcribed RNA can be directly purified by using spin-column-based methods, such as MEGAclear^™ Transcription Clean-Up Kit (Invitrogen^™ Ambion^®).

3. When more than one RNA species is present in the transcription reaction, separate the reaction products on a denaturing 5% urea:polyacrylamide gel (19:1), and after the run is complete, transfer gel from the glass plates on to a Fluor-coated TLC plate (covered with plastic wrap), illuminate it from overhead with short wavelength UV, and cut out the gel piece containing the RNA band of interest with a clean razor blade (see Note 9).

4. Crush the gel piece by passing it through a 5 mL syringe, soak the pieces in the elution buffer in a 15 mL Falcon^™ tube, and rock it overnight in a cold room, to elute the RNA.

5. Next morning (or after ~6–8 h), give the solution a quick spin to let the gel pieces settle to the bottom and pipette out the supernatant carefully. Precipitate the RNA from the supernatant solution by adding 100% ethanol (3× volume of the supernatant), incubating at −20 °C (or in dry ice) for 1 h, and spinning down the samples in a refrigerated tabletop centrifuge at 14,000 rpm for 1 h at 4 °C.

6. Rinse the pellet by spinning twice (14,000 rpm for 30 min at 4 °C) with 70% ethanol, dry it in a desiccator (SpeedVac), and dissolve it in 1× CE buffer to a concentration of 10–50 μM.

7. For RNA–protein studies, right before the AUC experiment, thaw an appropriate volume of the stored RNA sample and buffer-exchange it five times into the 1× AUC Phosphate buffer, using the small (0.5 mL) EMD Millipore UFC500324 Amicon Ultra Centrifugal Filters.

3.2. Protein Sample Preparation

1. For genes cloned with a His-tag in pET vectors or with an Intein-CBD-tag in pTYB vectors, pick a single colony from a plate of transformed BL21-Codon Plus (RIPL) cells, inoculate 5 mL LB broth (containing ampicillin 100 μg/mL) to set up a primary culture, and grow by shaking at 37 °C overnight (or until O.D.₆₀₀ is »1).

2. Inoculate a 1 L LB (+ampicillin) broth with ~100 μL of primary culture and grow cells by shaking at 37 °C until the cells are in the log phase of growth (OD₆₀₀ ~ 0.5–0.8).

3. Induce cells for protein expression by adding IPTG to a final concentration of 0.5 mM and incubate with shaking either at 37 °C for ~8 h or at 15 °C overnight (incubation conditions, including optimum IPTG concentrations, temperature, and duration of expression, to obtain best soluble protein yields should be tested beforehand).

4. Spin down cells in a JA-10 rotor (spin 15 min at 8000 rpm), discard the supernatant, resuspended the pelleted cells in ice-cold lysis buffer, and lyse the cells either by sonication (typically 10–15 cycles of 15–30 s pulse and 30 s gap between pulses, on ice) or using a French Press (two rounds, each at 1800 psi, for efficient lysis).

5. Spin down the lysate in SS-34 type rotors at 13,000 rpm for 30 min at 4 °C, collect the supernatant (always keep it at 4 °C), and proceed to the appropriate column-purification.

6. For general purification strategies of His-tagged proteins, cloned in pET vectors, refer to the manufacturer’s protocol for details (http://www.emdmillipore.com/US/en/product/pET-45b+-DNA-Novagen), and when using the HisPur^™ Ni-NTA Resin, refer to https://assets.thermofisher.com/TFS-Assets/LSG/manuals/MAN0011700_HisPur_NiNTA_Resin_UG.pdf. Briefly, equilibrate the desired amount (typically 5–10 mL) of Ni-NTA resin in a gravity-flow column with the equilibration buffer (2× the bed volume), mixing the lysate with an equal volume of equilibration buffer, and incubate at 4 °C for 30 min, with constant rolling, and drain the flow-through. Wash the resin with the Ni-NTA wash buffer (3× bed volume) and elute the protein by adding the Ni-NTA elution buffer (1× bed volume). If using an FPLC system, briefly, charge a regenerated 5 mL HiTrap chelating column (Amersham^®) with a freshly prepared 100 mM NiSO₄ (Sigma-Aldrich^®), equilibrate with 5× column-volume (CV) of the Ni-NTA equilibration buffer, and load the filtered lysate (filtered sequentially through 5 mm to 0.8 mm filters) onto the equilibrated nickel column at a flow rate of 3 mL/min. Wash the column with Ni-NTA wash buffer (5 CV) and elute with the Ni-NTA elution buffer (2 CV) at 5 mL/min.

7. For purifying Intein–CBD-tagged proteins, refer to the manufacturer’s protocol for details (https://www.neb.com/products/e6901-impact-kit#Protocols,%20Manuals%20&%20Usage).

   Briefly equilibrate ~5 mL of the chitin resin in a 20 mL gravity flow column, with 1 column volume (CV) of the lysis buffer, apply the clarified supernatant to the resin, incubate with gentle shaking at 4 °C for 1 h, drain the flow-through by gravity flow, and wash the resin with 3 CV of the Intein wash buffer. Then wash the resin with 1 CV of the Intein cleavage buffer (without DTT) and elute the protein with ~3× chitin bed volume of the Intein cleavage buffer supplemented with 50 mM DTT.

8. Concentrate the eluted proteins using an Amicon^® Ultra Centrifugal Filter of the appropriate molecular weight cutoff, carry out buffer exchange at least five times with the protein storage buffer, and further concentrate down to the desired concentration (typically 10–50 μM). Aliquot in small volumes, flash-freeze in liquid nitrogen, and store at −80 °C (see Note 10).

9. Right before the AUC experiment, thaw an appropriate volume of the stored protein sample and buffer-exchange it five times in to 1× AUC Phosphate buffer, using the small (0.5 mL) EMD Millipore UFC500324 Amicon Ultra Centrifugal Filters.

3.3. Setting Up Equilibrium Counterion/Protein-Mediated RNA Compaction Reactions

1. To study counterion (monovalent cations, divalent cations, polyamines, etc.) or protein-induced equilibrium RNA compaction, set up each titration point in a separate 1.5 mL micro-fuge tube, such that the total reaction volume is 500 μL, with a final RNA concentration of 0.1–1.0 OD (A₂₆₀). For monovalent and divalent cation titrations, use Tables 1 and 2, respectively, whereas for protein titrations, use Table 3, as a guideline to set up the reactions. We typically use 0.5–0.6 OD as the final RNA concentration, and the optimal loading volume in each sector-shaped compartment of an AUC cell is 460 μL (also see Notes 11 and 12).

2. Prepare sufficient volume of stock solutions of the counterions/proteins (e.g., 2 M KCl, 5 M NaCl, 1 M MgCl₂, 20 μM protein). Titrations should span an entire range over which the equilibrium compaction of the RNA is expected to occur. Generally, this range is 0 to 1.5 M for monovalent ions (M⁺), 0 to 100 mM for divalent ions (M²⁺), and 0 to 1 to 10 μM for proteins (varies considerably depending on the protein being studied).

3. Using Tables 1, 2, and 3 as guidelines, prepare eight titration points (for An-60 Ti rotors), for each SV-AUC run (see Note 7). DO NOT add the RNA to the tubes at this stage. Therefore, at the end of this step, one should have a total of eight reaction tubes containing ($500-{\rm Y}$) μL volume of each of the titration samples, with all the reagents added in except ${\rm Y}$ μL of the RNA. We typically use the CE buffer for M⁺ titrations, the CEK buffer for M²⁺, and the AUC Phosphate buffer for protein titrations (see Notes 3, 4, and 5). The following tables show examples of reaction setups for M⁺ (K⁺), M²⁺ (Mg²⁺) and protein titrations, using 2 M KCl, 1 M MgCl₂, and 10 μM protein stock solutions, respectively.

4. In a separate 1.5 mL tube, take slightly greater than $8{\rm Y}$ μL of the RNA stock solution (${\rm Y}$ is the amount to be added to each of the eight folding reaction points as described above). The slight excess volume accounts for evaporation during the following denaturation step.

5. Heat the RNA solution to 95 °C on a heating block for 1 min (to denature it) and cool it slowly to 50 °C (to allow equilibration to native secondary structure) (see Note 13). During this step, also incubate the reaction tubes containing all the reagents except the RNA, at 50 °C.

6. After briefly centrifuging the RNA solution in a tabletop centrifuge to spin down the condensates on the lid, put them back at 50 °C and pipette out ${\rm Y}$ μL of the RNA solution into each of the three reaction tubes containing the folding buffers at 50 °C. Mix the solutions gently by pipetting up and down and continue incubation at 50 °C for an additional 30 min (see Note 14).

7. Move the reaction tubes from 50 °C to another heat block that is set to temperature at which the equilibrium tertiary structure will be monitored (usually, 25 or 37 °C) and incubate for 1 h to allow the RNA molecules to equilibrate to their native tertiary structures at the given temperature (see Note 15).

Table 1.

Sample reaction setup for monovalent cation titrations (e.g., K⁺; using a 2 M stock)

Final K+ conc.	RNA (10–50 μM)	CE buffer (10×)	KCl (2 M)	H2O	Total volume
$X$ mM	${\rm Y}$ μL for (0.5 OD)	50 μL	0.25× μL	$[500-({\rm Y}+50+0.25X)]$ μL	500 μL

Table 2.

Sample reaction setup for divalent cation titrations (e.g., Mg²⁺, using a 1 M stock)

Final Mg2+ conc.	RNA (10–50 μM)	CEK buffer (10×)	MgCl2 (1 M)	H2O	Total volume
$X$ mM	${\rm Y}$ μL for (0.5 OD)	50 μL	0.5× μL	$[500-({\rm Y}+50+0.5X)]$ μL	500 μL

Table 3.

Sample reaction setup for protein titrations (e.g., using a 20 μM stock protein)

Final protein conc.	RNA (10–50 μM)	AUC phosphate buffer (10×)	Protein (20 μM)	H2O	Total volume
$X$ μM	${\rm Y}$ μL for (0.5 OD)	50 μL	25× μL	$[500-({\rm Y}+50+25X)]$ μL	500 μL

3.4. Setting Up Reactions to Probe Equilibrium Conformational Change of an RNA Upon Ligand/Protein Binding

1. To probe whether binding to a small-molecule ligand or a protein alters the global conformation of an RNA, whose hydrodynamic parameters in the folded state have already been determined, set up eight reactions at a time, each with the fully folded RNA molecule (in CEK buffer +5 mM Mg²⁺) and with eight different molar concentrations of the ligand, with respect to the RNA, to start with (e.g., 0.5:1, 1:1, 2:1) (see Note 16). Table 4, as presented below, may be used as a reference to set up the reactions:

2. For each reaction tube, first add and mix all the components except for the MgCl₂ (5 μL) and the ligand (5× μL), incubate the reactions on a heating block at 95 °C for 2 min, and slowly cool them down to ~ 50 °C. During this step, also incubate the MgCl₂ stock (1 M) at 50 °C and the ligand stock (100 μM) at the final reaction temperature (usually 25 or 37 °C).

3. After briefly centrifuging the reaction tubes in a tabletop centrifuge to spin down the condensates on the lid, put them back at 50 °C and pipette in 5 μL of the 1 M MgCl₂ solution into each of the three reaction tubes containing the RNA samples in CEK buffer at 50 °C. Continue incubation at 50 °C for an additional 15 min.

4. After briefly centrifuging the reaction tubes, transfer them to the heating block set at the final reaction temperature (at which the ligand is also being incubated), allow equilibration for ~ 5 min, and add 5× μL of the 100 μM ligand stock solution. Mix the solutions gently by pipetting up and down and continue incubation at the reaction temperature for an additional 1 h to allow the folded RNA to reach its new equilibrium conformation in the presence of the ligand. Treat the three corresponding blank solutions in exactly the same way.

Table 4.

Sample reaction setup to monitor changes induced by a ligand (100 μM stock)

Final ligand conc.	RNA (10–50 μM)	CEK buffer (10×)	MgCl2 (1 M)	Ligand (100 μM)	H2O	Total volume
$X$ μM	${\rm Y}$ μL for (0.5 OD)	50 μL	5 μL	5× μL	$[500-({\rm Y}+55+5X)]$ μL	500 μL

3.5. AUC Cell Assembly and Loading Samples into the Cells

1. Before each experiment, soak all the components of the cells with a mild detergent (e.g., RNaseZap^™; Invitrogen^™ Ambion^™), wash with ultrapure water, rinse with 70% ethanol, and dry carefully with a low-lint, nonabrasive tissue (e.g., low-lint Kimwipes^®), to prevent scratching of the quartz windows and the epon-charcoal centerpieces.

2. Refer to the instruction manual accompanying the AUC instrument or to Backman Coulter’s website under Product Documentation (“An-50 Ti and An-60 Ti Analytical Rotor, Cells, and Counterbalance”), for detailed instructions on AUC cell assemblies. Briefly place each quartz window in its corresponding holder, with the gasket and the liner, and then make a vertical assembly containing the centerpiece sandwiched between the two windows. Gently slide the metal cell housing over the vertical assembly (see Note 17).

3. Insert the O-ring and tighten the screw ring at the top (make sure the side with the marking “OUT” is facing outside), first by hand and then with the Analytical Torque Stand and Wrench (Beckman Coulter) to 120 inch-pounds, to prevent sample leakage during AUC runs.

4. With the screw ring facing the investigator, and the cell in a horizontal position with the fill holes at the top, load 460 μL of the reaction sample into each channel. Looking at the screw ring, and the fill holes at the top, the left channel will be marked “B” and the right channel will be marked “A” in the instrument (see Note 18). Pipette carefully to avoid air bubbles and be careful not to touch the channel surface with the hypodermic needle to avoid damaging the centerpieces.

5. After loading all four pairs of reaction samples, number the aluminum housing of each cell with a marker and note which cell and channel corresponds to each sample.

6. Cut small circular polyethylene gaskets and seal the loading ports of the centerpieces by first placing a gasket piece on each port and then gently tightening the brass screws over them.

7. Clean the windows on either side using brief bursts of compressed air.

3.6. Loading AUC Cells into the Rotor

1. While the experimental samples are incubating at the folding temperature, it is advisable to equilibrate the rotor to the experimental temperature to minimize delays at the beginning of the experiment. For experiments to be performed at 4 °C, it is recommended to temperature-equilibrate the rotor in a cold room or refrigerator, or inside the Optima^™ AUC set to the desired temperature and under vacuum.

2. Once the samples and blanks have been loaded, measure the weight of the sealed ultracentrifugation cells by balancing two cells against each other to minimize weight differences between opposing pairs of cells. Ideally, weight differences are between 0 and 0.2 g to maintain the rotor balance within tolerance, should one of the cells leak. If a leak occurs, the weight difference should not exceed 0.5 g and the experiment should be stopped immediately to prevent centerpiece damage.

3. Take out the rotor from the chamber, place it on the rotor stand (see Note 19), and insert the balanced cells across from each other, with at least one cell in hole “4” (for An-60 Ti rotors) or “8” (for An-50 Ti rotors). Place the remaining pair (s) of balanced cells at positions across from each other in the remaining slots. Make a note as to which samples (based on the numbering outside the ultracentrifuge cell) are placed into each rotor hole. The screw ring-sides should be facing upward, and the screw-capped fill-holes should point to the rotor center (see Note 20).

4. Very carefully tilt the entire loaded rotor sideways on a bench, and while looking from the bottom, use the supplier’s alignment tool to perfectly align the markings on the lower edge of the metal cell-housings with the lines below each slot in the rotor. Note the rotor serial number which is needed for the data acquisition protocol.

5. Place the rotor into the centrifuge chamber, and swivel the UV-vis periscope into position such that the magnet lock is engaged. Place the lid on top of the rotor chamber, close the sliding door, and turn on the vacuum pump (see Note 21).

6. Allow the rotor to re-equilibrate to the run temperature until the desired temperature has been reached and is stable before accelerating the rotor.

3.7. Experimental Design

A critically important step in the experimental process is the development of a carefully crafted experimental design that best addresses the questions of the experimentalist. AUC experiments, and in particular MWL-AUC experiments, are not trivial to perform and design and require detailed planning to maximize information content of the resulting data. In addition to the preparation of the samples, a number of instrument variables affecting the experimental outcome need to be optimized, including temperature, rotor speed, experiment duration, wavelength(s) of observation, frequency of scanning, and solute concentration. We outline here some of the important factors that an investigator must consider prior to setting up an AUC run:

1. Temperature: For temperature settings, typical considerations include the stability of the sample (4 °C favors stability) or the need to maintain physiological conditions (37 °C). Although it was difficult to conduct experiments at temperatures above 25 °C with the older oil-diffusion pumps of the Proteome Lab^™ instruments, the turbomolecular pumps used in the Optima^™ AUC make it easier to conduct higher temperature experiments. If no other requirements exist, a 20 °C temperature avoids the need to convert water density and viscosity corrections to standard conditions (although this is handled automatically by UltraScan for experiments run at temperatures other than 20 °C).

2. Rotor speed: The rotor speed can be modulated to enhance certain characteristics of the experiment. For example, a higher rotor speed will favor better resolution of the sedimentation coefficient, useful for the characterization of heterogeneity and also for the separation of similar species. On the other hand, a slower rotor speed allows more time for the sample to diffuse before it is pelleted, improving the observed diffusion signal [48]. High-quality diffusion information is required for density, molar mass, or anisotropy determinations. If sufficient sample and instrument time are available, both a high-speed and a low-speed experiment can be performed and globally fitted with UltraScan to combine both optimized signals.

3. Experimental duration: The duration of the experiment should be chosen such that at the selected rotor speed the sample is sedimented long enough to approach the equilibrium state. It is important to realize that longer solution columns and hence, longer sedimentation distances also improve resolution. Hence, it is often better to run a more dilute sample with a higher loading volume than a smaller volume of a more concentrated solution. The length of the solution column, of course, also affects the length of time until equilibrium is approached. This varies greatly depending on the experimental needs and the sample characteristics. For highly anisotropic molecules, like nucleic acids, additional time is required to reach an equilibrium state, whereas for small, globular proteins with larger diffusion coefficients, faster rotor speeds can be considered to obtain an acceptable resolution of the sedimentation coefficient as well as good diffusion signal. For multi-wavelength experiments, additional considerations come into play: The experiment needs to be collected over a sufficiently long time to support collection of a sufficient number of scans for each wavelength. In such a case, one may choose to extend the run-time and sacrifice sedimentation-resolution to increase the data quantity by collecting additional scans.

4. Frequency of optical scans: In most cases, the scan frequency should be set to collect as much data as possible. The intrinsic signal from the solutes in the experiment increases linearly with multiple observations, while the stochastic noise only increases with the square root of the number of observations, increasing the distance between the two metrics with additional observations. Hence, the signal-to-noise ratio (SNR) improves through the collection of more scans, although there is a limited return at some point when the SNR improves so little that it is not worth to extend the instrument time and expend the additional computational effort for fitting additional scans (larger datasets take more CPU time to fit).

5. Concentration of the analyte: Nucleic acids, which are polyanions that may carry unbalanced negative charges from backbone phosphates, depending on ionic strength, could exhibit considerable concentration-dependent nonideality arising from anisotropic shape and charge–charge repulsions between proximal molecules. If these charges are not shielded by cations, the repulsion leads to concentration-dependent nonideality which complicates analysis considerably and therefore should be avoided. Lowering concentration, or increasing ionic strength, or both, often remedies these problems.

6. Instrument optical settings: The factors to be considered for selecting scanning wavelengths include the light intensity produced by the Xenon flash lamp and the dynamic range of the detector (a photomultiplier system). The overall concentration of the analytes should be adjusted to match this range, with 0.5–0.6 OD typically producing the optimal signal-to-noise ratio (see Notes 11 and 12).

7. Molar extinction coefficients: For MWL experiments, the molar extinction coefficient of the analyte(s) as a function of wavelength should be carefully considered. Prior to mixing samples, pure spectra for each sample should be collected, ideally by diluting the analyte with a non-absorbing buffer like sodium phosphate, which allows experiments to extend to 215 nm without significant background absorbance (Fig. 3). Such pure spectra can later be used to deconvolute composite spectral patterns using UltraScan. For multi-analyte solutions, consider that the molar extinction coefficients at each wavelength are mostly additive, assuming that hypochromic effects are minimal (see Note 5).

8. Wavelength selection for RNA–protein interactions: For cases where protein and RNA interactions are to be studied, it is often difficult to match absorbances in the 260–280 nm range for proteins and RNA due to the strong discrepancy in their molar extinction coefficients. Especially if stoichiometric measurements are performed, the size of each molecule plays a significant role. Generally, due to the much higher molar extinction of nucleic acids around 260 nm compared with proteins at 280 nm, protein absorbance can be negligible in comparison with the nucleic acid and signals cannot be properly resolved. If a phosphate buffer is used, it is often possible to obtain sufficient protein absorbance by simply shifting the wavelength range to exploit the backbone peptide bond extinction. Each peptide bond adds significantly to the absorbance between 215 and 230 nm producing an absorbance that is proportional to the protein size (see Fig. 3 for a relative comparison between RNA and protein extinction spectral characteristics). Whenever samples are mixed at different ratios, it is important to consider the relative extinction coefficients for different molecules participating in the reaction as they typically add to potentially move the total extinction outside of the dynamic range of the detector. In the experimental design, it should be attempted to maintain total absorbance always below 1 OD at any wavelength to be scanned.

Fig. 3.

Estimating RNA and protein absorbance for wavelength selection. Typical RNA (blue) and RNA-binding protein (red) extinction profiles, showing good complementarity that aids spectral separation between RNA (260 nm) and proteins (280 nm). Due to smaller molar extinction coefficient of proteins at 280 nm, compared with nucleic acids at 260 nm, the absorbance signal at 280 nm is much smaller compared with that at 260 nm

3.8. Setting Up a Sedimentation Velocity Run and Data Acquisition

Start the UltraScan data acquisition software (US-DA) and make sure it can communicate with the Optima^™ AUC and the UltraScan LIMS database. Refer to the UltraScan user manual to configure the experimental protocol and submit it to the Optima instrument. Several important design choices should be considered as follows:

1. Make sure the rotor temperature is well equilibrated with the experimental run temperature before starting the experiment. The instrument is programmed not to start the rotor until the rotor temperature is within 0.4 °C of the target temperature. This is not sufficient for complete temperature equilibration. An additional 30–60 min of equilibration at the target temperature should be included in the protocol. This delay is set in the US-DA software.

2. Use the UltraScan adaptive space-time finite element method simulator (ASTFEM) [49, 50] to predict the appropriate run duration for your experiment. The US-DA software will automatically calculate the total number of scans available for the determined duration based on the chosen rotor speed. The number of scans that can be collected over a given duration varies significantly as a function of rotor speed since the flash lamp’s flash rate needs to be synchronous with the rotor hole positioning to achieve the maximum scan rate. Refer to the UltraScan ASTFEM simulator manual for proper operation of this software.

3. Select investigator, instrument, rotor, rotor calibration, cell type and window type, and solutions (containing preselected buffer and analyte entries) from the relevant database menus and populate the optical system and range settings for radial scan limits (typically 5.75–7.25 cm), and program the selected wavelength(s), depending on SWL or MWL operation. For MWL, design the experiment such that at least 30–40 scans are collected per wavelength to obtain a good statistical representation of the experimental data (see Note 22). It should be kept in mind that both channels in one cell can contain different samples, so even if only one cell is scanned, one can still compare two different conditions in a single multi-wavelength experiment.

4. Submit the experimental protocol to the Optima AUC and select the protocol from the Optima’s touch screen. Start the protocol and return to the computer to monitor the experiment remotely from the US-DA software terminal. In particular, monitor the meniscus position from each channel during the first 2–4 scans to detect any leaks in any channel. The telltale sign of a leak is that for a given channel, the meniscus will appear to move to the right with successive scans. If this is observed, immediately abort the experiment, since unequal menisci on either side of the centerpiece-septum can generate significant pressure differentials that could damage the centerpiece.

5. At the end of the experiment, the software will automatically retrieve the data from the instrument and convert it to the Open AUC data format used in the UltraScan software [21]. At this point, data analysis may commence.

6. After the SV run is complete and the rotor stops spinning, turn off the vacuum pump and let the pressure return to ambient conditions. Open the door and remove the lid and retrieve the rotor. Remove the cells from the rotor, disassemble the parts, and clean them thoroughly with RNaseZap^® and ultrapure water before assembling them for loading the next set of samples (see Note 17).

3.9. Analyzing SV Data to Obtain Hydrodynamic Properties

Several software are available for analyzing SV data. These software include DCDT+ [51], SEDPHAT [52, 53], LAMM [54], SVEDBERG [55], SEDFIT [56, 57], SEDANAL [58], and UltraScan [24]. We focus on the use of UltraScan because it is the only software that can take advantage of high-performance computing to handle the much higher data density generated by the Optima^™ AUC instrument for MWL-AUC experiments and fits intensity data directly without the requirement for scan-pair subtractions that amplify the stochastic noise by a factor of ~1.4 (see Note 23). Please note that investigators who are using the older Proteome Lab^™ XL instruments can also use UltraScan for data analysis as the software is compatible with both the new and the older data file formats.

3.9.1. Steps Common to MWL and SWL Data Analysis

1. After uploading of data by the US-DA software is complete, edit the data to define an initial estimate of the meniscus position and the outer limit of the data at the bottom of the cell. Very steep boundary regions should be excluded either by deleting such scans (toward the beginning of the experiment) or by limiting the data range near the bottom to avoid refractive artifacts occurring in these regions (Fig. 4a).

2. Process the edited MWL (and SWL) data in batch mode by submitting all wavelengths simultaneously to the UltraScan Science Gateway [59, 60] for analysis according to the two-dimensional spectrum analysis (2DSA) workflow [40] (Fig. 4a-d). This process is identical for SWL and MWL experiments and further described in the analysis flowchart at http://www.ultrascan3.aucsolutions.com/sed-veloc-flowchart.php. (For more information, you can also see http://www.ultrascan3.aucsolutions.com/documentation.php.)

3. Check the RMSD values of the processed data in the Finite Element Model Viewer implemented in UltraScan, to confirm the presence of random residuals fits to all the wavelength scans.

Fig. 4.

Steps in the processing of raw data at a single wavelength. (a) RNA data acquired in intensity mode (258 nm) on the Optima AUC with UltraScan. (b) Same data converted to pseudo-absorbance mode and edited to exclude unnecessary scans and meniscus defined, plus top and bottom data range. (c) Residuals from ASTFEM [49, 50] fit with UltraScan. (d) Data after fitting with ASTFEM/2DSA solutions with UltraScan

3.9.2. Steps Specific to MWL Analysis (RNA–Protein Interactions)

For MWL, some additional analysis steps are required. The Optima AUC acquires all wavelength scans sequentially, which interferes with spectral deconvolution, since wavelength scans are not on a synchronous time grid. To solve this problem, the obtained 2DSA iterative fitting models are mapped on to an identical time grid for all wavelengths with the Optima MWL.Fit Simulation module.

1. Replicate actual run conditions for the simulation (i.e., rotor speed, duration, meniscus position) to create a three-dimensional surface for the wavelength decomposition. This is performed with the intrinsic molar extinction profiles of all absorbing species in the solution using the MWL Species Fit module of UltraScan.

2. Generate a separate traditional dataset with dimensions of time, radius, and molar concentration for each species present in the previous step.

3. Analyze these datasets with the 2DSA-Monte Carlo analysis [42] to generate 2D hydrodynamic profiles for each species. If complexes were formed between the species, identical hydrodynamic species will be present in those profiles. To facilitate interpretation of the results, the resulting dC/ds distributions can be overlaid with the function “Combine Discrete Distributions” from the “Utility” menu in UltraScan.

4. Should additional refinement be required, perform a global genetic algorithm analysis by simultaneously fitting all species to a discrete model [15, 41].

3.9.3. Steps for SWL Analysis (CounterIon or Ligand-Induced RNA Compaction)

1. For each titration point, repeat steps 1–3 of Steps common to MWL and SWL data analysis, to obtain the equilibrium $s_{20,\mathrm{w}}$ and the $D_{20,\mathrm{w}}$ of the RNA at each counterion/protein/ligand concentration used for detecting tertiary folding or for measuring ligand/protein binding.

2. Plot the $s_{20,\mathrm{w}}$ values as a function of cation concentration to demonstrate that the RNA molecules assume an increasingly compact shape (higher $s_{20,\mathrm{w}}$) with increasing cation concentration (Fig. 5a). Also, plot the $s_{20,\mathrm{w}}∕D_{20,\mathrm{w}}$ value, at each cation concentration, to check for counterion-dependent RNA aggregation (Fig. 5e) (see Note 24).

3. Both $R_{\mathrm{H}}$ and $F$ report on tertiary compaction of an RNA induced by a counterion or a global conformational change induced by a ligand/protein; the midpoint of the transition and the cooperativity of the folding reaction can be obtained by phenomenological fits (see Note 25) of the plots of $R_{\mathrm{H}}$ (or $F$) vs. counterion concentration to the Hill equation: ${\rm Y}=\frac{K_{\mathrm{d}}^{n_{\mathrm{H}}}{[\mathrm{M}]}^{n_{\mathrm{H}}}}{1+K_{\mathrm{d}}^{n_{\mathrm{H}}}{[\mathrm{M}]}^{n_{\mathrm{H}}}}$, where ${\rm Y}$ denotes the signal quantitated ($R_{\mathrm{H}}$ or $F$), [M] is the counterion concentration, $K_{\mathrm{d}}$ is the midpoint of the equilibrium folding transition, and $n_{\mathrm{H}}$ is the Hill coefficient that indicates the cooperativity of the folding transition. Fit the data using the “nonlinear least squares fitting” routine implemented in standard software like Origin^® and report the best-fit values at 65% confidence intervals (see Note 26).

Fig. 5.

Counterion-induced folding of the Twort group I intron ribozyme by SV-AUC. (a) SV-AUC-derived $s_{20,\mathrm{w}}$ values for Mg²⁺-mediated (filled black circles) and K⁺-mediated (open squares) equilibrium folding of the Twort ribozyme in CEK and CE buffer, respectively. The lower and upper abscissa represent Mg²⁺ or K⁺ concentrations, respectively, while the ordinate represents $s_{20,\mathrm{w}}$ in Svedberg units (S = 10⁻¹³ s); increasing $s_{20,\mathrm{w}}$ depicts that the molecules assume a more compact, faster sedimenting conformation with increasing ionic strength. The insets in panel a depict representative models of the Twort ribozyme at three different ionic strengths, consistent with the measured hydrodynamic properties: (b) CE buffer, extended structures ($s_{20,\mathrm{w}}$ = 4.47 S, $R_{\mathrm{H}}$ = 72.5 Å); (c) CEK buffer, partly compact structures ($s_{20,\mathrm{w}}$ = 7.26 S, $R_{\mathrm{H}}$ = 44.8 Å); (d) 80 mM Mg²⁺ or 2 M K⁺, tertiary folded structure ($s_{20,\mathrm{w}}$ = 8.91S, $R_{\mathrm{H}}$ = 36.5 Å). The blue region around each model schematically depicts its global shape; ellipsoid structures become progressively spherical (f/f_s = 2.8, 1.76, 1.43 for b, c, and d, respectively). The data presented in a have been replotted from [45]. The structural models and the hydrodynamic measurements have been extracted from [75]. (e) Test for concentration-dependent aggregation of the Twort ribozyme, either in the folded state (CEK with 10 mM Mg²⁺; filled black circles) or in the unfolded state (CEK; open squares). The upward slope of the $s_{20,\mathrm{w}}$ values of the folded RNA, with increasing RNA concentration (measured in 260 nm absorbance units), indicates formation of higher molecular weight species, suggestive of aggregation. The constant $s_{20,\mathrm{w}}$ values of the unfolded ensemble, with increasing RNA concentration, suggests that the aggregation is specific to the folded state of the ribozyme [Reprinted by permission from Springer; RNA Folding: Methods and Protocols, Methods in Molecular Biology, Volume 1086, Pages 265–288, Detecting RNA Tertiary Folding by Sedimentation Velocity Analytical Ultracentrifugation, Somdeb Mitra, Pages 862–870. Copyright (2014)]

4. Notes

1. To obtain RNA molecules with precise and homogenous 5′ and 3′ ends, often the DNA sequence corresponding to the RNA of interest is cloned immediately downstream of a DNA sequence encoding a Hammerhead (HH) ribozyme and immediately upstream of a DNA sequence encoding a hepatitis δ virus (HδV) ribozyme. The HH ribozyme sequence is immediately preceded by the promoter sequence for the RNA polymerase used for the in vitro transcription reaction, such that transcription from this template generates the RNA of interest with a 5′ HH ribozyme and a 3′ HδV ribozyme. The ribozyme sequences usually cleave themselves out co-transcriptionally, thereby leaving a homogenous population of RNA molecules with well-defined 5′ and 3′ nucleotides [61]. However, often, after the transcription reaction is over, an additional 20 mM MgCl₂ is added, and the reaction is incubated at 60 °C for 15 min to ensure maximum self-cleavage of the both the ribozymes.

2. Heterogeneity at the 3′ end, which is a more pressing issue, especially when using T7 RNA polymerase, can be significantly reduced by using modified reverse primers during PCR amplification of the linear DNA template for transcription. Such modified primers generate templates in which the last two nucleotides are methylated at 2′-O position, which prevents addition of extra nucleotides at the 3′ end by the RNA polymerase [43]. Finally, for RNA sequences <100 nucleotides, high transcription yields are often obtained by using a single stranded antisense template DNA, containing a sequence at its 3′ end (immediately preceding the complementary sequence of the RNA of interest), complementary to the T7 promoter sequence. When hybridized to an oligonucleotide that encodes the sense-strand sequence of the T7 promoter, the double-stranded promoter sequence is restored, leading to recognition and transcription initiation by T7 RNA polymerase [43].

3. If the goal of the experiment is to measure the midpoint and cooperativity of an RNA tertiary folding transition induced by monovalent ions, the starting buffer of choice is CE buffer, which has approximately 8 mM of free Na⁺ ions. Although the amount of free Na⁺ ions from 10 mM Na-cacodylate is negligible (~8 mM) compared with what is required for either RNA backbone charge neutralization or tertiary folding, to be technically accurate while conducting monovalent ion titrations with other counterions (e.g., potassium), one might choose to use the corresponding salt of cacodylic acid for making the experimental buffer.

4. If Mg²⁺ (or other multivalent cation)-induced tertiary folding is being studied, it is recommended to use buffers like CEK, which have at least 100 mM monovalent ions. The 100 mM K⁺ ions in the CEK buffer neutralizes the negatively charged phosphodiester backbone of RNA molecules, thereby reducing intermolecular electrostatic repulsion that can generate artifacts during measurements of hydrodynamic properties. Furthermore, it also screens intramolecular repulsive forces between the backbones of the helices, thereby allowing for greater conformational flexibility (Fig. 5b vs. Fig. 5c) [62]. Finally, it helps avoid the idiosyncratic Mg²⁺-mediated folding behavior observed for some large structured RNA molecules, like the self-splicing group I intron ribozymes (Tetrahymena [63] and Azoarcus [64]), under conditions of very low ionic strength, most likely due to due to non-native docking of secondary structure modules in a rigid ensemble.

5. The choice of the AUC phosphate buffer RNA–protein MWL experiments minimizes the background absorption from buffer components and extends the optical detection range to 215 nm (suitable for the detection of the peptide bond absorbance of the protein backbone). Moreover, it is recommended to eliminate reductants in the buffer to avoid background absorbance in the 250–285 nm absorbance range. If a reductant is required, it is recommended to replace thiol-containing reducing reagents present in protein-storage buffers, like DTT and BME, with the non-thiol reducing reagent TCEP (tris(2-carboxyethyl)phosphine)), in the AUC phosphate buffer. However, TCEP does absorb below 250 nm so it should be added only if needed and preferably only when conducting experiments at wavelengths greater than 250 nm. Always check the absorbance of the buffer to make sure it is not contributing to the absorbance in the wavelength range to be examined.

6. For all RNA and most RNA–protein interaction studies, since sample detection is based on UV absorption, investigators can use the standard Optima^™ AUC instrument. For detection of unlabeled molecules that do not absorb significantly in the UV–visible range, investigators can also use the Rayleigh interference optics of the Optima^™ AUC (or the older Proteome Lab^™ XL-I instruments). Based on detecting differences in refractive indices between the sample and the reference solution, the interference method can also provide higher signal–noise ratio and a greater coverage of sample concentration range (beyond the 0.1–1 OD range). Finally, the ability of both the Optima^™ AUC and the older Proteome Lab^™ instruments to detect absorbance in the visible region (400–700 nm) extends the scope of analytical investigations to molecules that are either naturally absorbed in the visible region of the spectrum or labeled with fluorescent dyes/proteins that do so [38, 65].

7. The direct integration of data acquisition software within UltraScan-III for the Optima^™ AUC instrument permits the use of all 4 positions in the An-60 Ti rotor (or 8 positions in the An-50 Ti) for the collection of 8 or 16 samples, respectively, in UV–vis data collection mode only. A counterbalance is only used for an initial, one-time radial calibration of the UV–vis detection system (the Rayleigh interference optics still require a counterbalance in each experiment). Once calibrated, the information is stored in the Optima^™ AUC, and the instrument can be used without a counterbalance in positions 4 or 8 Occasional recalibration is advised.

8. When using a recombinant T7 RNA polymerase, overexpressed and purified in our laboratory, in vitro transcription can be carried out in standard reaction buffers like 1× transcription buffer (50 mM Tris HCl (pH 8.1), 30 mM MgCl₂, 2 mM spermidine, 0.1% Triton-X-100, 0.05 mg/ml BSA, usually made and stored as a 10× stock at 4 °C) [43]. A standard transcription reaction is set up in the desired preparative volume (1–10 mL) by adding an appropriate amount of RNA polymerase (optimized using smaller analytical scale reactions prior to setting up preparative scale reactions) and transcription buffer (to a final 1× concentration) to the following components: 5 mM DTT, 2.5 mM of each NTP, and 1 μM linear DNA template. The reactions are incubated at 37 °C for 3–5 h. Usually, a white precipitate of magnesium phosphate is formed as a side product, which should be spun down before purifying the RNA by spin columns or denaturing PAGE.

9. The standard gel-loading buffers (e.g., gel-loading buffer II; Invitrogen^™/Ambion^™) contain loading dyes like xylene cyanol and bromophenol blue, which appear as dark bands in UV-shadowed gels. To distinguish them from actual RNA bands, it is always recommended to run only the loading buffer in one of the lanes, which can be used as reference during UV-shadowing and band excision. In case of RNA transcripts containing ribozyme sequences, it is also advised to run either an RNA ladder or an RNA of known molecular weight in a separate lane, which can be used as a reference to distinguish the RNA of interest from the cleaved HH and/or HδV ribozyme bands in the gel.

10. Since the proteins are used in combination with RNA and it is not uncommon for protein preparations to be contaminated with RNases, an aliquot from each batch of protein preparation should be tested for the presence of RNases. Generally, this involves incubating the RNA with the highest concentration of protein used in the experiments, in the experimental buffer, at 50 °C, and at the experimental temperature (37 °C/25 °C), for 2–3 h, and running the samples in a denaturing Urea-PAGE. Presence of smeared bands or multiple bands in the RNA + protein lane, compared with an RNA only lane, is indicative of significant RNase contamination in the protein sample.

11. The optimal absorbance range for operating the Optima AUC is an absorbance range between 0.2 and 0.8 OD units. Samples should be scanned on a high-quality spectrophotometer with a 1 cm quartz cuvette (and not with a Nanodrop-like device which is too imprecise) prior to performing the AUC experiment. The RNA stock solution should be sufficiently concentrated to give an O.D. >0.1 upon dilution into the reaction buffer. The optical pathlength of the sector-shaped channels in the centerpieces of the AUC cells is 1.2 cm, as opposed to the typical 1 cm pathlength of generic cuvettes. This pathlength difference should be considered while calculating the amount of RNA required to achieve the desired OD (~0.5). For any RNA, the 0.1–1.0 A₂₆₀ range in a 1.2 cm optical path length spans a concentration range of approximately 4 to 48 μg/mL (using the conversion factor 1 A₂₆₀ is equivalent to 40 μg/mL of RNA). Therefore, for a 100-nucleotide-long RNA molecule, with a 50% G-C content and a molecular weight (MW) of approximately 32.66 KDa, this spans a molar concentration range of 0.12–1.47 μM.

12. Before beginning an actual titration experiment, one should verify by preliminary SV-AUC runs that there is no concentration-dependent RNA aggregation in the 0.1–1 OD range, both with and without the counterion/protein. In the absence of aggregation, the hydrodynamic properties of the RNA ($s$ and/or $D$ values) should remain invariant as a function of increasing RNA concentration. An upward slope of the $s$ vs. RNA concentration plot, for the counterion containing sample, is typically indicative of the RNA aggregating into larger molecular weight species (Fig. 5e). It is also typical to observe a downward slope of the $s$ vs. RNA concentration plot for the samples lacking counterions (especially those that are just dissolved in the CE buffer that lacks even moderate concentrations of monovalent cations), most likely due electrostatic repulsion between polyanionic molecules, resulting in deviation from ideal solute behavior [32].

13. RNA molecules can often form highly stable alternate (non-native) secondary structures [66]. Therefore, in the absence of divalent cations (or insufficient monovalent cations), the starting point for RNA tertiary compaction experiments can contain a heterogenous ensemble of RNA molecules with the native and non-native secondary structures. The heat denaturation, followed by slow cooling, helps in generating a relatively homogenous ensemble, as a starting point for the tertiary compaction reaction. The slow cooling from 95 to 50 °C is critical because it allows equilibration to the native secondary structure. It is always advisable to conduct chemical or enzymatic structure-probing experiments (e.g., SHAPE [67] DMS footprinting [68], RNase mapping [69]), either prior to or in parallel with SV-AUC experiments) to verify formation of native secondary structure under the experimental conditions of SV-AUC.

14. Addition of the counterion that mediates tertiary compaction at the higher temperature of 50 °C, followed by the short incubation at that temperature, helps RNA tertiary-folding intermediates overcome kinetic traps by using the higher thermal energy [70]. However, when working with divalent cations (e.g., Mg²⁺) as the folding counterion, the incubation time should be kept as short as possible (no more than 10–15 min) to avoid divalent cation-mediated RNA hydrolysis at higher temperatures.

15. Since RNA tertiary compaction happens on second to minute timescales [47], the hour long incubation time at the folding temperature is generally sufficient for RNA molecules to reach their native tertiary structures. However, in a few instances, very long-lived misfolded intermediates that can persist for several hours have been detected [71]. Tertiary structure-probing experiments (e.g., hydroxyl radical footprinting) [46] and/or functional assays (e.g., ribozyme activity) [72] often help in determining the appropriate time and solution conditions for equilibrium compaction into the native tertiary structure.

16. Magnesium ions heavily influence the dynamics of RNA structures [73, 74]. The end points of Mg²⁺ titrations, intended to maximize the population of fully folded molecules, have >10 mM Mg²⁺. Such high Mg²⁺ concentrations bias the equilibrium tertiary structure heavily toward the folded state, thereby compromising the structural dynamics of the RNA molecule. In the experiments designed to probe equilibrium conformational changes of RNA, mediated by small-molecule ligands or proteins, the Mg²⁺ concentration should be kept around the midpoints of typical Mg²⁺-mediated RNA tertiary folding reactions (1-5 mM Mg²⁺, when conducting the experiments in CEK buffer) to render the RNA sufficiently flexible for conformational switching.

17. Sometimes it might be hard to push the assembly into the cell housing. It is recommended to not apply too much force which can damage the windows and the centerpiece; instead, all the cell components, including the housing, are placed inside a refrigerator at 4 °C for about 15 min, taken out, reassembled, and then gently pushed into the housing; it should go in easily. Similarly, it is often hard to push the assembly out of the housing after an AUC run, in which case, the cells are again placed at 4 °C for about 15 min to allow the assembly to be gently pushed out.

18. Each centerpiece has two sector-shaped channels with 1.2 cm optical path length, both of which can be used for samples. In the older Proteome Lab^™ AUC instruments, the channel names are reversed (“A” is the left, “B” is the right channel when facing the screw-ring with the fill holes at the top). Also, for the Proteome Lab^™ instruments, it is important to never exceed a total OD of 0.4 in the “A” channel to avoid resetting the photomultiplier tube voltage in the middle of a run.

19. A photosensitive analytical overspeed detection disk, with embedded magnets, is attached to the bottom of the rotor. Along with a sensor in the chamber, this device is used by the instrument to time the flash of the Xenon lamps and to check the rotor speed and make sure it does not cross the upper speed limit. Investigators are urged to exercise extreme caution to place the rotor only on the rotor stand and not on any other surface, to avoid damage to the analytical disk.

20. If you encounter difficulties inserting the cells, confirm that the fill-hole screws do not extend beyond the cell’s surface and that the fill-hole screws and the screw ring were not tightened excessively, which will bulge the entire cell body and prevent a smooth fit in the rotor.

21. Special care should be taken while attaching the Proteome Lab^™ monochromator to its socket. The users should keep in mind that it is inserted into the socket at an acute angle. After securing the device in its socket, the base ring should be carefully hand-tightened to avoid damaging the thread. Before closing the chamber door, the monochromator should be thoroughly checked to ensure it is not wobbling. Serious damages to the instrument can occur if the monochromator gets dislodged during an AUC run. These considerations, however, do not apply to the Optima^™ AUC which uses a periscope that can be swiveled into place and is locked into position by a magnetic lock.

22. The number of wavelengths selected for a given experiment also determines the number of scans that can be recorded for each wavelength. For example, if 3000 scans are available for the entire experiment, and 60 wavelengths are scanned, only 50 scans can be collected at each wavelength for a single cell. If two cells are scanned, this number drops to 25 scans per cell. Therefore, it is often advantageous to only scan one cell in a multi-wavelength experiment.

23. Subtraction of pairwise scans as done in SedAnal and DCDT+ to eliminate time invariant noise introduces amplification of stochastic noise by a factor equal to the square root of two which is undesirable. The same problem exists when absorbance data are used instead of intensity data. Absorbance data incur higher stochastic noise due to the subtraction of the reference scan. The UltraScan workflow automatically eliminates time- and radially invariant noise and therefore effectively deals with the time-invariant noise typically observed from window imperfections or photomultiplier tube response variations across its surface.

24. UltraScan will automatically transform $s$ and $D$ measured under a given experimental condition to standard conditions (20 °C, water) and report them as $s_{20,\mathrm{w}}$ and $D_{20,\mathrm{w}}$ values. This standardization is especially important in counterion-mediated RNA folding studies as it reports on the changes in the intrinsic properties of an RNA molecule and therefore, facilitates comparison between folding behaviors studied in different solution conditions. Appearance of more than one peak in the s-value distribution computed by UltraScan could indicate the existence of distinct conformational states with different hydrodynamic properties or self-association of the RNA molecules, or both. Users are advised to carefully check the distribution to rule out the presence of multiple peaks. Since, $s∕D$ is proportional to the molecular weight, the plot of $s_{20,\mathrm{w}}∕D_{20,\mathrm{w}}$ versus counterion concentration should produce a straight line parallel to the abscissa, provided that the molecules do not aggregate at high ionic strengths. It is not uncommon to observe deviation from this behavior in samples containing very high concentrations of Mg²⁺.

25. Cooperative ligand binding to a molecule is quantitatively described by the Hill equation, ${\rm Y}=\frac{K{\mathrm{d}}^{n_{\mathrm{H}}}{[\mathrm{M}]}^{n_{\mathrm{H}}}}{1+K{\mathrm{d}}^{n_{\mathrm{H}}}{[\mathrm{M}]}^{n_{\mathrm{H}}}}$, in which $K_{\mathrm{d}}$ represents the equilibrium dissociation constant, [M] is the ligand concentration, and $n_{\mathrm{H}}$ is the Hill coefficient that depicts the cooperativity of the process. The fitting of equilibrium RNA folding data points to the Hill equation is purely phenomenological, and the fitted curves do not represent counterion binding to RNA; instead, the fitted curve at a given temperature, also called the folding isotherm, simply represents a transition between two states, mediated by the counterion. For the purpose of analyzing counterion-mediated RNA folding isotherms, $K_{\mathrm{d}}$ and $n_{\mathrm{H}}$ represent, respectively, the midpoint and the cooperativity of the folding transition, whereas [M] denotes the concentration of the counterion.

26. In some cases where the RNA folding reaction is not a simple two-state transition, the folding isotherms may not be adequately fit by a single-site binding model. In such cases, the data points should be fit to modified Hill equations that represent multiple-site cooperative ligand binding. For example, an isotherm that is best described by two transitions can be phenomenologically fit to a two-site binding Hill equation:

$${\rm Y}=f\left\{\frac{K\mathrm{d}{1}^{n_{\mathrm{H}1}}{[\mathrm{M}]}^{n_{\mathrm{H}1}}}{1+K\mathrm{d}{1}^{n_{\mathrm{H}1}}{[\mathrm{M}]}^{n_{\mathrm{H}1}}}\right\}+(1-f)\left\{\frac{K\mathrm{d}{2}^{n_{\mathrm{H}2}}{[\mathrm{M}]}^{n_{\mathrm{H}2}}}{1+K\mathrm{d}{2}^{n_{\mathrm{H}2}}{[\mathrm{M}]}^{n_{\mathrm{H}2}}}\right\}$$

where ${\rm Y}$ denotes the signal quantitated; $f$ represents the fraction of the first transition; $K_{\mathrm{d}1}$, $K_{\mathrm{d}2}$ and $n_{\mathrm{H}1}$, $n_{\mathrm{H}2}$ are the midpoints and cooperativities of the two transitions, respectively; and [M] is the counterion concentration.