Frictional Effects on RNA Folding: Speed Limit and Kramers Turnover
FFrictional Effects on RNA Folding: Speed Limit and Kramers Turnover
Naoto Hori, ∗ Natalia A. Denesyuk, and D. Thirumalai † Department of Chemistry, University of Texas, Austin, Texas 78712, United States Biophysics Program, Institute for Physical Science and Technology,University of Maryland, College Park, Maryland 20742, United States (Dated: September 1, 2018)We investigated frictional effects on the folding rates of a human telomerase hairpin (hTR HP)and H-type pseudoknot from the Beet Western Yellow Virus (BWYV PK) using simulations ofthe Three Interaction Site (TIS) model for RNA. The heat capacity from TIS model simulations,calculated using temperature replica exchange simulations, reproduces nearly quantitatively theavailable experimental data for the hTR HP. The corresponding results for BWYV PK serve aspredictions. We calculated the folding rates ( k F ) from more than 100 folding trajectories for eachvalue of the solvent viscosity ( η ) at a fixed salt concentration of 200 mM. By using the theoreticalestimate ( ∝√ N where N is the number of nucleotides) for folding free energy barrier, k F data forboth the RNAs are quantitatively fit using one-dimensional Kramers’ theory with two parametersspecifying the curvatures in the unfolded basin and the barrier top. In the high-friction regime( η (cid:38) − Pa · s), for both HP and PK, k F s decrease as / η whereas in the low friction regime, k F valuesincrease as η increases, leading to a maximum folding rate at a moderate viscosity ( ∼ − Pa · s),which is the Kramers turnover. From the fits, we find that the speed limit to RNA folding atwater viscosity is between 1 and 4 µ s , which is in accord with our previous theoretical prediction aswell as results from several single molecule experiments. Both the RNA constructs fold by parallelpathways. Surprisingly, we find that the flux through the pathways could be altered by changingsolvent viscosity, a prediction that is more easily testable in RNA than in proteins. INTRODUCTION
The effects of friction on barrier crossing events, with arich history [1, 2], have also been used to obtain insightsinto the dynamics and folding of proteins. For exam-ple, in a pioneering study, Eaton and co-workers estab-lished that accounting for the internal friction is neededto explain experiments in the ligand recombination tothe heme in myoglobin [3]. Only much later, the im-portance of internal friction, a concept introduced in thecontext of polymer physics [4], in controlling the dynam-ics of folded and unfolded states of proteins has beenappreciated in a number of experimental [5–8] and the-oretical [9–13] studies. The presence of internal frictionis typically identified as a deviation in the viscosity ( η )dependence of reaction rates from the predictions basedon Kramers’ theory [14]. The timeless Kramers’ theory ∗ [email protected] † [email protected] showed that the rate should increase linearly with η atsmall η and decrease as / η at large η . The change fromsmall η behavior to / η dependence with a maximum atintermediate viscosity values is often referred to as theKramers turnover [1, 2, 15]. Theoretical studies [16, 17]also showed that folding rates of the so-called two-statefolders are in accord with the theory of Kramers [14].Kramers’ theory has been used to understand frictionaleffects of the solvent in various reactions, from diffusionof single particles to folding of proteins that are morecomplex with the multidimensional folding landscape.Although Kramers’ theory was originally developed forbarrier crossing in a one-dimensional potential with asingle barrier, experiments and simulations suggest thetheory holds for dynamic processes in biomolecules. In-terestingly, following the theoretical study, establishingthat folding rates ( k F ) of proteins vary as k F ∼ / η [16],experiments on cold shock protein [18], chymotrypsin in-hibitor [19], and protein L [20] confirmed Kramers’ high- η predictions. Although these studies showed that therate dependence on η follows Kramers’ prediction, this a r X i v : . [ q - b i o . B M ] S e p was most vividly demonstrated in single molecule stud-ies only recently by Chung and Eaton [6]. The success ofthe Kramers’ theory, which views the complex process ofpolypeptide chain organization as diffusion in an effectiveone-dimensional landscape, is surprising. However, it hasbeen shown using lattice models [21] that diffusion in anenergy landscape as a function of a collective coordinates,such as the fraction of native contact ( Q ), provides anaccurate description of the folding rates obtained in sim-ulations. Subsequently, computational studies [22] usingG¯o model for a helix bundle further showed that the ratedependence follows the theoretical predictions includingthe Kramers turnover, providing additional justificationthat Q is a good reaction coordinate for protein sequencesthat are well optimized.In contrast to several studies probing viscosity effectson protein folding and dynamics, frictional effects on nu-cleic acid folding have been much less studied. A vex-ing issue in experiments is that common viscogens suchas glycerol may significantly alter the stability of RNAmolecules. Thus, in order to isolate the frictional effects,a condition of isostability has to be established by ma-nipulating other experimental parameters such as tem-perature to compensate for the stability change causedby adding viscogens [23]. Ansari and Kuznetsov showedthat, when corrected for stability changes, the rates ofhairpin formation of a DNA sequence are proportional to / η [23]. Kramers’ predictions at high η were also borneout in the folding of G-quadruplex DNA [24], and mostrecently in a tetraloop-receptor formation in RNA [25].These studies show that nucleic acid folding might also beviewed as diffusion in an effective one-dimensional foldinglandscape.In this paper, we consider frictional effects on RNAfolding using coarse-grained (CG) simulations. We inves-tigate the variations in rates of folding of a human telom-erase hairpin (hTR HP) and an H-type pseudoknot frombeet western yellow virus (BWYV PK) as a function of η .Because both the HP and PK fold by parallel pathways,our study allows us to examine whether frictional effectsaffect the flux through parallel pathways in RNA fold-ing. Despite the differences in sequences and the foldedstructures, the dependence of k F on η is quantitatively fitusing Kramers’ theory including the predicted turnover.The excellent agreement between theory and simulationsallows us to estimate a speed limit for RNA, which wefind to be ∼ µ s . Surprisingly, we find that the fluxthrough the pathways may be altered by changing solventviscosity for both the HP and PK. The change in the fluxis more pronounced for HP, especially at a temperaturebelow the melting temperature. We argue that this pre-diction is amenable to experimental tests in RNA eventhough it has been difficult to demonstrate it for proteinfolding. FIG. 1. (A)
Secondary representation and sequence of humantelomerase hairpin (hTR HP). The folded hairpin structure ison the right. Note that there are four noncanonical base pairsbetween S1 and S2. (B)
Secondary structure of Beet WesternYellow Virus pseudoknot (BWYV PK). The tertiary structureof the PK is shown on the right. In the secondary structures,blue lines represent canonical base pairs (thick lines) and non-canonical pairs (dotted lines).
MATERIALS AND METHODS
RNA Molecules:
We choose a sequence that formsa hairpin (HP) with no tertiary interactions from thehuman telomerase (hTR) and an H-type BWYV pseudo-knot (PK), which is a minimal RNA motif with tertiaryinteractions. The folding mechanisms of PKs are diverse[26], and they often reach the native structure by parallelpathways [27]. The use of two RNA molecules with differ-ent folded states, with both HP and PK folding occurringby parallel pathways, allows us to examine many conse-quences of viscosity effects on their folding. The struc-ture of hTR HP (PDB ID 1NA2) has been determinedusing NMR (see Figure 1A) [28]. The folded structureof the BWYV PK is taken from the crystal structure(PDB ID 437D) [29]. The PK has 28 nucleotides formingtwo stems. The two loop regions have hydrogen bondinginteractions with the stems (Figure 1B). We added anadditional guanosine monophosphate to both the (cid:48) and (cid:48) terminus to minimize end effects. Thus, the simulatedPK has 30 nucleotides. Three Interaction Site (TIS) Model for RNA:
We employed a variant of the TIS model, which hasbeen previously used to make several quantitative pre-dictions for RNA molecules ranging from hairpins to ri-bozymes [26, 30–32]. We incorporated the consequencesof counterion condensation into the TIS model, allow-ing us to predict the thermodynamic properties of RNAhairpins and PKs that are in remarkable agreement withexperiments [33]. Because the details of the model havebeen reported previously, we only provide a brief de-scription here. In the TIS model [30], each nucleotideis represented by three coarse-grained spherical beadscorresponding to phosphate (P), ribose sugar (S), anda base (B). Briefly, the effective potential energy (fordetails see Ref. [33]) of a given RNA conformation is U TIS = U L + U EV + U ST + U HB + U EL , where U L ac-counts for chain connectivity and angular rotation of thepolynucleic acids, U EV accounts for excluded volume in-teractions of each chemical group, and U ST and U HB are the base-stacking and hydrogen-bond interactions, re-spectively.Electrostatic interactions between the phosphate (P)groups are given by U EL . The repulsive electro-static interactions between the P sites are taken intoaccount through the Debye-Hückel theory, U EL = (cid:80) i,j q ∗ e πε ε ( T ) r ij exp (cid:16) − r ij λ D (cid:17) , where the Debye length is λ D = (cid:113) ε ε ( T ) k B T e N A I . In the present simulations, salt con-centration (monovalent ions) is set to 200 mM, whichis close to the physiological value. The ionic strength I = (cid:80) c i z i where c i is the molar concentration, and z i is the charge number of ion i , and the sum is takenover all ion types. Following our earlier study [33], weused an experimentally fit function for the temperature-dependent dielectric constant ε ( T ) [34]. To account forcounterion condensation, we used a renormalized chargeon the phosphate group, − q ∗ e ( q ∗ < . The renormal-ized value of the charge on the P group is approximatelygiven by − q ∗ ( T ) e = − bel B ( T ) , where the Bjerrum lengthis l B ( T ) = e πε ε ( T ) k B T , and b is the mean distance be-tween the charges on the phosphate groups [35]. Weshowed elsewhere [33] that a constant value of b = 0 . nm accounts for the thermodynamics of several RNAmolecules, and is the value adopted here. All the force-field parameters used here are the same as in our earlierstudy [33]. Simulation Details:
We performed Langevin dy-namics simulations by solving the equation of motion, m ¨ x = − ∂U TIS ∂ x − γ ˙ x + Γ , (1)where m is the mass of the particle, x is the coordi-nate, and Γ is a Gaussian random force that satisfies thefluctuation-dissipation relation given by (cid:104) Γ i ( t ) Γ j ( t (cid:48) ) (cid:105) =6 γk B T δ ( t − t (cid:48) ) δ ij . The friction coefficient follows theStokes-Einstein relation, γ = 6 πηR , where R is the ap-propriate size of the coarse-grained bead (P, S and B)and η is the solvent viscosity. The numerical integrationis performed using the velocity-Verlet algorithm [36].In the high friction regime where η = 10 − to − Pa · s , we performed Brownian dynamics simula-tions [37] by neglecting the inertial term, since the dy-namics is overdamped. In this limit, the equation of mo-tion is, ˙ x = − γ ∂U TIS ∂ x + Γ . (2)We used reduced units in the analysis of data [16].In the TIS representation, we chose the mass of a bead m =
116 g/mol, the typical length scale a = 0 . nm, and the energy scale ε = 1 kcal/mol. Thus, the naturalmeasure for time in Eq. 1 is τ = ( ma ) / ∼ . In theoverdamped condition (Eq. 2), the natural unit of timeis τ = γa k B T . We used this measure to obtain τ ≈
300 ps for converting the simulation times to real times at theviscosity of water, η w = 10 − Pa · s [38].We confirmed that both Langevin dynamics and Brow-nian dynamics simulations give identical results at η =10 − Pa · s, using simulations of hTR hairpin. The dif-ference between the two simulations method is in therange of statistical error estimated by the jack-knifemethod. For example, the folding rate for hTR HP is k F = 5 . ± . ms − calculated from 100 trajectoriesgenerated using Brownian dynamics simulations and is k F = 6 . ± . ms − obtained from another set of 100trajectories generated using Langevin dynamics simula-tions at η = 10 − Pa · s. Hydrodynamic Interactions:
In order to ensurethat the results are robust, we did limited simulations offolding by including hydrodynamic interactions (HI). Totake into account the effects of HI, we performed Brow-nian dynamics simulations using the following form withconformation-dependent mobility tensor, ˙ x i = − (cid:88) j µ ij ∂U TIS ∂ x i + Γ , (3)where µ , the mobility tensor, is computed using theRotne-Prager-Yamakawa approximation [37], µ ij = (cid:40) πηR ( i = j ) πηr ij (cid:104)(cid:16) + r ij r ij r ij (cid:17) + R r ij (cid:16) − r ij r ij r ij (cid:17)(cid:105) ( i (cid:54) = j ) . (4)In the above equation, r ij is a coordinate vector betweenbeads i and j. Because coarse-grained beads in our TISmodel have different radii ( R ) depending on the type ofbeads (phosphate, sugar, and bases) [33], we employed amodified form of µ developed by Zuk et al. [39]. Thermodynamics Properties:
We performedtemperature-replica-exchange simulations (T-REMD)[40] to calculate the heat capacity. Temperature wasdistributed from 0 to 120 °C with 16 replicas at 200mM salt concentration. The T-REMD simulation isperformed using a lower friction ( η = 10 − Pa · s) toenhance the efficiency of conformational sampling [36]. Order Parameter:
In order to determine if a fold-ing reaction is completed, we used the structural overlapfunction [41] χ = 1 N p N p (cid:88) i,j H (cid:0) d − (cid:12)(cid:12) r ij − r ij (cid:12)(cid:12)(cid:1) , (5)where H is the Heaviside step function, d = 0 . nmis the tolerance, and r ij is the distance between parti-cles i and j in the native structure. The summationis taken over all pairs of coarse-grained sites separatedby two or more covalent bonds, and N p is the number TABLE I. Thermodynamic Properties of the RNAs.
N T L T m1 T m2 a (cid:104) χ (cid:105) T L ∆ G ‡ ab hTR HP 31 22 55 (54) c
81 (74) c a The temperatures are in the units of ◦ C and ∆ G ‡ in k B T . b The free energy barrier ∆ G ‡ is estimated based on thenumber of nucleotides, N [42]. c For hTR hairpin, meltingtemperatures measured in experiments [28] are shown inparentheses. of such pairs. The structural overlap function quanti-fies the similarity of a given conformations to the nativeconformation. It is unity if the conformation is identi-cal to the native state. In T -quench kinetics simulations,if the value of the structural overlap function exceeds athreshold, (cid:104) χ (cid:105) T L , the trajectory is deemed to be com-pleted, and the folding time τ i is recorded; (cid:104) χ (cid:105) T L is thethermodynamic average at the lower simulation temper-ature at which RNA molecules are predominantly folded(Table I). In addition to χ , we also calculated the aver-age value of (cid:104) R g (cid:105) , measurable in scattering experiments(SAXS or SANS), to assess the temperature dependenceof compaction of the RNA molecules. T -Quench Folding: To prepare the initial struc-tural ensemble for T -quench simulations, we first per-formed low-friction Langevin dynamics simulations. Thesimulation temperatures are chosen to be 1.2 times higherthan the second melting temperature (in Kelvin unit) toensure that completely unfolded conformations are pop-ulated. After generating a sufficiently long trajectory toensure that the chain has equilibrated, the unfolded con-formations are sampled every time steps. Finally, wecollected hundreds of conformations, which were used asinitial structures in the T -quench folding simulations.In order to initiate folding, starting from an unfoldedstructure, we quenched the temperature to T S and gen-erated folding trajectories using Langevin or Browniandynamics simulations by varying the solvent viscosityfrom η = 3 . × − to − Pa · s (cf. water viscosity η w ≈ − Pa · s). The viscosity is directly related tothe friction coefficient as γ = 6 πηR where R is the ra-dius of coarse-grained beads. For each condition, at least100 folding trajectories are generated. Folding time τ i ismeasured by monitoring the overlap function, χ , in eachtrajectory i , and folding rates were calculated by averag-ing over M trajectories, k F = τ − MFPT = (cid:16) M (cid:80) Mi τ i (cid:17) − [43]. We used two values of T S . One is T S = T m1 , which isthe lower melting temperature in the heat capacity curve(Figure 3), and the other is T S = T L < T m1 , which is ob-tained by multiplying a factor 0.9 to T m1 in Kelvin unit.The values of T L and T m1 are listed in Table I. Data Analysis Using Kramers’ Rate Theory:
The simulation data for RNA is analyzed using Kramers’theory [14] in which RNA folding is pictured as a barriercrossing event in an effective one-dimensional landscape
FIG. 2. Schematic of effective one-dimensional landscape ofRNA folding from the unfolded state (U) to the folded state(F). Parameters, ω a and ω b , determine curvatures of the freeenergy surface at the reactant basin (U) and the saddle point. ∆ G ‡ is the height of the free energy barrier. (see Figure 2). For our purposes here, it is not relevant todetermine the optimal reaction coordinate because we es-timate the values of the barrier heights theoretically andobtain the two frequencies ω a and ω b (see below for thedefinition) by fitting the simulation data to the theory.According to transition state theory (TST), the reactionrate is expressed as k TST = ω a π exp (cid:18) − ∆ G ‡ k B T (cid:19) , (6)where k B is the Boltzmann constant and T is the tem-perature. The rate, k TST , gives us an upper bound of thetrue reaction rate since in the TST there are no recrossingevents once RNA reaches the saddle point [1, 2].The Kramers’ folding rate in the intermediate to high η range may be written as k KR = ω a πω b (cid:32)(cid:114) γ ω b − γ (cid:33) exp (cid:18) − ∆ G ‡ k B T (cid:19) , (7)where γ is the friction coefficient and ω a and ω b are theparameters that determine curvatures of the free energysurface at the reactant basin and saddle point (maximumin the free energy surface), respectively (Figure 2). It isassumed that, in the vicinity of the saddle point, thefree energy may be approximated by a parabola G ( x ) = G ( x b ) − mω b ( x b − x ) where x is an unknown reactioncoordinate, x b is the position of the saddle, and mω b = − ∂ G ( x ) ∂x .In the high friction limit, k H KR ∼ ω a π ω b γ exp (cid:18) − ∆ G ‡ k B T (cid:19) (cid:16) γ (cid:29) ω b (cid:17) , (8)which shows that the folding rate should depend on theinverse of the friction coefficient. When the friction issmall, the rate linearly approaches the TST limit, k M KR ∼ ω a π exp (cid:16) − ∆ G ‡ k B T (cid:17) = k TST ( γ (cid:28) ω b ) . If we further consider the time scale at which localequilibrium is achieved (very weak damping limit), therate for the effective one-dimensional landscape becomes[1] k LKR ∼ γ ∆ G ‡ k B T exp (cid:18) − ∆ G ‡ k B T (cid:19) . (9)In this regime, barrier crossing is controlled by energydiffusion [44], and the TST is no longer valid. Theseextremely well-known results, used to analyze the sim-ulations, can be summarized as follows: Kramers’ the-ory predicts that k LKR ∝ γ in the low friction regime, k KR reaches a maximum in moderate friction ( k MKR ), and k HKR ∝ γ − in the high friction regime. The folding ratesover the entire range of γ can be fit using [45] k − = k LKR − + k TST − + k HKR − . (10) Barrier Heights in the Folding Landscape:
Inorder to use Eq. 7–10 to analyze simulation data, the freeenergy barrier to folding has to be calculated. However,estimating barrier heights is nontrivial in complex sys-tems because the precise reaction coordinate is difficultto calculate or guess, particularly for RNA in which ioneffects play a critical role in the folding reaction. In or-der to avoid choosing a specific reaction coordinate, weappeal to theory to calculate the effective barrier height.One of us has shown [46], which has been confirmed byother studies [47], that for proteins the free energy bar-rier ≈ √ N where N is the number of amino acids. In thecontext of RNA folding, we showed that there is a robustrelationship between the number of nucleotides ( N ) andthe folding rates, k F ≈ k exp( − αN . ) [42]. Experimen-tal data of folding rates spanning 7 orders of magnitude(with N varying from 8 to 414) were well fit to the theoryusing α = 0 . and k − = 0 . µ s (speed limit for RNAfolding) [42]. Therefore, in this study, we estimate thefree energy barrier based on N alone. A clear advantageof using the theoretical estimate is that it eliminates theneed to devise a reaction coordinate. The values of thebarrier height for the two RNAs are given in Table I.In summary, our strategy in this study is to (1) con-duct folding simulations using the TIS RNA model to ob-tain the folding rates by varying the solvent viscosity andthen (2) examine the applicability of the Kramers’ theoryto RNA folding by fitting the rates using Eqs. 7 to 10.In order to calculate the reaction rates in Kramers’ the-ory, the barrier height ∆ G ‡ and frequencies ω a and ω b are needed (Eq. 7). This would require an appropriatereaction coordinate to characterize the folding landscape.We eliminate the need for creating a specific reaction co-ordinate by estimating ∆ G ‡ from the length of RNA ( N ) based on a previous study [42] and by using the other twoquantities ω a and ω b as fitting parameters. RESULTS
Thermal Denaturation:
We calculated the heatcapacities of the HP and the PK at the monovalent salt
FIG. 3. Temperature dependence of thermodynamic prop-erties at 200 mM of monovalent salt concentration. (A, B)
Heat capacity of (A) hTR HP and (B) BWYV PK. The redlines are heat capacities, C ( T ) , computed from the T-REMDsimulations. Black lines in panel A are UV absorbance melt-ing profiles ( δA / δT ) at 260 nm (dotted line) and 280 nm (dot-dash line) experimentally reported elsewhere [28]. The scaleof δA / δT is not relevant because we only compare the posi-tions of the peaks. The melting temperatures for the HP aregiven in Table I. For the PK, the values of T m are predictions. (C, D) Structural overlap functions ( χ , red solid) and radiusof gyration ( R g , blue dashed). (E, F) Populations of folded(purple), intermediate (green and cyan), and unfolded states(yellow) as functions of T . concentration of 200 mM (Figure 3). The heat capaci-ties have two distinct peaks, which indicate there is atleast one intermediate between the unfolded and foldedstates. This finding is consistent with previous experi-mental and simulation studies [28, 48, 49]. From the posi-tion of the peaks, we determined the two melting temper-atures, T m1 and T m2 , whose values are listed in Table I. Itshould not go unnoticed that the melting temperatures, T m1 and T m2 , for the hTR HP are in excellent agree-ment with experiments, demonstrating the effectivenessof TIS model in predicting the thermodynamic proper-ties of RNA. The experimental heat capacity curve is notavailable for BWYV PK at 200 mM salt concentration,and hence the values reported in Table I serve as predic-tions. Viscosity Dependence of the Folding Rates,Kramers Turnover and Absence of Internal Fric-tion:
Friction dependent folding rates obtained fromthe T -quench simulations are shown in Figure 4. In FIG. 4. Friction (viscosity) dependence of folding rates at simulation temperatures T S = T L and T m1 for hTR HP (A, B)and BWYV PK (C, D). See Table I for the numerical values of T S . Folding rates (blue circles) are normalized by the valuesfrom the transition state theory, k TST (Eq. 6). Error bars, which are presented with 95% confidence level for each data point,lie within the size of the circles. The data in the moderate to high friction regime ( η ≥ − Pa · s ) were fit to Eq. 7 (lines incyan). The data in the low friction regime ( η ≤ − . Pa · s ) were fit to Eq. 9 except rates of BWYV PK at T L . With use of ω a and ω b by fitting to Eq. 7, the rates for the entire range of η are well represented by the connecting formula, Eq. 10 (dashedline). The results of Brownian dynamics simulations with hydrodynamic interactions at the water viscosity ( η = 10 − Pa · s) areshown in red in panels B and D for hTR HP and BWYV PK, respectively.TABLE II. Fitting Parameters. ω a ω b α ω a ω b πγ at η w hTR HP T L (22 ◦ C) 0.38 0.87 0.75 3.1 µ s − T m1 (55 ◦ C) 0.062 2.6 0.86 1.5 µ s − BWYV PK T L (20 ◦ C) 0.018 1.2 0.094 0.19 µ s − T m1 (52 ◦ C) 0.017 1.7 0.65 0.27 µ s − the high friction regime, η (cid:63) − Pa · s, the foldingrates k F decrease as the friction is increased. This be-havior is found in both HP and PK at both temper-atures, T L and T m1 . In the moderate friction regime, − (cid:62) η (cid:62) − Pa · s, the folding rates reach maximumvalues. For η ≥ − Pa · s, we fit the values of k F toEq. 7 with ∆ G ‡ ≈ . √ N k B T . By adjustment of thetwo free parameters, ω a and ω b , Eq. 7 quantitatively ac-counts for the simulation data (lines in cyan in Figure 4,parameters are summarized in Table II). Thus, the vari-ation of k F ∝ η − in the high friction regime shows thatKramers’ theory accurately describes the dependence ofthe folding rates on η of these two RNA constructs. Weconclude that even for RNA, driven by electrostatic inter-actions, folding could be pictured as a diffusive processin an effective one-dimensional landscape. The quanti-tative account of simulation data on k F using Kramers’theory at high η shows the absence of internal friction inthe folding process of these RNA constructs.As η decreases, there is a maximum in the rate followedby a decrease in k F at low η , which shows the expectedKramers turnover (Figure 4). For η ≤ − Pa · s, the de- pendence of the rates is k F ∝ η α with a positive α (linesin purple in Figure 4, and values of α are in Table II).In contrast to the high friction case, the low η depen- FIG. 5. Variations in the flux through the two pathways as a function of viscosity: (A) hTR HP and (B) BWYV PK. Thedefinition of states for each RNA is schematically shown on the right. (A) hTR HP structure naturally splits into two helices,S1 and S2 (Figure 1A). The folding pathways are classified if either S1 forms first or S2 forms first. The fraction Φ is thenumber of trajectories in which S1 forms first divided by the total number of trajectories. (B) BWYV PK has two hairpinstems, S1 and S2, allowing us to classify the pathways in the same manner as in panel A. The error bars indicate 95% confidenceintervals. dence, that is, α value, varies with each RNA molecule.When the ω a and ω b obtained from the fitting to Eq. 7 inthe high friction regime are used, the rates for the entirerange of η are well described by the connection formula,Eq. 10 (dashed line in Figure 4). At T = T m1 , the low η dependence is in quantitative accord with the theory.This is remarkable because there is no additional fittingparameter in Eq. 10 to account for the dependence of η α in the low η regime. Although there are some deviationsat T = T L case, the overall rate dependence showingturnover at moderate friction is well characterized by theKramers’ theory. Viscosity Effects on Hairpin Folding Pathways:
The hTR HP has two regions of consecutive canonicalbase pairs, which we label stem 1 (S1) and stem 2 (S2)(Figure 1A). Four noncanonical base pairs are flanked byS1 and S2. Because of the differences in base pairingbetween these regions, the folding pathways may be vi-sualized in terms of formation of S1 and S2 separately.It is clear that S1 is more stable than S2, and we expect the former to form first in the folding process accordingto the stability principle suggested by Cho, Pincus, andThirumalai (CPT) [26]. In order to assess if the differ-ence in stability leads to friction-induced changes in theflux between the two pathways (S1 forms before S2 or vice versa ), we calculated the fraction of pathways ( Φ )from the folding trajectories, which is obtained by count-ing the number of trajectories that reach the folded stateby first forming S1. We found different trends betweenthe two temperatures (Figure 5). At T m1 , the dominantpathway (labeled I) is characterized by formation of themore stable S1 at all values of η . The flux through I is ≈ . at η values close to η w (the water viscosity) andthat of minor pathway (II) (1 − Φ) ≈ . in which S2forms first followed by S1 (figure 5A). This finding is inaccord with the expectation based on the relative stabil-ities of S1 and S2 [26]. The dominance of pathway I at T m1 suggests that folding starts away from the loop withthe formation of a base pair between nucleotides G1 andC29 and the HP forms by a zipping process.Interestingly, at the lower temperature T L , we find that Φ changes substantially as η increases (Figure 5). At η in the neighborhood of η w , Φ is only ≈ . , which im- plies that at T L folding predominantly occurs in the lessdominant pathway (II), by first forming the less stableS2. This finding may be understood using our previous FIG. 6. A typical folding trajectory of hTR HP simulated at high viscosity ( η = 10 − Pa · s ) with T S = T L . Time seriesin panel A shows the structural overlap function, and that in panel B shows the number of base pairs formed in each stemregion (blue, stem 2; cyan, noncanonical; green, stem 1) along with several snapshots of representative conformations. In thistrajectory, S2 transiently forms three times before the RNA folds. The folding was initiated with the formation of S2, followedby the non-canonical base pairs and S1 at last. (B, inset) Averaged number of the transient S2 formations before hTR HPreaches the folded state depending on η . study on P5GA, a 22-nucleotide RNA hairpin contain-ing only WC base pairs [38]. We found that, althoughthere are multiple ways for P5GA to fold, the most prob-able route is through formation of a short loop (SL) thatinitiates nucleation of base pair formation involving nu-cleotides close to the loop. With that finding in mind,we can rationalize the flux changes at T L . The entropyloss ( ∆ S ) due to loop closure, which in hTR HP wouldbring the two uracil bases (Figure 1) close enough to ini-tiate a G–C base pair (nucleation step), would be small( T ∆ S ≈ k B T ln 5 ). Once the G–C base pair near theloop forms, zipping occurs leading to HP formation. At T m1 > T L , S1 formation occurs first, which necessarilyinvolves long loop (LL) formation that brings (cid:48) and (cid:48) ends close. At high temperature this process is facile eventhough T ∆ S ≈ k B T ln 30 . When the (cid:48) and (cid:48) are close,the highly favorable enthalpy gain due to the formationof a number of favorable WC base pairs compensates forthe entropy loss due LL formation. The argument given above to explain the Φ values at T L can be substantiated by analyzing a typical foldingtrajectory at η w = 10 − Pa · s shown in Figure 6. Be-fore folding occurs, there are several (three times in thisparticular trajectory) attempts to form S2 involving thefavorable SL, as found in P5GA hairpin [38]. This stepis the expected initiation step in helix nucleation. How-ever, formation of S2, needed for growth of the helix,is disrupted because S2 is inherently unstable. Conse-quently, I2 unfolds and pauses in that state for a longtime (Figure 6). In the fourth attempt, the formationof two base pairs near the loop is followed by formationof the noncanonical base pairs, followed by S1, resultingin the folding of the HP. Interestingly, the transient S2formation is only observed at the higher friction regime(Figure 6B, inset). At η > − Pa · s , there are, on av-erage, 5 ∼
10 attempts of S2 formation before the RNAfolds, whereas it does not apparently occur at lower η ,which is dominated by energy diffusion. Viscosity Alters the Flux through the ParallelRoutes in BWYV PK Folding:
In BWYV PK fold-ing, there are two potential intermediates, I1 character-ized by the formation of the more stable stem 1 or I2where only stem 2 is formed. In Figure 5B, we show Φ as a function of viscosity. In contrast to the hTR HPcase, the pathway through I1 is always dominant at allvalues of η at both the temperatures. At the viscosityof water, the fraction Φ ≈ . . This result is consistentwith experimental studies indicating that I1 is the majorintermediate [50, 51]. Our previous study also showedthat the thermal and mechanical (un)folding occur pre-dominantly through the I1 state [49]. The present resultsshow that I1 state is not only thermodynamically stable,but also the major kinetic intermediate. Folding of thePK, which occurs by parallel pathways, with the domi-nant one being U → I1 → F ( Φ ≈ . at η w , for example).In contrast to hTR HP, the loop entropy in the PK iscomparable (Figure 1), and hence the flux between thetwo pathways is determined by the CPT stability princi-ple [26].In the dominant pathway, the folding occurs by thefollowing two steps (see Supplemental Movie): (i) stem1 folds rapidly after T -quench ( (cid:104) τ U → I (cid:105) = 0 . ms at η = 10 − ) forming the intermediate (I1) state, andthen (ii) stem 2 folds after a substantial waiting time( (cid:104) τ I → F (cid:105) = 0 . ms). Since there is a large gap in thetime scale between the two transitions, the rate of thewhole process (U → F) is dominated by the second ratedetermining phase ( τ MFPT ≈ ms). Frictional Effects on Individual Steps in Fold-ing:
We have already shown that the rates for the wholefolding process (U → F) of BWYV PK depend on the vis-cosity in accord with Kramers’ theory (Figure 4). Sincethere is a major intermediate, I1, in the reaction process,we analyzed the folding rates by decomposing folding intotwo consecutive reactions, U → I and I → F. Figure 7shows the frictional dependence of the folding rates forthe two transitions; k I → F shows almost the same behav-ior as k U → F since the two time scales are essentially thesame (compare Figure 7 and Figure 4 C, D). It is inter-esting that the rate of the faster transition, k U → I , alsoexhibits the Kramers-type dependence especially in thehigh friction regime, that is, k F ∝ η − for η (cid:63) − Pa · s.This result indicates that, even if the folding reaction in-volves intermediates, (i) the entire rate still exhibits theKramers-type dependence, at least in a case that one ofthe substeps is rate limiting, and (ii) a substep that isnot rate determining to the entire rate constant may alsoshow Kramers-type viscosity dependence. DISCUSSION
Effect of Hydrodynamic Interactions:
In orderto ensure that our conclusions are robust, we also ex-amined the effect of hydrodynamic interactions by per-forming simulations only at the water viscosity ( η w =10 − Pa · s) for both hTR HP and BWYV PK. As shownin Figure 4 (red circles in B and D), the hydrodynamicinteraction (HI) accelerates the folding rates, but its ef-fect is not as significant as changing the viscosity. At η w , hTR HP folds with k F ∼ . − with HI, whereas k F ∼ . − without HI. Thus, the reaction is about1.5 times faster if HI is included. In BWYV PK case, k F ∼ . − with HI, whereas k F ∼ . − withoutHI, leading to a factor of ∼ FIG. 7. For BWYV PK, folding rates are individually cal-culated for two sequential structural transitions through theintermediate, U → I (upper panels) and I → F (lower pan-els). The results of the whole process, U → F, are shown inFigure 4C, D.
Changes in Viscosity Alter the Flux betweenParallel Assembly of RNA:
It is well accepted thatRNA in general and PK in particular fold by parallelpathways [26, 27, 52]. Recently, it was shown unambigu-ously that monovalent cations could change the flux tothe folded state between the two pathways in the VPKpseudoknot. Surprisingly, we find here (see Figure 5) that Φ could be also altered by changing the viscosity for boththe HP and the PK. Although the same prediction wasmade in the context of protein folding [16], it is difficult tomeasure η dependence of Φ because the secondary struc-tures in proteins are not usually stable in the absence oftertiary interactions. This is not the case in RNA. Forinstance, S1 and S2 are independently stable and hencetheir folding could be investigated by excising them fromthe intact RNA. Consequently, Φ as a function of η canbe measured. Based on the results in Figure 5 showingthat by varying η or η and T , our prediction could betested either for the hTR HP or the extensively studiedPK (BWYV or VPK). For example, at T L we find that Φ changes from 0.2 to 0.4 as η is varied over a broadrange for hTR HP. Although not quite as dramatic, thechanges in Φ are large enough for BWYV PK to be de-tectable. The stabilities of the independently folding S1and S2 constructs can be also altered by mutations. Forinstance, by converting some of the non-canonical basepairs neighboring S2 to WC base pairs in the hTR HPwould increase the stability of S2. Because there are avariety of ways (concentration of ions, temperature, andmutations) of altering the independently folding units ofRNA, our prediction that Φ changes with η could bereadily tested experimentally. Speed Limit for RNA Folding:
Based on the ideathat a protein cannot fold any faster than the fastest timein which a contact between residues that has the largest0probability of forming, it has been shown that the speedlimit ( τ SL ) for protein folding is τ SL ≈ µ s [53]. With theobservation that the typical folding barrier height scalesas √ N (see Eq. 10 in Ref. [46]) and analyses of experi-mental data [54], it was shown that τ SL ≈ τ ≈ (1 − µs ,where τ = πγω a ω b is the inverse of the prefactor in Eq. 8.A similar style of analysis of the experimental data showsthat for RNA τ SL ≈ µ s [42]. Here, an estimate of τ SL ≈ πγω a ω b using the values of ω a and ω b in Table IIand γ corresponding to water viscosity yields 0.7 µ s forthe HP and 3.7 µ s for the PK. Alternatively, the value of τ SL = k − where k = k F exp(0 . N . ) ( k F is the foldingrate obtained using simulations) gives τ SL ≈ µ s for theHP and τ SL ≈ µ s for the PK. If τ SL is equated withthe transition path time, then we can compare estimatesmade for DNA hairpins [55] and for RNA constructs (sev-eral PKs and the add riboswitch) [56] obtained using sin-gle molecule experiments. The values range from about1 to 10 µ s . Thus, there are compelling reasons to as-sert from the present and previous theoretical and ex-perimental studies that an RNA cannot fold any fasterthan about 1 µ s . Influence of Dielectric Friction:
In this article,we have treated the electrostatic interactions implicitly,and hence only systematic and viscous dissipative forcesact on the interaction sites of RNA. We have not con-sidered the effects of dielectric friction, which could besignificant even for an ion moving in an electrolyte solu-tion [57–60]. In RNA folding, the many body nature ofthe problem makes it difficult to estimate the magnitudeof the dielectric friction. There are multiple ions, withsignificant ion–ion correlations, that condense onto theRNA in a specific manner dictated by the architecture ofthe native fold [32]. The magnitude of dielectric frictionin this many body system of highly correlated ions couldbe significant, which in turn could affect the kinetics ofRNA folding. Despite this important issue, which hasnot been investigated to our knowledge, it is comfortingto note that experiments as well as simulations reportingviscosity effects on RNA folding appear to be in accordwith Kramers’ theory.
Transmission Coefficients:
The ratio κ = k F k TST shown in Figure 4 can be as small as ≈ − , in thehigh viscosity region. Recently, based on transition pathvelocity as a measure of recrossing dynamics [61], the val-ues of κ have been measured in single molecule pullingexperiments [62] for several DNA hairpins. By fixing themechanical force at the transition midpoint, where theprobability of being folded and unfolded are equal, thefolding trajectories were used to estimate that κ ≈ − [62]. For RNA hairpins, it is known that folding timesobtained by T -quench are larger by at least 1 order ofmagnitude relative to times obtained by quenching theforce [30]. Thus, the calculated values of κ are not incon-sistent with experiments on DNA hairpins under force.It would be most interesting to examine the viscosity de-pendence of k F by maintaining the RNA molecules undertension. CONCLUSIONS
Using the TIS coarse-grained model, we investigatedthe thermodynamics and folding kinetics of a hairpinand an H-type pseudoknot RNA molecule, focusing onthe dependence of the folding rates on the solvent vis-cosity. From temperature-quench folding simulations,we showed that the folding rates follow the so-calledKramers turnover; the rate increases in the low frictionregime and decreases at high friction, with a maximumrate at moderate friction. For both the hairpin and thepseudoknot, the dependence of the folding rates betweenmoderate and high friction regime is robust and is inaccord with the Kramers’ theory. We find clear η − de-pendence in the folding rates, leaving little doubt thatRNA folding involves a diffusive search in an effectivelow dimensional folding landscape.A major potentially testable prediction is that in the η values that are accessible in experiments the flux betweenpathways by which RNA folds depends on η . Becausethe stabilities of the individual stems could be altered inRNA easily, our prediction is amenable to experimentaltest. SUPPORTING INFORMATION
Movie of a representative trajectory of BWYV PKfolding.
ACKNOWLEDGEMENT
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