Force unfolding kinetics of RNA using optical tweezers. II. Modeling experiments

Optical tweezers have been used to study folding/unfolding (F-U) of RNA hairpins (14,15,16,17). The experimental setup consists of the RNA molecule flanked by doublestranded DNA/RNA handles; the entire molecule is tethered between two polystyrene beads via affinity interactions. The handles are poly mer sp acers required to screen interactions between the RNA molecule and the beads and to p revent direct contact of the beads. One of the beads is held in the op tical trap; the other bead is controlled by a piezoelectric actuator to app ly mechanical force to the ends of the RNA molecule. In hopping experiments a given constraint, i.e. a fixed force or a fixed extension, is applied to the exp erimental system while both the force and the extension of the molecule are monitored as a function of time. Close to the transition force (around 10-20 pN for RNA or DNA hairpins at room temperature (13,14)), a hairpin molecule can transit between the folded (F) and the unfolded (U) states, as indicated by the change in the molecu lar extension: the longer extension represents the unfolded single-stranded conformation; the shorter one to the folded hairpin. From the lifetimes of the single RNA molecule in each of the two states, we can obtain the rates of the F-U reaction (1,14). Both the unfolding and folding rate constants are force-dependent following the Kramers-Bell theory (17,18,19). From their ratio the force dep endent equilibriu m constant for the F-U reaction can be obtained.

In order to obtain accurate information about the molecule under study it is important to understand the influence of the experimental setup, including the handles and the trapped bead, on the measurements. In a recent simulation, Hy eon and Thirumalai (20) examined the relationship between the amplitude of the F-U transition signal and the magnitude of its fluctuations at various handle lengths. On the other hand, experimental results have shown that the F-U kinetics was dependent on the trap stiffness (21). Several questions then arise: how different is the measured rate from the intrinsic molecular rate, i.e., the F-U rate of the RNA in the absence of handles and beads? What are the optimal working conditions to obtain the intrinsic molecular rates? To address such questions, we previously p roposed a model (22), which considered the effect of the trapp ed bead and the handles on a two-state RNA folding mechanism. In this work, we further advance our simulation by incorporating a mesoscopic model introduced by Cocco et al. (23) that takes into account the sequence-dep endent foldin g en ergy. We have then ap p lied this model to a simple hairpin, P5ab (14). We investigate how the measured rates vary with the characteristics of the exp erimental setup and how much they differ from the intrinsic molecu lar rates of the individual RNA molecule. In a comp anion p ap er, we have also measured the F-U kinetics of the RNA hairpin by op tical tweezers (1). The theoretical and experimental results agree well.

The organization of the paper is as follows. In Sec. 2 we introduce the model for the experimental setup and describe its thermodynamic properties. We also analyze the characteristic timescales of the sy stem. In Sec. 3 we d iscuss the influen ce of the different elements of the experimental setup on the kinetic rates. Limitations of the instrument which affect the measured F-U rates, such as the force-feedback time lag and the data acquisition bandwidth, are also consid ered. B ased on the various timescales of the different dynamical processes described in Sec. 2, we develop a kinetic model for the RNA hairpin and a numerical algorithm used to simulate the hopping dynamics in Sec. 4. In Sec. 5 we carry out a detailed analysis of the dependence of the kinetic rates on the characteristics of the exp eriment (such as the len gth of the handles and the stiffness of the trap ), and compare our simulation results with the exp erimentally measured F-U rates. A search of the best fit between theory and exp eriments allows us to predict the value of the intrinsic molecular F-U rate of the RNA molecule. Finally , we discuss what are the optimal experimental conditions to minimize the effect of the instrument and to obtain the intrinsic molecu lar rates.

Hopping experiments (1) were done with a single RNA hairpin P5ab, a derivative of the L-21 Tetrahymena ribozyme. The kinetics of this RNA with 1.1 Kbp handles had been studied previously (14). In Fig. 1 we show a schematic picture of the setup used in such experiments. To manipulate the RNA molecule two RNA/DNA hybrid handles are attached to its 5’-and 3’-ends. The free ends of the handles are attached to micron-sized poly sty rene beads. One bead is held fixed in the tip of a microp ip ette while the other bead is trapped in the focus of the laser, which is well described by a harmonic potential of stiffness ε b . The configurational variables of the system x b , x r , 1 h x and 2 h x are the extensions of each element (trapp ed bead, RNA molecule and handles resp ectively) along the reaction coordinate axis (i.e. the axis along which the force is ap p lied). The external control parameter X T is the distance between the center of the optical trap and the tip of the microp ip ette. In an exp eriment, the app lied force f and the distance x b are measured. From the value of x b the chan ges in the distance between the two beads x, corresponding to the end-to-end distance of the molecular construct formed by the two handles and the RNA molecule (Fig. 1), can be obtained; x = X Tx b -R b1 -R b2 , where R b1 and R b2 are the beads radii. A more detailed description of the experimental setup is given in (1).

In hopping experiments, the force f and the changes in the extension x as a function of time are recorded. The structural changes of the RNA molecu le can be identified with the sudden changes in force and extension, here referred to as ∆f and ∆x respectively.

Experiments are carried out in two different modes: the passive and constant-force modes. In the passive mode (PM ) the distance X T between the center of the trap and the tip of the micropipette is held fixed. In PM hopping experiments both the extension x and the force f hop when the molecule switches from one state (F or U) to the other. In the constant-force mode (CFM ) the force is maintained constant by implementing a forcefeedback mechan ism. In CFM hopp ing exp eriments the changes in the state of the RNA molecu le can be identified with the measured chan ges in the extension x of the molecu lar construct. Experimentally, P5ab folds and unfolds with no apparent intermediates (1,14). The experimental traces show jumps in force and extension, ∆f and ∆x, that correspond to the full unfolding or foldin g of the RNA hairp in. From the data we can extract the mean lifetimes of the F and the U states of the molecule, τ F and τ U , at a given force. The foldin g and unfolding rates, kF and kU, are the recip rocal of τU and τF, respectively.

The experimental setup is modeled as previously described (22). The bead confined in the optical trap is considered as a bead attached to a spring whose stiffness equals the trap stiffness, ε b , and the double-stranded DNA/RNA handles are modeled by the worm-likechain (WLC) theory (24,25), which describes the elastic behavior of polymers by two characteristic parameters: the contour (L) and the persistence (P) length. In our previous model (22), we considered a two-state model for the F-U of an RNA molecule. Here we extend that approach by including intermediate configurations of the hairpin where a partial number of bp s are op ened sequentially starting from the end of the helix. In this descrip tion, the molecule can only occup y intermediate configurations in which the first n bp s are unp aired and the last N-n are p aired, where N is the total number of bps in the native hairp in. The index n is used to denote such intermediate configurations (Fig. 2), e.g. the F state corresp onds to n = 0 and the U state to n = N. This rep resentation excludes the existence of other non-sequential breathing intermediate configurations that might be relevant for thermal denaturation (26). For a given value of the control parameter (gen erically denoted by y, e.g. X T or f), and for each configuration n of the RNA molecule, we can define the thermodynamic potential G(y,n) as ( 22):

where G 0 (n) is the free energy of the RNA hairp in at the configuration n and G’ y (n) describes the energetic dep endence of the experimental sy stem on the control p arameter. Note that the term G’ y (n) is sequence indep endent, so all information about the sequence is included in the term G 0 (n). The critical control p arameter (F c or X T c ) is the valu e of the control parameter at which the F and U states are equally p opulated. For the P5ab hairp in, the value of the critical force measured in the experiments is around 14.5 pN (1).

In the ideal-force ensemble (hereafter referred as IFE), the force exerted up on an RNA hairp in is the control p arameter (y = f) and the system reduces to the naked RNA molecu le without beads and handles. The contribution G’ to the free energy of the RNA molecule is given by (27):

where W r (z) is the work required to stretch the molecular extension x r of the ssRNA from x r = 0 to x r = z. In our exp eriments (Fig. 1), where handles, b eads and the RNA molecu le are linked, the natural control p arameter in Eq 2.1 is y = X T . T his defines what has been denoted as the mixed ensemble (h ereafter ref erred as M E) (28). In such case the contribution G’ in Eq. 2.2 has been deriv ed in (22):

r, is the mean value of x α for a given valu e of the control p arameter X T and for a given configur ation n of the RNA hairp in. V b represents the optical trap p otential,

where ) (x f α is the equ ilibrium force extension curv e for the element α (22). These different contributions to the thermody namic p otential are free-energies corresp onding to the trapp ed bead, the handles and the ssRNA molecu le. Therefore, in the ME, the thermodynamic potential given by Eq. 2.1 depends not only on the RNA properties but also on the characteristics of the different elements of the setup , such as the stiffness of the trap and the contour and p ersistence lengths of the handles. In order to extract thermody namic information of the RNA molecule from the exp erimental results, we need to take into account the contribution from each of the elements formin g the setup (22). In the Supp lementary M aterials we show how the shap e of the thermody namic potential (Eq. 2.1) is modified for different values of the stiffness of the trap and the length of the handles. The characteristics of the experimental setup change the value of the maximum of the free-energy along the reaction coordinate, which is related to the kinetic barrier sep arating the F and U states, and thus influences the kinetics of the F-U reaction. The dependence of the F-U rates of a DNA hairpin on the stiffness of the trap has been already reported (21).

In p articular, when the exp erimental sy stem gets softer, the fluctuations in force decrease and the M E app roaches the IFE. The free energy landscap e G(X T c ,n

corresponding to the IFE, in the limit where the effective stiffness ε eff of the whole exp erimental sy stem vanishes. The effective stiffness ε eff is comp uted as:

where ε b is the stiffness of the trap and ε x is the rigidity of the molecular construct (i.e. the molecule of interest plus handles, see Fig. 1). Therefore the thermodynamics of systems with longer h andles (i.e. softer handles) and softer trap s app roaches to the IFE case, as shown in Fig. S1 in the Supplementary M aterials. However, thermodynamics alone is not sufficient to understand the influence of the experimental setup on the kinetics. For this we have to consider a kinetic description of the system. This is the subject of the next sections.

The dy namics of the glob al sy stem presented in Fig. 1 involves processes occurrin g at different timescales. Therefore, in order to study the kinetics, it is essential to analyze the different characteristic times of the system: the relaxation time of the bead in the trap τ b; the relaxation time associated with the elastic lon gitudinal mod es for the handles and the ssRNA, denoted by τ handles and τ ssRNA respectively; the time k F-U -1 in which the RNA hairpin folds and unfolds; the base-pair (bp) breathing time k bp - 1 . Table 1 reviews the different characteristic times of the experimental sy stem.

• Bead: The time at which the bead in the optical trap relaxes to its equilibrium position is given by ( 22):

where γ (γ = 6πR b1 η, where η is the viscosity of water) is the frictional coefficient of the bead, ε b and ε x are the stiffness of the trap and the molecular construct resp ectively . Typ ical exp erimental values are: ε b ≈ 0.02 -0.15 pN/nm for the trap stiffness; R b1 ≈ 0.5 -1.5 µm for the bead radius; L h ≈ 130 -1300 nm and P h ≈ 10 -20 nm for the contour and persistence lengths of the handles respectively, which result in values for the stiffness of the molecular construct of ε x ≈ 0.15 -1.5 pN/nm (computed by using the WLC (24,25) theory at forces about 15 pN). For these values, τ b lies in the range 10 -5 -10 -3 s. The corner frequency of a tethered bead is defin ed as the recip rocal of τ b . Events that occur at frequencies higher than the corner frequency of the bead cannot be followed by the instrument.

• Handles and ssRNA: The relaxation time associated with the longitudinal modes of the handles and ssRNA when a given force f is app lied to their ends can be estimated from polymer theory (29) as:

where η is the viscosity of the water, (η≈ 10 -9 pNs/nm 2 ), P is the p ersistence length of the double-stranded or single-stranded nucleic acids respectively, T is the temperature of the bath and k B is the Boltzmann constant. For the handles used in the exp eriments (P handles = 10-20 nm) the relaxation time lies in the range 10 -8 -10 -6 s. For the ssRNA corresponding to the unfolded P5ab hairpin (N ss RNA = 49 bases), τ ssRNA is approximately 3.5•10 -9 s.

• RNA molecule: There are two different timescales associated with the k inetics of the RNA molecule. The first timescale is the overall kinetic rate k F-U giv en by :

where k F and k U are the folding and unfolding rates. The rate k F-U depends on the sequence and structural features. Under tension at which a hairpin hops, typical values of k F-U are in the range 0.1-100 Hz. The second timescale corresponds to the characteristic frequ ency for the op ening/closing of sin gle bp s,

, which is estimated to be around 10 6 -10 9 Hz (30,31).

In summary , the dy namics of the system p resents the following hierarchy of timescales:

Ap art from the timescales associated with each of the different elements in the sy stem there are also intrinsic characteristic timescales of the instrument. It is imp ortant to consider them in order to understand and correctly analy ze the results obtained from the exp eriments.

• Instrumental times: There are three characteristic timescales that limit the performance of the instrument. The first timescale is defin ed by the bandwidth B which is the rate at which data are collected in the exp eriments. Collected data rep resent an average of the instantaneous data measured over a given time window of duration 1/B. Typ ical values for the bandwidth used in the exp eriments lie in the ran ge from 10 to 1000 Hz. The second imp ortant characteristic timescale is giv en by the time lag of the feedback mechanism, T lag , implemented in the CFM . In our exp eriments (1), typ ical values for T lag are 100 ms. In order to app roach the IFE one would like Tlag as small as p ossible. Recently , a new dumbbell dual-trap op tical tweezers instrument has been develop ed (21,32). This design op erates without feedback and can maintain the force nearly constant over distances of about 50 nm. Nevertheless, regard less of the sp ecific instrumental design, there is a limitation in the measurement that is imp osed by the corner frequency of the bead: the b ead does not resp ond to force chan ges that occur faster than τ b . In our exp erimental setup this limiting time is ap p roximately 10 -4 s.

The third timescale ranges from seconds to minutes and corresp onds to the drift of the instrument. The drift is a low frequency noise due to mech anical and acoustic vibrations, air currents, thermal exp ansion in resp onse to temp erature chan ges and other causes. Since the drift does not affect the occurrence and detection of F-U transitions, we did not take into account drift effects in our model.

From the force and extension traces recorded in hopp ing experiments (1), we can extract the rates of the F-U reaction. These traces reflect the resp onse of the whole exp erimental system (Fig. 1) not only the individual RNA molecule. In add ition, data collected are averaged ov er a bandwidth B, and the mechanism imp lemented in the CFM has a finite resp onse time, T lag . In this section, we analy ze the effect of the experimental setup on the measured F-U rates as comp ared to the intrinsic molecular rates.

There are d ifferent experimental mod es and different way s of analyzing the exp erimental data which result in different valu es of the rates of the reaction; ultimately , we wish to obtain values as close as p ossible to the intrinsic molecular rates. The intrinsic molecu lar rate, 0 F-U k , corresp onds to the rate measured in an IFE where a fixed force F c (i.e., the critical force value where the F and U states are equally p opulated) is applied directly to the RNA molecule. In the followin g paragraphs, we introduce the different rates that are exp erimentally measurable, i.e. under the CFM and PM . These rates have been defin ed in our comp anion p ap er (see (1) for details).

• CFM rates: The CFM rates are the foldin g and unfo ldin g rates measured when the instrument op erates in the CFM at a given force. In what follows we will consider the critical rate CFM c k , which is the F-U rate (Eq. 2.8) measured at the critical force valu e where the molecule sp ends the same amount of time in the F and U states.

• PM rates: The force traces in the PM show that the folding and unfo ldin g transitions occur at different forces, f F and f U resp ectively (Fig. 4). f F and f U are the mean forces in the up p er and lower bounds of the square-like force traces resp ectively . The PM unfolding (fo ldin g) rate at f F (f U ) is then identified with the unfolding (foldin g) rate measured in such PM traces from the lifetime of the folded and the unfold ed states resp ectively . The PM critical rate PM c k is the F-U rate at the force value where the unfoldin g and foldin g PM rates are equal (Fig. 6).

To study the relation between the measured and intrinsic molecular rates, we now consider the different effects that influence the kinetics in the CFM and PM as comp ared with the IFE. Under the exp erimental cond itions, the force exerted directly on the RNA molecu le (f RNA ) is subject to fluctuations due to the dy namic evolution of the different elements in the exp eriment (Fig. 1). There are at least three contributions to these fluctuations: 2 ):

where ε x is the stiffness of the molecular construct. As shown in the Sup p lementary M aterials the effect of the fluctuations given in Eq. 3.1 is to increase the kinetics of the F-U reaction as comp ared with the IFE.

(ii) Base-pair hopping effect: At the timescale at which bps attempt to open and close, k bp -1 , the bead hardly moves (τ b » k bp -1 ). Hence, when a bp forms (dissociates) the handles and the ssRNA stretch (contract), and corresp ondingly there is an increase (decrease) in the force exerted upon the RNA molecule, f RNA . The change in the force f RNA after a bp opens or closes, assumin g that in the timescale k bp -1 the p osition of the bead is fixed, is given by : bp

where ∆x bp is the difference in extension between the formed and dissociated bp . Therefore after the formation (rup ture) of a new bp the force increases (decreases) by an amount given by Eq. 3.2 and the p robability to dissociate (form) it again increases as comp ared with the IFE case. Therefore, the base-pair hopping effect slows the overall F-U kinetics of the RNA molecule.

In the PM the average force exerted up on the system depends on the state of the RNA molecule (Fig. 4). Therefore at the timescale k F-U -1 associated to the F-U reaction, the average force exerted on the RNA molecule will ch an ge by:

where ∆x r is the chan ge on the RNA extension when the molecule unfolds and ε eff is the effective stiffness of the whole exp erimental sy stem given by Eq. 2. 5. The force differen ce (Eq. 3.3) is a consequen ce of the p articular design of exp erimental setup (Fig. 1). For longer handles (i.e. softer handles) or softer trap s the value of the effective stiffness, and hence the force difference (Eq. 3.3), decreases. In this latter case the thermody namic p otential of the whole exp erimental system (Eq. 2.1) in M E app roaches to the IFE case as shown in the Sup p lementary M aterials (Fig. S1).

The overall effect of such fluctuations in the hopp ing kinetics is not straightforward because the F-U rates might be increased due to (i) and (iii), but also decreased due to (ii). In the limitin g case of very soft handles, i.e. when the stiffness of the molecular construct ε x approaches zero (and therefore ε eff =0), all p revious effects (i), (ii) and (iii) tend to disapp ear and the experimental conditions get closer to the IFE. However, the temp oral and the sp atial/force sensitivity are also expected to decrease for softer handles (1). The reason is two-fold. On the one hand, in order to measure the F-U rates, the resp onse of the trapp ed bead must be faster than the F-U reaction, i.e. τb « kF-U -1 . The corner frequ ency of the trapp ed bead (given by the inverse of Eq. 2.6) becomes lower for softer handles, decreasin g the temp oral resolution of the exp eriment. On the other hand, in order to detect accurately enough the force/extension jump s that characterize the F-U transition the handles should be stiffer than the trap or ε x ≥ ε b . Otherwise the signal-tonoise ratio (SNR) would become too low and the exp erimental sign al giv en by the force/extension jump s could be masked by the handles (1). In the current exp erimental conditions ε b ∈[0.035 -0.1] p N/nm whereas ε x ∈[0.15 -1.5] pN/nm so the inequality ε x ≥ ε b is satisfied. It can be shown that when ε x ≥ ε b the magn itude of the force fluctuations described in (i) and (ii), is quite insensitive to the p articular value of ε b . The main effect of ε b is to modify the value of the force difference (iii) (Eq. 3.3), which is minimized by taking ε b as small as p ossible. Therefore, to get estimates closer to the intrinsic molecular rate softer trap s should be used.

The resolution and limitations of the instrument are also imp ortant when acquiring the exp erimental data. In particular, measurements are sensitive to the bandwidth B at which data are collected and to the time lag of the feedback mech anism, T lag :

(iv) Limited bandwidth: If the bandwidth B is not higher than the F-U rates, the time resolution of the measurement beco mes too low to detect the F-U reaction and the measured kin etic rates will be affected.

In the CFM the force-feedback mechanism op erates to comp ensate for the force difference given by Eq. 3

is verified and the feedback mech anism can eff iciently keep the force constant. Otherwise, the feedback mechanism cannot maintain the force constant on timescales where the molecule folds/unfolds. In the latter case, the feedback mech anism leads to distorted rates. We call this the distortion effect (1).

-1 and T lag are much shorter than k F-U -1

, only the effects (i)-(iii) remain. By using longer (i.e. softer) handles and softer trap s these effects are also minimized and the measured r ates should ap p roach the ideal molecular rates. For all the exp erimental setup s we have investigated in this work the sp atial/force resolution is h igh enou gh to detect the F-U reaction (1) and the condition τ b « k F-U -1 holds. Therefore, the op timal conditions to carry out measurements would be to use handles and trap s as soft as p ossible within the limitin g resolution imposed by the experimental setup (SNR>1,

).

Even though current exp eriments (1) do not reveal the p resence of intermediates of the F-U reaction, most of the kinetic effects observed in the exp eriments are not captured by a simp le two-state model that does not include intermediate configurations. In fact, the two-state model only considers the dy namical effects (i) and ( iii), which in crease the RNA F-U kinetics as comp ared with the IFE case. To rep roduce the observed dep endence in the kinetics it is necessary to take into account the base-pair hopp ing effect (ii) in the dy namics. Therefore a multi-state model, as the one p rop osed here, is needed to cap ture the effect of the exp erimental setup on the measured kinetics.

In this section we study the RNA F-U kinetics under the exp erimental conditions by simulatin g the dy namics of the whole sy stem in the PM and CFM . In Sec. 4.1 we describe the model we use for the F-U kinetics of the RNA hairp in. The simulation algorithm is p resented in Sec. 4.2.

To model the kin etics of the RNA hairp in we ad ap t the model by Cocco et al. (23) to our exp erimental setup ; we assume the dy namics of the hairpin to be sequential (see Sec. 2.1). Therefore one-step transitions connect each conf iguration n with its first nearest neighbors in the configur ational sp ace, n+1 and n-1. The dy namical process is then govern ed by the kinetic r ates to go from n to n’ with n’ = n-1, n+1. This kinetic model is schematically dep icted in Fig. 2. The evo lution in time of the conf iguration n of the hairp in is described by a set of coup led master equations:

with n = 0,…N and ( ) 0, (0) 0 k N k denotes the rate of the reverse reaction. The exp erimental system includes different elements such as the handles, the trapp ed bead and the RNA molecule, therefore the F-U kinetics is described by the rates associated to the transitions, ( ) ( )

. Because the bead relaxes much slower than the handles and the ssRNA the kinetics of the hairp in is slaved to the relaxational dy namics of the bead. Consequently the kinetics r ates can be f actorized in two terms:

is computed by using Eq. 2.4 as:

where f α (x) with α = h 1 ,h 2 ,r corresp onds to the equilibrium force extension relation as giv en by the WLC model (24,25). The variables

x + x r , corresp ond to the extension of the handles 1 and 2 and the released ssRNA for the RNA configuration with n opened bps. The choice of the op ening and closin g rates

is based on two assumptions ( 23): (i) The transition state corresp onding to the formationdissociation reaction of a giv en bp is located very close to the formed state. Therefore the op ening rate → k for a given bp dep ends on the p articular bp and its neighbor (i.e., GC versus AU), but does not dep end on the value of the control p arameter y ( e.g.

The rate of closing k ← is indep endent of the sequence and is determin ed by the work required to for m the bp starting fro m the d issociated state. The rates

ar e of the Arrhenius form and are given by : 0 ( ) ( )

The constant k a is a microscopic rate that does not dep end on the particular bp sequence and is equal to the attempt frequency of the molecular bond. The kinetic process defined by → k , ← k is of the activated typ e. The value of → k is a function of the free energy difference ∆G 0 (n) = G 0 (n+1) -G 0 (n) between the two adjacent conf igurations, n and n+1.

Whereas the value of ← k depends on the value of the control p arameter X T and on the value of x b ,

is the free ener gy defined in Eq. 4. 4. The choice of these rates has the advantage that there is only one free parameter, k a , while the rest of p arameters can be obtained from measured thermody namics. This model is an extension of the on e p roposed by Cocco et al. (23), as giv en in Eq. 4.1, by considerin g the ap p rop riate kinetic rates (Eq. 4.5) ad apted to rep roduce the exp erimental CFM and PM .

To simulate the hop p ing exp eriment we benefit from the large separation of timescales between the different elements of the system: τ b » τ ha ndle s ,τ ssRN A (Table 1). We consider that during the time of an iteration step in the simulation, dt = 10 -8 s, the handles and the ssRNA are in local equilibriu m, but the bead in the trap is not. Note that the timescale of iteration is smaller than the relaxation time τ ha ndle s . However, we do not expect our results to change mu ch by taking into account the microscopic dy namics of the hand les, because the most imp ortant dynamical eff ect either in the simulations or the exp eriments comes from the bead in the trap . In fact, the bead is the element of the sy stem with largest dissip ation and slowest relaxation rate as comp ared to the elastic and bendin g modes of the handles and the ssRNA. In our simulation we implement the following algorithm:

• At each iteration step dt:

1. The p osition of the bead trapp ed in the optical p otential x b evolves according to the Lan gevin dy namics of an overd amp ed p article (22):

where ε b is the stiffness of the trap , γ is the friction co efficient of the b ead and f x is the force exerted by the molecular construct on the bead. f x is comp uted as the force need ed to extend the molecular construct (the handles and the ssRNA)

The stochastic term ξ(t ) is a white noise with mean valu e 〈ξ (t)〉 = 0 and v arian ce

From the evolution of the bead p osition, giv en by Eq. 4.6, we obtain the instantaneous values of the molecu lar extension x = X T -xb -Rb1 -Rb2 and force f = εb xb.

, we comp ute the equilibrium value of the extension of the handles and the released ssRNA for the configurations with n and n-1 op ened bp s, by using the WLC mod el (24,25). We then compute the function

where the rates → k and ← k are defin ed in Eq. 4.5.

Return to 1.

• We average the instantaneous data over a bandwidth B.

• In the CFM , at every 1 ms of time, we increase (decrease) the value of the total end-to-end distance X T by 0.25 n m if the m easured for ce differs by more than 0.1 pN below (above) the set p oint force value at which the feedback mechanism op erates.

In this section we comp ute the rates of the F-U reaction fro m the hop p ing traces corresp onding to the CFM and PM simulations. We then comp are them with the exp erimental results (1). We use the free ener gy p arameters given in (33,34,35) to comp ute the free ener gy landscap e at zero force G 0 (n) of the P5ab hairp in at 25 ˚C and in 1M NaCl * . We consider that the mechan ical resp onse of the handles and the ssRNA is characterized by a p ersistence len gth (P) and contour len gth (L) equal to P h = 10 nm, Lh = 0.26 nm/bp for the handles and P ssRNA = 1 nm, L ssRNA = 0.59 nm/base for the ssRNA. In order to analyze the effect of the instrument on the measured rates we study different exp erimental setup s by considering handles of sever al len gths and op tical trap s characterized by different stiffness.

Figs. 3 and4 show examp les of CFM and PM traces obtained from the simulations of the exp erimental system. The distributions of lifetimes obtained either from the exp erimental traces or from the simulations have an exp onential decay (Supp lementary M aterials, Fig. S2), as exp ected for a two-state sy stem. T o extract the rates of the F-U reaction in each mode we have analyzed the simulated data using the same methods as for the exp erimental data (1). We then comp are these rates with the experimentally measured rates p resented in our comp anion p aper (1). The value of the free parameter k a is chosen to optimize the fit between the rates extracted from the simulation traces and the ones measured in the exp eriments. For this fit we used the PM data as explained in the Sup p lementary M aterials (Fig. S3). Notice that the value of k a fixes the timescale unit of the simulation allowing us to establish the connection between the real microscop ic dy namics of the molecule and the mesoscopic descrip tion. We get the char acteristic bp attempt frequency , k a = 2. In what follows we comp are this value with the measured rates in the CFM and PM in order to infer the op timal conditions to obtain rates as close as p ossible to the intrinsic molecu lar rate 0 F-U k .

In Fig. 5 we show the values of the CFM critical rates, C FM c k , obtained from the simulations as a function of the len gth of the h andles (f illed sy mbols) compared with the exp erimental ones (1) (emp ty sy mbols connected by lines). T he agreement between the exp erimental and simulation results in the CFM is reasonable. T he analy sis done in Sec. 3.2 p redicts that the measured critical r ates should converge to the value of the intrinsic molecu lar rate 0 F-U k for softer handles, i.e. lon ger handles. T his is true when the instrument has enough time resolution to resolve the force/extension jumps, i.e. k F-U -1 » T la g , B -1 . However, we do not observe this conver gence, n either in the simulations nor the exp eriments (Fig. 5), probably because in our instrument k F-U -1 ~ T lag ~ 0.1 s, such that the condition k F-U -1 » T la g is not satisfied. In this situation, the measured rates highly dep end on the bandwidth and on the criteria used to an aly ze the data, i. e. the so-called distortion effect discussed in the comp anion p ap er (1). We think that the non-conver gence of the measured r ates for lon g handles to the intrinsic molecular rate 0 F-U k arises fro m distortion effects due to the finite resp onse time of the instrument, T lag . T o validate this hypothesis and to obtain better estimates for the rates, we p rop ose to use the PM data to extract the PM critical r ate PM c k (see Sec. 3.1). In the PM there is no feedb ack mech anism; therefore PM data does not suffer from the distortion effect. Also, by using a bandwidth high enough, i.e. B » k F-U , we waive the dep endence of the m easured rates on the b andwidth. Therefore, the PM critical rate PM c k should p rovide a better estimate of the F-U rate at the critical for ce.

From the PM data, we extract the PM rates. By doing numer ical simulations at different values of X T , we obtain the PM folding and unfolding rates at d ifferent forces. As shown in Fig. 6 the logarithm of the PM folding and unfold in g rates as a function of the force fits well to a straight line, as predicted by the Kramers-Bell theory for two-state systems (18). T he exp erimental measured rates show the same dependence on the force as the simulation results (Fig. 6), suggesting that the model p rop osed p redicts well the location of the transition state (17). T he PM critical rate PM c k is obtained from the intersection of the linear fits to the comp uted data for ln(kU) and ln(kF) as a function of the force. In Fig. 7 we show the measured PM critical rates from the simulation traces (filled sy mbols) as well as the experimental results (1) (emp ty sy mbols connected by lines) as a function of the length of the handles. Two sets of data at the trap stiffness ε b = 0.1 pN/nm and ε b = 0.035 p N/nm are shown. Both exp erimental and simulation results agr ee p retty well. T he bandwidth used, B = 1 KHz, is much greater than k F-U . Hence, the time resolution is sufficient to follow the F-U reaction, and the measured rates are not aff ected by the average of the data over the time window B -1

. Better estimates are obtain ed for the softer trap case as expected.

Finally , in Fig. 8 we comp are the critical PM (filled sy mbols connected by lines) and CFM rates (empty symbols connected by lines) measured in the experiments. T he discrep ancy between the critical r ates PM c k and CFM c k is lar ger for the stiffest trap results (εb = 0.1 p N/nm, upp er p anel) and the longest handles, case in which distortion effects in the CFM are more imp ortant (1). Moreover, the values of the rates PM c k obtained from the PM data in both experim ental setup s (upper and lower p anels) increase for lon ger handles and show a tendency to app roach to the ideal molecular rate value of 13Hz as exp ected (see Sec. 3.2). T hese results confirm our initial exp ectations that, then k F-U -1 is of the order of T la g , the measured rates in the CFM are strongly affected by the distortion effect.

To compare different estimates for the cr itical rates, it is useful to def ine a p arameter that characterizes the reliability of the measurement. We define the quality factor Q as the relative diff erence between the measured r ate (k est. ) and the intrinsic molecular rate 0

As a comp endium of all the results, we show in Fig. 9 the value of Q obtained for the different estimates for the critical rates as extracted from the exp erimental data. T he factor Q is shown as a three-dimensional p lot as a function of the length of the handles and the trap stiffness. We show 3 surfaces, each corresp onding to a different estimate of the rates: the CFM critical r ates ( CFM Dep ending on the RNA molecule (sequence, len gth, foldin g and unfoldin g rates) and the characteristics of the exp erimental setup (trapp ed bead, hand les, feedback time lag and bandwidth), the quality factor of each estimate may change. As a general result we infer that better measurements are obtained for softer traps and longer hand les as lon g as the transition signal is detectable. For fast hopp ers (which have F-U rates which are not much slower than the force-feedback fr equency , as h ap p ens in our study of the P5ab hairp in where k F-U -1 ~Tl a g , ~0.1s), PM rates p rovide better estimates than CFM rates. On the other hand, for slow hopp ers, the PM becomes impractical due to the p resence of drift effects. In the latter case, the CFM is efficient and the CFM critical r ates should be a good estimate for the intrinsic molecu lar rates.

In this work we have introduced a mesoscopic mod el for the study of the foldin g/unfold ing (F-U) force-kin etics of RNA hairp ins in hop p ing exp erim ents using optical tweezers. T he model incorp orates the different elem ents of the exp erimental setup (bead, handles and RNA sequence) and limitations of the instrument (time lag of the constant-force mode and finite bandwidth). We carry out numerical simulations of the prop osed model and comp are them with hopp ing exp eriments in the P5ab RNA hairp in rep orted in our comp anion article (1). T his analy sis allows us to extract the value of the microscop ic attempt frequency k a for the dissociation kinetics of individual base-p airs. The estimate for k a is then used to extract the intrinsic molecular rate for the RNA hairp in, 0 F-U k . We then compare the estimate of the intrinsic molecular rate with the values for the different rates (constant-force mode (CFM ) and p assive mode (PM) rates) obtained under different exp erim ental conditions. T he goal of the research is to infer the optimal cond itions to extract the intrinsic molecular rate of the RNA molecule using data obtained in the different experimental modes: p assive and constant-force. We h ave considered diff erent values of the stiffness of the trap and different len gths of the handles. Due to the comp lexity of the sy stem the quality factor Q (defined as the relative difference between the measured rate and the intrinsic molecular rate) will critically dep end on various p aram eters of the instrument (experimental setup and the instrumental limitations) and the molecule.

Through our analy sis we are able to find the optimum exp erim ental conditions to measure hop p ing rates. Even though our study has been carried out for an RNA hairp in with a fixed sequ ence in an op tical tweezers setup , the methodology and rationale presented here can b e ap p lied to other exp erim ental setup s, such as dumbbell dual-trap optical tweezers (21,32,36,37), other acid nucleic sequen ces, or p roteins (12) ]. In particular, in order to detect the force/extension jump s that characterize the F-U transition the trap should not be stiffer than the handles. Handles. For all exp erim ental mod es it is advisable to use handles as lon g as possible within the resolution limit of the instrument [ 3 Kbp 10 Kbp h L ≤ ≤ ]: (i) the SNR of the extension/force signal must be lar ge enough to follow the F-U reaction and (ii) the corner frequen cy of the bead ( equal to the inverse of its relaxation time) must be much high er than the F-U rate of the hairp in (see the discussion in (1)). Bandwidth. For all experimental modes it is advisable that the bandwidth of data collection is as lar ge as p ossible. Force-feedback frequency. In the CFM it is imp ortant that the frequency of the force-feedb ack mechanism is as high as p ossible. In particular, the force-feedb ack frequency must be higher than the F-U rate, otherwise distortion effects are big and the force-feedback mechanism b ecom es inefficient. If the latter restriction is not satisfied (as happ ens in our study of the P5ab hairp in where k F-U -1 ~ T lag ~ 0. , bp breathing time k bp -1 ; and intrinsic times of the instrument: average samp lin g time B -1 and time lag of force feedback mechanism T la g .

1-10 . The bandwidth used is 1 KHz. We show the m ean for ces, f F and f U , in the up p er and lower p arts of the square-like-sign force traces, corresp onding to the forces at the folded and unfold ed states, resp ectively . Note that the value of such forces, f F and f U , is high er than the on es measured in exp erim ents (1) by about 1-1.5 p N. T his discrepancy is consistent with the fact that the free ener gy p arameters for P5ab used in simulations (32,33,34) corresp onds to higher salt concentrations than the exp erimental ones * . We show the quality factor Q (defined as the closen ess between the measured rate and the intrinsic molecular rate) obtained from differ ent measured critical rates in exp eriments as a function of the len gth of the handles and the trap stiffness. For the CFM exp erimental results we show the Q corresp onding to the CFM critical rates at two different values of the bandwidth, B=200Hz (red) and B=10Hz (black). In blue it is also shown the Q for the PM critical rates extracted from PM exp eriments. Generally , better measurem ents (higher Q values) are obtain ed from softer trap s and longer handles. * Our experiments were performed at 250mM NaCl while our simulations used the free en ergy paramet ers obtained from (33,34,35) at 1M NaCl. T he presence o f s alt in the solvent stabilizes folded con form ations of RNA molecules due to the larger screening o f the elect rostatic repulsion between the phosphates groups. Therefore, the RNA native structure at higher salt concent rations has a lower free energy (i.e it is more stable) and the critical value o f the fo rce for the folding/un folding reaction is larger. T his is in agreement with the fact that the values o f the critical fo rce that we obtain in our simulations are about 1-1.5 pN above the ones measured in the experiments. In order to reproduce the thermodynamic properties of the RNA molecule from our simulations we must shift the forces by 1-1.5 pN downward. In addition, the salt concentration might also affect the F-U kinetics. T he value we estimate for the attempt frequen cy k a by fitting simulations and experiments already incorpo rates the salt correction.

TABLES