Salt Effects on the Thermodynamics of a Frameshifting RNA Pseudoknot under Tension

Naoto Hori; Natalia A. Denesyuk; D. Thirumalai

Determination of the phase diagram in the [C, f ] plane requires an appropriate order parameter that can distinguish between the various states. In single-molecule pulling experiments, the variable that can be measured is the molecular extension, R ee , which is conjugated to the applied force. The overall size of the RNA molecule is assessed using the radius of gyration, R g . Therefore, we characterized the states of the RNA using both R g and R ee as order parameters. The values of the predicted R g at f = 0 can be measured using scattering experiments. Using these two parameters, we obtained the [C, f ] phase diagram (Fig. 4) . Comparison of the diagram of states in Figs. 4D and 4E reveals common features and some differences. From Figs. 4D and 4E, we infer that at f > 12.5 pN, extended (E) conformations are dominant at all values of C. As the force decreases, the PK forms compact structures. The boundary separating the extended and relatively compact phases depends on the salt concentration and the value of the mechanical force. The critical force to rupture the compact structures increases linearly as a function of logarithm of salt concentration ( Fig. 4D ; boundary between red and green regions). At low forces (f < 2.5 pN), the diagram of states based on R ee shows that the extension is relatively small as C changes from a low to a high value. From this finding, one might be tempted to infer that the PK is always folded, which is not physical especially at low (C ≈ 10 mM) ion concentrations. In contrast, Fig. 4E shows that below 5 pN, there is a transition from compact structures (R g ≈ 1.3 nm in the blue region) at high C to an intermediate state (R g > 2.2 nm in the green region) at C ≈ 100 mM. The differences between the diagrams of states in the [C, f ] plane using R ee and R g as order parameters are more vividly illustrated in terms of the free energy profiles G(R α ) = −k B T log(R α ) where R α is R ee or R g (Fig. 5 ). The profiles G(R ee ), at three values of C and different f values, show two minima at most. At f = 0, there is only one discernible minimum at R ee ≈ 4.2 nm at C = 10 mM. The minimum moves to R ee ≈ 3 nm at C = 1200 mM corresponding to a folded PK. At f = 5 pN there are two minima at C = 10 mM corresponding to a compact structure and a stretched conformation (see the cyan profile in Fig. 5A ). As f exceeds 5 pN, there is essentially one minimum whose position increases as f increases. In the force regime (f > 5 pN), only the E state is visible at all C.