The Snakelike Chain Character of Unstructured RNA

Abstract

In the absence of base-pairing and tertiary structure, ribonucleic acid (RNA) assumes a random-walk conformation, modulated by the electrostatic self-repulsion of the charged, flexible backbone. This behavior is often modeled as a Kratky-Porod “wormlike chain” (WLC) with a Barrat-Joanny scale-dependent persistence length. In this study we report measurements of the end-to-end extension of poly(U) RNA under 0.1 to 10 pN applied force and observe two distinct elastic-response regimes: a low-force, power-law regime characteristic of a chain of swollen blobs on long length scales and a high-force, salt-valence-dependent regime consistent with ion-stabilized crumpling on short length scales. This short-scale structure is additionally supported by force- and salt-dependent quantification of the RNA ion atmosphere composition, which shows that ions are liberated under stretching; the number of ions liberated increases with increasing bulk salt concentration. Both this result and the observation of two elastic-response regimes directly contradict the WLC model, which predicts a single elastic regime across all forces and, when accounting for scale-dependent persistence length, the opposite trend in ion release with salt concentration. We conclude that RNA is better described as a “snakelike chain,” characterized by smooth bending on long length scales and ion-stabilized crumpling on short length scales. In monovalent salt, these two regimes are separated by a characteristic length that scales with the Debye screening length, highlighting the determining importance of electrostatics in RNA conformation.

Introduction

Once thought to act as a mere messenger within the central dogma of molecular biology (1), ribonucleic acid (RNA) is now known to fill a diversity of roles, including as an enzyme (2) and genetic regulator (3). These functions depend on an RNA’s folded, three-dimensional structure (4). In addition to the effects of secondary (base pairing) and tertiary structural elements, folding is strongly ion-dependent (5,6). Ions, which can interact specifically by binding to the RNA or nonspecifically through condensation (7) and the formation of an ion “atmosphere,” serve to neutralize the strong, negative charge of the backbone. It is this screening that allows charged monomers to come into close contact, facilitating stable folding. Thus, to fully understand the folding behavior of RNA, we must understand the coupling of electrostatics and conformation. This relationship can be studied in unstructured RNA (i.e., lacking secondary and tertiary structure), which behaves as a flexible polymer and therefore adopts a random walk topology (8).

To treat the random conformation of unstructured RNA in an analytically tractable way, many studies, e.g., (9–12), have applied the Kratky-Porod, or wormlike chain (WLC), model of polymer conformation (13). This model, intended for semi-flexible polymers, treats the RNA as a continuously deformable rod. The random conformation is characterized by an exponential decay, with separation distance, in the correlation between two vectors tangent to the polymer; the decay constant is the persistence length, $l_p$, of the polymer.

A significant advantage of the WLC model is that it provides analytical predictions that can be compared with experimental data, notably the interpolation formulae for elastic response, from Marko and Siggia (14), and for the scattering structure factor, from Pedersen and Schurtenberger (15). However, application of the WLC model to highly charged, flexible polymers, such as RNA, is questionable because of the long-range nature of electrostatic interactions, which undermines the assumption of exponential correlations. Alternative theories exist: e.g., the de Gennes-Pincus-Velasco-Brochard (dGPVB) scaling model for the conformation of weakly charged polymers (16), which are predicted to occur as chains of blobs. Within each blob, electrostatic interactions are insignificant and the polymer adopts an ideal random walk conformation, whereas between blobs there exist strong repulsive interactions that straighten the chain and impart a long-range persistence length (17). The strongly charged case, of direct relevance to RNA, was probed in more recent simulations (18–21), which also found the polymer to take on a blob-chain conformation. Following Ullner (22), we term the blob-chain model of a highly charged, flexible polyelectrolyte the “snakelike chain” (SLC). In the SLC model, as in the dGPVB model, the polymer conformation is characterized on long length scales by blob-blob repulsion, leading to exponential correlations (18). However, on short length scales, where the dGPVB model predicts ideal behavior, the SLC is swollen and stretched by electrostatic repulsion. This is manifest in the Flory exponent of ∼0.75 obtained from simulation of a charged, flexible polyelectrolyte in monovalent salt (23). Additionally, it has been shown (24) that the SLC phenomenology can be explained by power-law decay in tangent vector correlations at length scales smaller than the blob size, at odds with the single, exponential decay regime of the WLC model.

Based on its strongly charged, flexible nature, we predicted that unstructured RNA would exhibit SLC conformation. We tested this hypothesis by measuring the end-to-end extension, $L$, of polyuridine RNA as a function of applied force, $f$, in solutions of varying salt concentration. We interpret such force-extension measurements in terms of conformational length scales by noting that applied force mechanically segregates structural elements separated by more than a tensile screening length (25),

$$\xi \equiv \frac{k_{B}T}{f}.$$ (1)

Since the ability to probe smaller forces translates into the ability to probe longer length scales, the magnetic tweezers technique we employed, with its ability to reliably apply smaller forces than other single-molecule force manipulation methods (26), is best-suited to examining the effect of the inherently long-range Coulomb interaction on RNA conformation. In doing so, we found strong evidence for the SLC nature of unstructured RNA.

Materials and Methods

RNA synthesis

Polydisperse poly(U) (approx. 4000 to 10,000 bases) was synthesized by elongation of 20 bp poly(U) oligonucleotides, 5′-labeled with a protected thiol group (Integrated DNA Technologies, Coralville, IA), using polynucleotide phosphorylase from E. coli (Sigma-Aldrich, St. Louis, MO). The polymerization reaction conditions were those of (27). The polymerization product was 3′-labeled with biotin by incorporation of biotin-dUTP using terminal deoxynucleotidyl transferase (Life Technologies, Carlsbad, CA). Purification by ethanol precipitation (28) was performed following each reaction. Product concentration was quantified by absorption spectroscopy at 260 nm. Before surface coupling, the thiol group was deprotected by treatment with Tris(2-carboxyethyl)phosphine (TCEP).

Elasticity measurements

Experiments were performed in a flow-cell assembled by forming a channel between two pieces of double-sided adhesive tape placed between two glass cover slips. The bottom cover slip surface was passivated with polyethylene glycol and decorated sparsely with maleimide groups (Microsurfaces, Englewood, NJ). The heterobifunctionally labeled poly(U) were coupled to the surface by covalent binding of the deprotected thiol groups of the polymer to the maleimide sites of the surface, in 50 mM phosphate buffer solution. Amine-labeled latex beads were similarly coupled for use as fiducial markers. Subsequent steps were performed in 10 mM Tris-HCl (pH 7.5) buffer with 250 μM Tween 20 surfactant added to prevent adhesion of magnetic beads to the flow-cell surface (29). Micron-diameter, streptavidin-functionalized paramagnetic beads (Dynabeads myOne, Life Technolgies) were attached to the 3′-end of the RNA tethers by the biotin-streptavidin linkage.

Elasticity measurements were performed on an existing magnetic tweezers apparatus (30). Bead tracking was accomplished as in (30). Applied force, $f$, was quantified by fitting the Allan variance of bead fluctuations to the solution of the relevant Langevin equation with two free parameters: a spring constant, $k$, and a dissipation, $\alpha$ (31). Since the quality of the fit of $\alpha$ breaks down at the high end of the force range probed, the obtained values of $\alpha$ were fit, weighted by confidence, to a power law in $f$. The $\alpha$ obtained from this fit were then used to again fit the Allan variance, this time with only $k$ free. Finally, forces were obtained from Hooke’s law ($f=kL$).

Due to the applied magnetic field, the beads have only one rotational degree of freedom. Rotational fluctuations can lead to underestimation of the true end-to-end extension of the RNA (32). We correct for this effect by employing an equipartition argument to find the expected underestimate, $\delta$, given by:

$$\delta =k_{B}T/2f.$$ (2)

We then add $\delta$ to the measured bead height to give the true polymer extension, $L$. This expression for $\delta$ is valid in the $fR\gg k_{B}T$ limit, where $R$ is the bead radius. Over the forces we probed, $fR$ is at least 50 pN nm, compared with the room temperature value of $k_{B}T$ : 4.1 pN nm. We note that this is a fairly minor correction, affecting the lowest-force data by ∼5% and the high-force data by much less.

Due to strand breakage and other experimental concerns, we could not always collect elasticity data across the full spectrum of salt concentrations from the same molecule; instead, we collected data from different molecules and rescaled by the extension at $f=10$ pN to account for variations in overall contour length. For each salt species, we studied at least five molecules, in the case of NaCl, or three molecules, in the cases of MgCl₂ and CaCl₂, per concentration. The data plotted in Results are a representative subset but are consistent with the full dataset. When quantitative values are reported, the entire dataset was used to generate them. Throughout, uncertainties reported are standard errors unless otherwise indicated.

Determination of crossover force

The above elasticity measurements showed two scaling regimes in length with applied force, separated by a salt-dependent crossover force, $f_c$. These crossover forces were quantified by least-squares fitting of each force-extension curve to a continuous, piecewise function with different functional forms above and below $f_c$, as in (33). In monovalent salt, a power-law was used for $f<f_c$ and a logarithm for $f>f_c$. In divalent salt, a power-law was used in both regimes. Application of this method caused force-extension curves in the good-solvent regime to collapse to a single master curve for each valence; however, a constant multiplicative factor had to be introduced to achieve good low-force agreement between monovalent and divalent salt. Since we are interested only in the scaling of $f_c$ with ionic strength, and not its absolute value, this offset is unimportant.

Ion excess measurements

A previously described method (34) was used to extract the ion excess (number of ions associated with the RNA compared with an equivalent volume of bulk solution) from the applied force, $f$, extension, $L$, and bulk salt concentration, $c$. Force-extension curves of a particular poly(U) molecule were collected for $f$ ≈ 0.1 to 10 pN and $c$ = 20, 50, 100, 200, 500, 1000, 2000 mM. For this analysis, the forces measured at a particular magnet position were averaged across all salt concentrations (weighted by confidence). A force-extension curve was also collected at the Θ condition and fit to a WLC model to give the polymer contour length, $L_0$. We have previously found (34) that $L(f,c)$ data of this form are well fit by a second-order polynomial in $\text{log}\left(c\right)$ and a third-order polynomial in $\text{log}\left(f\right)$; such a surface was fit to these data. Equation 4 was then applied to this surface to give the change in ion excess, $\Delta n$, as a function of force. The excess per base was found by dividing $\Delta n$ by the number of bases: $L_0$ from the WLC fit at Θ divided by 0.59 nm per base (35). Due to potential edge effects in the fitting, ion excess curves for the highest and lowest $c$ are ignored. Error in the reported values of $\Delta n$ arises from two sources: uncertainty in the measured values of $f$, which is reflected in the error bars we report, and uncertainty in the fitted value of $L_0$ at $\Theta$, which introduces a 10% error common to all of the ion excess curves.

Results

Elasticity measurements

Since we sought to probe the elastic-response of RNA in the absence of higher-order structure, we studied the polyuridine homopolymer (poly(U)), which does not undergo base-pairing or tertiary structure formation, which possesses an unfavorable free energy for base-stacking interactions (36), and which showed no signature of base-stacking in earlier elastic measurements (37). Nonetheless, poly(U) regions do occur in biologically significant RNAs, such as regulatory sRNA (38).

Enzymatically synthesized poly(U) was heterobifunctionally labeled and tethered, at one end, to a functionalized glass cover slip and, at the other end, to a paramagnetic bead (see Materials and Methods). Application of a magnetic field gradient to the bead generates a force, which is quantified by comparing the bead fluctuations with those predicted from the Langevin equation for an overdamped, Brownian harmonic oscillator (31). The height of the bead above the cover slip is measured and gives the end-to-end extension of the RNA tether.

We measured end-to-end extension of poly(U) RNA as a function of applied force in the presence of varying concentrations of three ion species (NaCl, MgCl₂, CaCl₂). A representative subset of these data, rescaled by the extension at 10 pN applied force, is shown in Fig. 1. All of these curves exhibit the same qualitative behavior (Fig. 2 A): two elastic-response regimes separated by a salt-dependent crossover force, $f_c$. Over a range of salt concentrations (20 to 2000 mM NaCl, 0.2 to 20 mM MgCl₂, 0.2 to 10mM CaCl₂), and for $f<f_c$, the extension, L, obeys $L\sim f^{0.6}$, approximately the $f^{2/3}$ power-law elasticity expected of a self-avoiding chain (25); these salt concentrations thus correspond to the good-solvent regime for the RNA. For $f>f_c$, the response depends on the screening ion valence, a phenomenon that is further described below.

Figure 1.

Elasticity measurements of poly(U) RNA in various concentration solutions of (A) NaCl, (B) MgCl₂, and (C) CaCl₂. Dashed lines indicate WLC fits at the solution Θ points. To see this figure in color, go online.

Figure 2.

Analysis of elastic regime crossover force, ${\mathit{f}}_{\mathit{c}}$. (A) Force-extension curves, in good solvent, rescaled by ${\mathit{f}}_{\mathit{c}}$ and ${\mathit{L}}_{\mathit{c}}$. Lines are drawn to highlight the low-force power-law behavior common to both valences and the high-force logarithmic behavior in monovalent salt. (B) ${\mathit{f}}_{\mathit{c}}$ plotted as a function of ionic strength (same symbols as in (A)). Lines corresponding to a power-law dependence of ${\mathit{f}}_{\mathit{c}}$ on $I$ are shown for both valences; in monovalent salt this relationship is consistent with inverse ${\kappa }^{-1}$ scaling. To see this figure in color, go online.

For each salt species, there is a concentration for which the low-force elasticity is linear: $L\sim f$. Such a linear elastic response is expected of an ideal (not swollen) chain, so we interpret this concentration as a Θ-concentration, where ideal behavior results from the exact counterbalance of monomer-monomer repulsion and attraction. At Θ, the RNA elasticity is well fit by the Marko-Siggia model of WLC elasticity (14). WLC fits at Θ are shown in Fig. 1 and the fitted values of persistence length, $l_p$, and corresponding Θ concentrations, $c_{\Theta }$, are listed in Table 1. We also list $c_{\Theta }$ and $l_p$ for single-stranded DNA (ssDNA). Values of $l_p$ listed were obtained by fitting all curves at the Θ-concentration, not just the representative curve plotted in Fig. 1. The values of $l_p$ are in general agreement across all salt species tested, consistent with the marginalization of electrostatic effects at Θ.

Table 1.

Persistence length of poly(U) RNA at Θ conditions in various salt solutions, compared with ssDNA results from (39)

Salt	RNA		ssDNA (39)
	${\mathit{c}}_{\mathit{\Theta }}$ (mM)	lp (nm)	${\mathit{c}}_{\mathit{\Theta }}$ (mM)	lp (nm)
NaCl	≈ 4,000	0.83 ± 0.05	≈ 3,500	0.60 ± 0.02
MgCl2	≈ 120	0.78 ± 0.08	≈ 50	0.64 ± 0.03
CaCl2	≈ 15	0.83 ± 0.07	≈ 20	0.61 ± 0.02

Regime crossover analysis

From each force-extension curve well below the Θ concentration (Fig. 1), we extract the interregime crossover force, $f_c$, and extension. We then use those values to rescale each curve and find that they collapse to a single, universal curve for a given salt species (Fig. 2 A). Although poly(U) in all three species conforms to an $L\sim f^{2/3}$ dependence for $f<f_c$, for $f>f_c$ the behavior is valence-dependent: in monovalent NaCl, $L\sim \text{log}\left(f\right)$, whereas in the divalent salts we observe additional compliance.

Since an applied force corresponds to a tensile screening length (Eq. 1), the crossover force $f_c$ corresponds to a crossover length scale, ${\xi }_c=k_{B}T/f_c$, separating conformational regimes (33). To investigate the salt-dependence of that length scale, we plot in Fig. 2 B the relation between $f_c$ and the solution ionic strength, $I$. Ionic strength is used because it is the relevant parameter in the Debye-Hückel treatment of solution electrostatics and provides a framework to compare mono- and divalent salt. For monovalent salt, we find $f_c\sim I^{0.58\pm 0.09}$, consistent with crossover length scaling as ${\xi }_c\sim I^{-1/2}\sim {\kappa }^{-1}$, where ${\kappa }^{-1}$ is the Debye screening length. We thus identify ${\kappa }^{-1}$ as an important conformational length scale of unstructured RNA in monovalent salt, corresponding to the SLC blob size separating short- and long-scale conformations. In divalent salt, $f_c\sim I^3$ scaling is observed, which does not correspond to an obvious physical length scale.

Quantification of associated ions

Force-extension data permit quantification of the ion atmosphere surrounding the molecule as a function of both $f$ and bulk salt concentration, $c$ (see Materials and Methods and (34)). We use such results both to study how the number of associated screening ions is modulated by RNA conformation and to directly test WLC-derived models of salt-dependent RNA elasticity. Briefly, the independent experimental variables are $f$ and $\mu$, the chemical potential; these are related to the thermodynamically conjugate, dependent variables $L$ and $n$, the ion excess, respectively. The ion excess corresponds roughly to the surfeit of ions in the vicinity of the RNA compared with those that would be present in an equivalent volume of bulk solution (40). The ion excess is nonzero because the negative charge of the RNA attracts additional cations to, and repels some anions from, its vicinity. We can relate $f$, $c$, $L$, and $n$ by a Maxwell relation (34,41) as follows:

$${\frac{\partial \text{L}}{\partial \mu }|}_f={\frac{\partial n}{\partial f}|}_{\mu }.$$ (3)

From this, we derive an integral expression giving the difference in ion excess ($\Delta n$) between a state with applied force $f$ and some reference state with applied force $f_o$ :

$$\Delta n=\frac{c}{2k_{B}T}\underset{f_o}{\overset{f}{\int }}{\frac{\partial L}{\partial c}|}_{f^{\text{'}}}df\text{'}.$$ (4)

This equation assumes an ideal solution; accounting for actual ion activities changes the results by less than 10% (34) and does not affect key conclusions. To perform the partial differentiation and integration required by Eq. 4, we fit a smooth surface to $L\left(f,c\right)$. Applying Eq. 4 to this surface, we can quantify changes to the polymer-associated ion excess as the RNA is stretched.

The results of this analysis, for a representative RNA molecule, are shown in Fig. 3 A for poly(U) in the presence of 50 to 1000 mM NaCl and subject to 0.1 to 10 pN applied force. As the RNA is stretched, we observe that associated ions are driven away ($\Delta n<0$). For sufficiently large forces, the number of ions driven away increases with increasing salt concentration, a result we explain by arguing that $\Delta n$ per SLC blob is constant and that the number of blobs per length increases with salt concentration, since blob extent scales with ${\kappa }^{-1}$. This phenomenon is seen directly in Fig. 3 B, where we plot the ion excess per approximate blob size, ${\kappa }^{-1}$, versus rescaled force, $f/f_c$. The excess at $f_c$, the force corresponding to the blob size, was chosen as a natural zero-reference. Under this rescaling, all of the ion excess curves collapse to a single, master curve, indicating that the variation between curves was attributable to the disparate number of SLC blobs per length. Because most of the ion excess change occurs when $f/f_c>1$, corresponding to $\xi <{\xi }_c$, we conclude that the ions are predominately associated with the intrablob crumpling and not with the longer-length-scale chain of blobs. This analysis, which confirms the SLC picture of ion-stabilized crumpling on short length scales (see Discussion), was repeated for two other RNA molecules and yielded comparable results.

Figure 3.

Determination of RNA-associated ion excess as a function of applied force and monovalent salt concentration. (A) Change in ion excess, per RNA base, as a function of force, compared with a low-force reference state, for various NaCl concentrations (dashed line: lowest concentration; heavy line: highest concentration). Error bars reflect propagated uncertainty from force measurement; an additional 10% error common to all curves arises from uncertainty in polymer contour length. Inset: Expected ion excess curves for a WLC model with scale-dependent persistence length (14,42), using parameters of (9) (same ordering of line weight with concentration). (B) Change in ion excess, per Debye length, as a function of $\mathit{f}/{\mathit{f}}_{\mathit{c}}$, compared with a reference state at ${\mathit{f}}_{\mathit{c}}$ (values of ${\mathit{f}}_{\mathit{c}}$ from Fig. 2B) for the same NaCl concentrations.

These data also provide a test of the modified WLC model with scale-dependent persistence length (14,42) sometimes used to account for the polyelectrolyte character of RNA. Using such a modified WLC model, with parameters obtained from another study of poly(U) elasticity (9), we generated force-extension curves at various salt concentrations and, from them, computed the change in ion excess. The results, shown in the inset to Fig. 3 A, show an opposite trend in $\Delta n$ with increasing $c$ compared with what we observe experimentally.

Discussion

Elastic-response

Interpreting force measurements in terms of tensile screening lengths, $\xi$, the crossover force, $f_c$, separating observed elastic-response regimes (Fig. 2 B) corresponds to a crossover length scale, ${\xi }_c,$ separating spatial conformational regimes. In monovalent salt solution, we observe that $f_c\sim I^{1/2}$, corresponding to ${\xi }_c\sim I^{-1/2}$. In the Debye-Hückel theory, the Debye length varies as ${\kappa }^{-1}\sim I^{-1/2}$; thus ${\xi }_c\sim {\kappa }^{-1}$. This indicates that, in monovalent salt solution, there are separate conformations of unstructured RNA on short and long length scales, and that they are separated by the Debye screening length, as illustrated in Fig. 4. This result is in agreement with the finding that the polymer’s tangent vector correlation transitions from power-law to exponential decay at ${\kappa }^{-1}$ in the simulation studies of (24). In divalent salt solution, however, this scaling is not observed. Instead, we see $f_c\sim I^3$ for $c\le 5$ mM ($I\le 25$ mM) and nonpower-law scaling for the full range of tested salt concentrations in the good solvent regime. Thus, ${\kappa }^{-1}$ is not the important length scale separating conformational regimes in the presence of appreciable divalent salt, as is the case under physiological conditions. Instead, ${\kappa }^{-1}$ of Fig. 4 is replaced with a length scale that decreases more rapidly with increasing $I$, consistent with the enhanced efficiency of divalents (Fig. 1 B and C, compared with 1 A) and the breakdown of the Debye-Hückel picture in this case (39).

Figure 4.

Diagrammatic representation of SLC conformation. In monovalent salt, the long length scale bending and short length scale, ion-stabilized crumpling regimes are separated by the Debye screening length, ${\kappa }^{-1}$. In divalent salt, ${\kappa }^{-1}$ is replaced by a crossover length scale that decreases more rapidly with increasing ionic strength and the conformation on short length scales is more tightly packed than in the monovalent case.

Having identified the length scale that separates the two conformational regimes of unstructured RNA, we now characterize those regimes. In the limit of long length scales ($f<f_c$), in both monovalent and divalent salt, we observe, approximately, the $L\sim f^{2/3}$ scaling characteristic of a chain of blobs swollen by excluded volume interactions (25), akin to the dGPVB theory of weakly charged polyelectrolytes (16), but with a different blob size. Unlike in uncharged polymers, where steric effects dominate, here the primary mechanism of swelling is the Coulomb repulsion between charged monomers. This is seen by noting the absence of an intermediate regime of ideal, $L\sim f$ elasticity (43) in the force-extension data. It has been argued previously that such a regime occurs only for chains with rod-like, not sphere-like, statistical monomers (44). Thus, our results indicate that RNA, on long length scales and at moderate salt concentrations, acts as a chain of spherical blobs, consistent with the dominance of the spherically symmetric Coulomb repulsion.

That the exact numerical exponent we measure for this low-force scaling, $L\sim f^{\gamma }$ with $\gamma =0.6$, differs in detail from the $\gamma =2/3$ (Pincus scaling) result for an infinite chain of swollen blobs can be attributed to the finite length of the polymers under study (4000 to 10,000 nucleotides). Simulation studies have shown that Pincus scaling does not arise in flexible, neutral polymers with degree of polymerization less than $N\approx {10}^5$ monomers (45). For polymers shorter than this, the applied force is felt within the blobs. More recent simulations have shown that Pincus scaling occurs in flexible, charged polymers for $N=25,000$, but not for $N=5000$ ($\gamma =0.60$ scaling is seen in that case) (46).

In the limit of short length scales ($f>f_c$), we observe an elastic response that depends strongly on salt valence. In monovalent salt, we find $L\sim \text{log}\left(f\right)$, a more compliant response than the WLC behavior expected for an uncharged polymer; in divalent salt we observe even more compliance (Fig. 2 A). These results support the SLC picture of short length scale polymer crumples stabilized by transient interactions with screening ions: contour length sequestered in the crumples is liberated by applied force and manifests as increased compliance. That additional high-force compliance, and therefore additional local crumpling, is observed for divalent ions is consistent with this picture: the more highly charged ions have stronger interactions with the RNA backbone, allowing it to adopt sharper bends and thus sequester additional contour length. This short length scale crumpling, and its variation with ion valence, is also seen in molecular dynamics simulations of a charged, flexible polymer in the presence of explicit screening ions (23).

Associated ion excess

The existence of an ion-associated, crumpled conformation on short length scales in unstructured RNA is further supported by our measurements of the force- and salt-dependent ion excess (Fig. 3). The decrease in ion excess with increasing force is consistent with ion-associated crumples: if the contour length sequestered in crumples is stabilized by co-localization of screening ions, then the liberation of that contour length by an applied force should concomitantly liberate those ions. The decrease in ion excess change with increasing salt concentration, for sufficiently high forces, is consistent with the liberation of a constant number of ions per SLC blob, but is directly at odds with the prediction of a modified WLC model (9,14,42) employing a scale-dependent persistence length (Fig. 3 A). That all of the ion excess curves collapse to a single, master curve when plotted as $[n\left(f\right)-n\left(f_c\right)]$ per ${\kappa }^{-1}$ vs. $f/f_c$ (Fig. 3 B) indicates that the salt-dependence enters only through changing the size of the Debye-length-scale SLC blobs. This is consistent with our measurements of elasticity, which indicate (Fig. 2 A) universal behavior across all salt-concentrations modulo a salt-dependent crossover force that corresponds to the SLC blob size.

Compared with previously published ion excess data for ssDNA (34), we see fewer ions liberated ($\Delta n$) when unstructured RNA is stretched. This could be explained by the differences in $l_p$ between those species (0.65 nm for DNA vs. 0.83 nm for RNA, at Θ; see discussion below): the more rigid RNA will form bends with a larger radius of curvature, implying a shallower electrostatic potential and weaker ion affinity.

Comparing ssRNA and ssDNA

The SLC conformation of unstructured RNA described above is in good qualitative agreement with findings from earlier studies of ssDNA using the same techniques (34,39). However, there exist quantitative differences in the persistence lengths and Θ conditions between the two polymers. Although RNA and DNA are structurally analogous, they differ in the substituent of the 2′ sugar carbon. The added hydroxyl group in RNA forces the pentose ring into the C_3′-endo conformer (DNA is in equilibrium between C_3′-endo and C_3′-exo) (36), possibly because of the formation of a water bridge between the 2′-hydroxyl and 3′-phosphate (47). This structural constraint is expected to increase the rigidity of RNA compared with DNA, which is indeed borne out by the longer persistence lengths we observe in RNA compared with DNA (Table 1). Base stacking has been proposed as an alternate origin of this rigidity (12), but we see no signature of such stacking in these elasticity measurements or those of (37).

The $l_p$ comparison we report is especially illuminating given that the two measurements were made using the same technique (single-molecule elasticity at low force under Θ conditions); the same was true in a study of poly(U) and poly(dT), using small-angle x-ray scattering and single-molecule Förster resonance energy transfer, which found a 34% difference in $l_p$ between ssDNA and RNA when extrapolated to Θ (12), in general agreement with the 38% difference we report. Away from Θ, reported measurements of $l_p$ vary widely ((48), and references therein); given the results reported here, we attribute this to the non-WLC nature of the single-stranded nucleic acids.

Additionally, the extra hydroxyl group of RNA, acting as a hydrogen bond donor, is expected to increase water solubility. Since the Θ condition occurs at the transition between the good solvent and poor solvent regimes (8), this increased solubility of RNA explains the higher observed Θ concentrations of NaCl and MgCl₂ compared with ssDNA. That the Θ concentration of CaCl₂ is not similarly higher in ssRNA than in ssDNA is not explained by this argument and could be due to some other effect, perhaps a specific chemical interaction between the salt ions and the RNA.

Implications for interpreting biophysical results

The SLC results we report differ markedly from those of a WLC for applied forces below ∼5 to 10 pN at near-physiological salt concentrations (Fig. 1). Since WLC-derived models are often used to interpret the results of biophysical studies of RNA, the question arises as to whether appreciable error is introduced into such studies because of low-force, non-WLC behavior. Through a tensile screening length argument, the low forces, $f<f_c$, where the SLC/WLC divergence is most pronounced correspond to length scales longer than the SLC blob size. Thus, properties associated with the molecule on long length scales are most likely to differ between the models. For example, the Benoit-Doty expression for the radius of gyration of a WLC, which exhibits ideal polymer behavior on long length scales, differs by a positive multiplicative factor from the expression accounting for excluded volume (50), which governs the SLC conformation above the SLC blob size.

The WLC model is frequently used to calculate the free energy associated with stretching ssRNA as a way of determining the free energy of folding of a structured RNA, e.g., (51). In particular, the free energy of folding measured at a particular applied force, $\Delta G\left(f\right)$, can be written as the sum of the free energy of folding at zero force, $\Delta G\left(0\right)$, and the free energy change associated with stretching the ssRNA, $\Delta G^{\text{ssRNA}}\left(f\right)$ :

$$\Delta G\left(f\right)=\Delta G\left(0\right)+\Delta G^{\text{ssRNA}}\left(f\right),$$ (5)

where the free energy of stretching, in the constant-force ensemble, is obtained from integration of a force-extension curve:

$$\Delta G^{\text{ssRNA}}\left(f\right)=-\underset{0}{\overset{f}{\int }}{L}^{\text{ssRNA}}\left(f^{\text{'}}\right)d{f}^{\text{'}}.$$ (6)

To quantify the error incurred in using the simpler WLC model in this integration instead of the SLC model, we fit both to our force-extension results for poly(U) in 100 mM NaCl and compared the resulting values of $\Delta G^{\text{ssRNA}}\left(f\right)$. For the WLC, we fit the high-force data with the Marko-Siggia interpolation formula (14). For the SLC, we fit all of the data with a piecewise, continuous function capturing the relevant phenomenology: $L\sim f^{\gamma }$ below $f_c$, $L\sim \text{log}\left(f\right)$ above $f_c$, as in (33). Polymer theory predicts another regime of entropic-spring, $L\sim f^1$ scaling at forces even lower than those we were able to measure in this study (52); because it occurs at such low force, this regime contributes negligibly to the $\Delta G^{\text{ssRNA}}\left(f\right)$ integral and is thus left out of the model. Both fitted curves were integrated over force values from zero to 14.2 pN, the unfolding force of a simple RNA hairpin (35). The difference in $\Delta G^{\text{ssRNA}}\left(f\right)$ values obtained was 2%; this discrepancy increases at lower forces and for lower salt concentrations.

Conclusion

Full understanding of RNA’s diverse biological functions can only be obtained through a detailed understanding of its three-dimensional conformation and structure. Many factors contribute, including the interactions responsible for secondary and tertiary structure. However, in the absence of these phenomena, the conformation of RNA is dominated by the solution-screened Coulomb interaction between the charged phosphate groups of the backbone. From the results presented above, we conclude that unstructured RNA exhibits a SLC conformation (Fig. 4), characterized by bending on long length scales and ion-associated crumpling on short length scales. These two regimes are separated by the salt-dependent SLC blob size that, in monovalent solution, scales with the Debye screening length. The existence of two conformational regimes is supported by our experimental observation of two elastic-response regimes in the force-extension curves, separated, in monovalent salt, by a force scaling inversely with ${\kappa }^{-1}$. That the short-range conformation is stabilized by ion association is supported by both the enhanced high-force compliance observed in divalent salt solution and the observation that excess ions are liberated from association with the RNA when it is pulled apart.

Although the WLC model of polymer elasticity has long served as an empirically effective, and analytically tractable, model for unstructured RNA conformation, our results indicate that it does not accurately describe the microscopic details. In particular, it predicts $L\sim f$ scaling in the low-force limit, an $(L_0-L)\sim 1/f^{1/2}$ approach to the contour length, $L_0$, in the high-force limit (14), and, when adjusted to account for electrostatics-dependent persistence lengths, a decrease in $\Delta n$ with increasing salt. In contrast, the SLC behavior we report in this study occurs as $L\sim f^{2/3}$ scaling at low force, $L\sim \text{log}\left(f\right)$ at moderately high force, and increasing $\Delta n$ with increasing salt. Better characterization of SLCs requires analytical models; whereas Toan and Thirumalai provide an elastic prediction for the SLC (24) that replaces the Marko-Siggia WLC elastic theory (14), it is based on the phenomenological assumption of power-law correlations rather than microscopic physical details. Additionally, we are unaware of an SLC-motivated prediction for the scattering structure factor that would replace the results of Pedersen and Schurtenberger (15).