Efficient methods for determining folding free energies in single-molecule pulling experiments
Aurelien Severino, Alvaro Martinez Monge, Paolo Rissone, Felix Ritort
EEfficient methods for determining folding freeenergies in single-molecule pulling experiments
A Severino , A M Monge , P Rissone and F Ritort , Dept. F´ısica de la Mat`eria Condensada, Universitat de Barcelona, C/ Mart´ı iFranqu`es 1, 08028 Barcelona, Spain CIBER-BBN de Bioingenier´ıa, Biomateriales y Nanomedicina, Instituto de SaludCarlos III, 28029 Madrid, SpainE-mail: [email protected]
Abstract.
The remarkable accuracy and versatility of single-molecule techniquesmake possible new measurements that are not feasible in bulk assays. Among these, theprecise estimation of folding free energies using fluctuation theorems in nonequilibriumpulling experiments has become a benchmark in modern biophysics. In practice, theuse of fluctuation relations to determine free energies requires a thorough evaluationof the usually large energetic contributions caused by the elastic deformation of thedifferent elements of the experimental setup (such as the optical trap, the molecularlinkers and the stretched-unfolded polymer). We review and describe how to optimallyestimate such elastic energy contributions to extract folding free energies, using DNAand RNA hairpins as model systems pulled by laser optical tweezers. The methodologyis generally applicable to other force-spectroscopy techniques and molecular systems.
Keywords: stochastic thermodynamics, single molecule experiments, nucleic acidsthermodynamics a r X i v : . [ q - b i o . B M ] D ec etermination of folding free energies in single-molecule experiments
1. Introduction
Predicting free-energy differences is a central problem in molecular biophysics. Proteinfolding [1], DNA hybridization [2], ligand binding, CRISPR–Cas9 RNA editing [3, 4],are molecular reactions whose fate is determined by the free-energy difference betweenreactants and products. Finding methods to extract free-energy, enthalpy andentropy differences is an essential task in biochemistry, where most of these quantitiesare measured by employing bulk techniques such as calorimetry, UV absorbance,fluorescence, surface plasmon resonance, among others [5]. Bulk methods yield resultsthat are incoherent temporal averages over a large population of molecules that are indifferent states. The signal is masked by the dominant species and reactions, limitingthe capability of detecting rare non-native states and reaction pathways. Moreover, bulkmolecular transformations often exhibit strong hysteresis effects rendering equilibriumdifferences inaccessible.By monitoring molecules one at a time, techniques such as single-moleculefluorescence [6], single-molecule translocation across nanopores [7] and single-moleculeforce spectroscopy [8] overcome the previous limitations and therefore have becomekey experimental tools in many laboratories worldwide. In particular, force-spectroscopy techniques using atomic-force microscopy, magnetic tweezers, acoustic-force spectroscopy and laser optical tweezers (LOT) have been extremely fruitful,revolutionizing biophysics over the last three decades ‡ .The main advantage of force-measuring techniques (as compared to fluorescence andother non-invasive optical technologies) lies in the possibility to measure simultaneouslyforce and displacement, giving direct access to mechanical work measurements insingle-molecule pulling experiments. Similarly to bulk assays, pulling experimentsare often carried out under irreversible conditions, in principle providing bounds(rather than direct estimates) of equilibrium free-energy differences. The developmentof the non-equilibrium thermodynamics of small systems (also known as stochasticthermodynamics) [10, 11, 12, 13] during the past three decades has provided thetheoretical concepts and methods needed to extract free-energy differences from repeatedirreversible work measurements. Exact results such as the Jarzynski equality [14]and the Crooks fluctuation theorem [15] are now commonly employed to extract free-energy differences from single-molecule pulling experiments [16, 17, 18, 19, 20, 21]. Aparticularly useful application is the measurement of the folding free energy of nucleicacids and proteins (∆ G ) which is equal to the free energy difference between the foldedstructure and the unstructured random coil in the solvent. This quantity can be obtainedfrom pulling experiments by measuring the free energy difference (∆ G ) between thefolded and unfolded-stretched states of the considered experimental system taken attwo force values, and by deriving from it the value of ∆ G . However, a general problemin the manipulation of small systems using single-molecule techniques is that we cannot ‡ LOT invention revealed to be a breakthrough in laser physics and has been awarded with the NobelPrize in Physics in 2018 [9]. etermination of folding free energies in single-molecule experiments G directly. It is insteadthe free energy difference ∆ G between the folded and unfolded-stretched states of the entire considered experimental system taken at two force values. In order to retrieve the’bare’ molecular properties such as the value of ∆ G in a single molecule, we thereforeneed to retrench from ∆ G some contributions stemming from the experimental set-up(e.g. optical trap in LOT or cantilever in AFM and the linkers used to manipulate themolecule under study). These so-called stretching corrections play a crucial role becausetheir contribution to the total free energy difference ∆ G are much larger than the freeenergy one wants to extract ∆ G , making the accurate estimation of the latter a difficulttask. Although there are several studies on the influence of the instrumental artifactson the folding kinetics in single-molecule experiments [22, 23, 24, 25, 26, 27, 28, 29],their influence regarding the determination of the folding free energies at zero force has,to the best of our knowledge, never been addressed in detail.In this work we will rigorously examine these experimental contributions in LOTshowing how to efficiently and reliably estimate the free energies of formation of DNAand RNA hairpins in unzipping assays. The same methodology is applicable to proteinsand ligand binding interactions using LOT or other force measuring techniques aswell (AFM, magnetic tweezers and so on). The development of novel and refinedstatistical analysis methods to extract differences in thermodynamic potentials (freeenergy, enthalpy, entropy, chemical potential, ...) will become crucial with the recentboost of high-throughput single-molecule techniques (magnetic tweezers, acoustic forcespectroscopy) that will require fast and efficient algorithms.The content of this paper is organized as follows. In sections 2 and 3 we describethe typical experimental setup of LOTs and then define and discuss the differentcontributions to the total free energy. The two following sections (4 and 5) feature howto estimate these contributions when analyzing DNA and RNA molecules. Section 4 firstcovers the situations in which it is possible to introduce the so-called effective stiffnessapproximation, which considerably simplifies the computation of the large stretchingterms. When this approximation fails, a more elaborate approach requiring a carefulevaluation of the elastic response of the linkers and of the force probe is needed, andthis is the focus of section 5. Finally, in section 6 we present the conclusions.
2. Model of the experimental setup
We consider the case of a nucleic acid (DNA or RNA) hairpin pulled by LOT. In LOT,the total distance λ between the tip of the micropipette and the center of the optical trapis the control parameter of the experiments. As shown in figure 1(a),(b) the distance λ can be decomposed as: etermination of folding free energies in single-molecule experiments λ ( f ) = (cid:40) x b ( f ) + x h ( f ) + x d ( f ) + const (folded state) ,x b ( f ) + x h ( f ) + x ss ( f ) + const (unfolded state) , (1)depending on whether the molecule is folded or unfolded. Here x b ( f ) is the displacementof the bead from the center of the optical trap, x h ( f ) = x h ( f ) + x h ( f ) accounts forthe sum of the elongations of the two double-stranded handles, x ss ( f ) is the end-to-endextension of the single-stranded unfolded molecule, and x d ( f ) is the average extensionof the folded hairpin. This last term is defined as the distance between the attachmentpoints of the handles to the 5’ and 3’ ends of the hairpin and is usually called ’hairpindiameter’ (whence the index d ). All these extensions are evaluated against the x -(pulling)axis and at a given force f . The ’const’ stands for an arbitrary shift in thetotal distance λ which does not affect the analysis.In general, a small perturbation δλ generates a small change in the applied force δf . The extent of this variation is the effective stiffness of the system k eff = δf /δλ andit equals the slope of the experimental force-distance curve (FDC). Therefore, accordingto the above definition and to the prescription given in (1), the inverse effective stiffnessof the folded (F) and unfolded (U) branches are respectively given by:1 k Feff ( f ) = 1 k b ( f ) + 1 k h ( f ) + 1 k d ( f ) , (2a)1 k Ueff ( f ) = 1 k b ( f ) + 1 k h ( f ) + 1 k ss ( f ) . (2b)where k b ( f ) corresponds to the stiffness of the bead in the optical trap, k h ( f ) isthe sum of the two handles stiffness and k d ( f ), k ss ( f ) stand for the molecular stiffnessof the folded and unfolded molecule, respectively.In particular, k d ( f ) is modelled as the stiffness to orient a dipole of diameter d (typically d = 2 nm for DNA and RNA hairpins [30]) along the force axis [31]. Recallingthat in general k − = δx/δf , the dipole stiffness can be derived from the well-knownrelation between a dipole average extension (which is here equal to the average extensionof the folded hairpin) and the force f to which it is subjected: x d ( f ) = d (cid:20) coth (cid:18) f dk B T (cid:19) − k B Tf d (cid:21) (3)where T is the temperature of the heat bath around the dipole and k B is Boltzmannconstant.An analytic expression for k ss and k h can be obtained by describing the elasticresponse of nucleic acids in their single-stranded and double-stranded form with theWorm-Like Chain (WLC) polymer model and its interpolation formula [32], f ( x ) = k B T P (cid:34)(cid:18) − xL c (cid:19) − − xL c (cid:35) (4) etermination of folding free energies in single-molecule experiments Figure 1. (a,b). Laser optical tweezers (LOT) experimental setup.
Themolecule is tethered between two polystyrene beads using two dsDNA (or dsRNA oreven dsDNA/DNA hybrids) handles. Arrow towards the center of the optical trapindicates the direction of the force. λ = x b + x h + x m (with x h = x h + x h ) is therelative distance between the center of the optical trap and the tip of the micropipette. x m equals x d when the molecule is folded (a) or x ss when the molecule is unfolded(b). (c). Sketch of the force versus relative extension (extension divided by contourlength) for each elastic element showing their respective energy contributions (shadedareas). (d).
Elastic energy contribution of each element vs force and comparison withthe typical energy of formation (dashed line, ∆ G ) for a 20bp DNA or RNA hairpin. where x is the average extension of the molecule ( x = x ss for the unfolded hairpin, x = x h for the double-stranded handles) and P is the persistence length, i.e. the typicaldistance along the polymer backbone over which there is an appreciable bending dueto thermal fluctuations. L c is the contour length, i.e. the end-to-end distance of thefully straightened polymer, which can also be written as L c = nd b with n being thetotal number of monomers in the polymer and d b the length per monomer. In general,inverting (4) to get x ( f ) is not an easy task (the full computation is reported in theAppendix A) and the solution depends on the system parameters.Finally, the stiffness of the polymer can be obtained by differentiation of (4): k ( x ) ≡ ∂f ( x ) ∂x = k B T L c P (cid:34)(cid:18) − xL c (cid:19) − + 2 (cid:35) . (5)Given (4), it is also possible to further take into account the elastic deformation ofthe stretched polymer by performing the substitution L c → L c (1 + f /Y ), with Y theYoung modulus of the stretchable polymer [33, 34], i.e. the resistance to deformationof the system to an applied uniaxial stress. In this case the contour length becomesforce-dependent and the corresponding model is called the extensible WLC. By contrastequation (4) where L c is constant is known as the inextensible WLC. The latter has etermination of folding free energies in single-molecule experiments P is a measure of the mechanical stiffness of the polymerbeing strongly sensitive to environmental conditions (e.g. ionic strength, temperature,solvation, etc..). Polymers with P (cid:29) L c effectively behave as rigid rods, whereas if P ≤ L c polymers are bent at the scale of the contour length by thermal forces. Itis important to mention that P does not only depend on the ionic concentration andtemperature [36] (as predicted by polyelectrolyte theories) but also on experimentalparameters such as contour length [35]. For example, at 1 M NaCl, recent single-moleculestudies have shown that, for short (a few tens bases) ssDNA molecules, P = 1 .
35 nm[37] whereas for long ssDNAs ∼
13 kb P = 0 .
76 nm [38]. On the other hand, for shortssRNA molecules P = 0 .
75 nm [39] and for long ∼ P = 0 .
83 nm [35].These values are significantly lower than for double-stranded nucleic acids (dsDNA anddsRNA) where P = 50 nm for dsDNA [40] and P (cid:39)
60 nm for dsRNA molecules [41].
3. Stretching contributions and free-energy recovery
Let us suppose that initially at t = 0 we have a molecule in thermal equilibrium atthe folded (or native, N ) state at a given value λ of the control parameter. Then, weperturb the system by applying a predetermined time-dependent forward (F) protocol, λ F ( t ), that starting at λ at t = 0 ends at an arbitrary λ at a time t . The mechanicalwork W done along this process equals to: W = (cid:90) λ λ f dλ . (6)The Crooks Fluctuation Theorem (CFT) [15] relates the mechanical work done ona system in a set of arbitrary irreversible measurements with the equilibrium free-energydifference of this system between λ and λ , ∆ G = G ( λ ) − G ( λ ). It reads: P F ( W ) P R ( − W ) = exp (cid:18) W − ∆ Gk B T (cid:19) , (7)where P F ( W ) is the probability distribution of the work done in the F process and P R ( − W ) is the probability distribution of the work measured in the time-reversed (R)process (i.e. starting in thermal equilibrium in λ and performing the time-mirroredprotocol so that: λ R ( t ) = λ F ( t − t )). The derivation of the CFT has become amilestone for single-molecule experimentalists, allowing the measurement of free-energydifferences in conditions where traditional bulk experiments are unfeasible. By pullingsingle molecules using LOT or magnetic tweezers it is possible to recover molecularfree-energy differences from irreversible work measurements [17, 42]. The CFT (Eq.7)implies the well-known Jarzynski equality [14]: etermination of folding free energies in single-molecule experiments (cid:28) exp (cid:18) − Wk B T (cid:19)(cid:29) F = exp (cid:18) − ∆ Gk B T (cid:19) , (8)Note that the average (cid:104)· · · (cid:105) F is evaluated over P F ( W ) (an analogous equality holdsfor the reverse process). It is important to bear in mind that the free energy ∆ G obtainedusing the CFT (7) (or the Jarzynski equality (8)) contains several contributions due tothe stretching of the different parts forming the experimental setup. These are themolecule under study, the molecular handles and the optically trapped bead (figure1(a),(b)): ∆ G = ∆ G + ∆ W m + ∆ W b + ∆ W h . (9)∆ G is the free energy of formation of the molecule at zero force, which is equalto the free energy difference between the folded and unfolded hairpin conformations insolution (i.e. without optical trap and handles and without any applied force). Thequantities ∆ W i ( i = m, b, h) are the reversible work differences between the state ofthe i th setup element (optical trap, handles or molecule) at λ (where the hairpin isfolded and subjected to a minimum force f min ) and λ (where the hairpin is unfoldedand subjected to a maximum force f max ). Mathematical definitions of these quantitiesfor the LOT setup are given in the subsections below.As depicted in figure 1(c,d), for typical unfolding forces in DNA and RNA hairpins(15 - 25 pN), (9) is dominated by the trap contribution, while the other terms havethe same order of magnitude. Therefore, an accurate measurement of ∆ G requiresprecise knowledge of all the different energetic contributions involved in the mechanicalunfolding of the molecule. The molecular contribution ∆ W m in (9) accounts for the reversible work needed tostretch the molecule under study and it can be written as:∆ W m = (cid:90) x ss ( f max )0 f ( x ss ) dx ss − (cid:90) x d ( f min )0 f ( x d ) dx d , (10)where f ( x ss ) and f ( x d ) are the equilibrium force-extension curves of the unfoldedand folded molecule, respectively (albeit different mathematical functions the same letter f will be used to lighten the notation). The first term in the right-hand side of (10)corresponds to the reversible work needed to stretch the unfolded molecule from itssingle-stranded random coil conformation at f = 0 up to f max and it can be computedfrom the WLC model, Eq. (4). The second term in the right-hand side of (10) is thereversible work required to orientate the molecular diameter along the force axis. It canbe written as: etermination of folding free energies in single-molecule experiments (cid:90) x d ( f min )0 f ( x d ) dx d = f min · x d ( f min ) − (cid:90) f min x d ( f ) df . (11)where x d ( f ) is given by (3). The term ∆ W b + ∆ W h , which corresponds to the sum of the reversible work requiredto displace the bead from the center of the optical trap (∆ W b ) plus the reversible workneeded to stretch the handles (∆ W h ), can be generally written as:∆ W b + ∆ W h = (cid:90) x b ( f max ) x b ( f min ) f ( x b ) dx b + (cid:90) x h ( f max ) x h ( f min ) f ( x h ) dx h = (cid:90) f max f min f (cid:18) ∂f∂x b (cid:19) − df + (cid:90) f max f min f (cid:18) ∂f∂x h (cid:19) − df = (cid:90) f max f min fk b ( f ) df + (cid:90) f max f min fk h ( f ) df . (12)Note that each element in the setup is substantially different. In particular, thebead in the optical trap can be well approximated by a Hookean spring, whereas theelastic response of the handles and the single-stranded molecule (plus the diameter)is strongly nonlinear (see below). The contribution of these two terms in Eq.(9) isoften large. In particular, the energy required to displace the bead from the center ofthe optical trap is considerably higher as compared to the other terms. A schematicdepiction of this fact can be seen in figure 1(c), where the shaded areas below the curvesrepresent the work W obtained according to (6) using realistic elastic parameters forDNA and RNA hairpins. A further important simplifaction can be carried out when the FDC along the foldedbranch is approximately linear over the integration range of forces. Such a situationcorresponds by definition to a scenario where the slope (or stiffness) is constant, i.e. k Feff (cid:54) = k Feff ( f ). It allows one to readily perform the integration in eq. (12) which is nowreduced to the simple task of integrating an affine function:∆ W b + ∆ W h = (cid:90) f max f min f (cid:18) k b + 1 k h (cid:19) df ∼ = (cid:90) f max f min fk Feff df = f − f k Feff , (13)where we used the fact that the stiffness of the dipole modelling the folded hairpin isconsiderably larger than the other terms in (2a), so that k Feff = ( k − + k − + k − ) − ∼ =( k − + k − ) − , and where the constant stiffness assumption is used in the last equalityof the right hand side of (13). Linearity of the FDC is a good approximation if theintegration range is not too large (for example, when f max − f min ≈ etermination of folding free energies in single-molecule experiments F o r ce [ p N ] Distance [nm]
12 15 k effF k effU
12 15 18 F o r ce [ p N ] Distance [nm]
20 6040 G G G G G G G G G G GCCCCCCCCCC A A A A A A A A AAATTTTTTTTTT A A
16 pN/s6 pN/s − slope = 0.93(7)
16 pN/s6 pN/s W [k B T] P F ( W ) , P R ( - W ) W [k B T] l og P F ( W ) P R ( − W )
285 290 295 300 (( W = ¢G (a) (b) (λ ,f min ) (λ ,f max ) Figure 2. Free-energy recovery of CD4 DNA with short handles (a).
Sequence of CD4 DNA (top panel). FDCs and integration range for the work W (bottom panel). Demonstration of the linearity of the FDCs in the integration range(inset) plus linear fits to the folded (solid line in the inset) and the unfolded branches(dashed line in the inset). (b). Forward (solid lines) and reverse (dashed lines)work distributions for two different pulling speeds calculated in the integration rangeindicated in (a) panel. Crossing points between work distributions are tagged as solidpoints. The CFT verification is shown as inset. Error bars have been obtained usingthe Bootstrap method. the DNA and RNA hairpins considered in the next section.). Above all, linearity of theFDC certainly requires a linear optical trap of constant stiffness [31]. We will refer tothis approximation as the effective stiffness method .
4. The effective stiffness method
The effective stiffness approximation discussed in section 3.3 provides an easy methodto treat the elastic contributions of the experimental setup. Here we provide two typicalscenarios where (13) provides a reliable estimation of the free energy of formation ∆ G of DNA and RNA hairpins. In section 4.1 the case of the CD4 DNA hairpin with shorthandles is reported. Then in section 4.2 we discuss the case of the CD4 RNA hairpinwith long handles. The use of short dsDNA handles ( ∼
29 bp each) in single-molecule experiments has beenshown to increase the precision of kinetic measurements due to their enhanced signal-to-noise ratio as compared to long handles [31]. Short handles also makes easier theevaluation of the stretching contributions. In fact, the large stiffness of short handles ascompared to the trap stiffness, k h (cid:29) k b , implies that k eff (cid:39) k b to first order. As the trapstiffness itself can be considered nearly force independent k Feff is, therefore, constant alongthe folded branch, and the effective stiffness approximation (13) becomes applicable. etermination of folding free energies in single-molecule experiments λ and λ . In the forward (reverse) process the systemstarts in thermal equilibrium at λ ( λ ) and it is driven out of equilibrium following apredetermined protocol λ F ( t ) ( λ R ( t )) until λ ( λ ) is reached. For each experimentalrealization the work W is calculated according to (6). Note that, in the force range atwhich the molecule typically unfolds and refolds (12 - 17 pN in figure 2(a)), the FDCsare linear in force (inset of figure 2(a)). Therefore, the conditions required to use theeffective stiffness method are fulfilled (13).In figure 2(b) we show the F and R work distributions calculated for two pullingspeeds (6 and 16 pN/s) in the same integration range. According to the CFT (7), thework value at which both distributions cross (black solid points) equals to ∆ G . Notethat, since the integration range is the same, ∆ G does not change with pulling speed,as it is required for an equilibrium quantity. We emphasize the validity of the CFT byplotting the function log P F ( W ) /P R ( − W ) as a function of W in k B T units. Accordingto (7), this function is linear in W with slope 1 and with a y-intercept equal to ∆ G (bothin k B T units). As expected, the experimental data (solid points) satisfy the previousrelation (see inset of figure 2(b), where the solid line is a linear fit to the experimentaldata).Once we have measured ∆ G using the CFT, we subtract the stretchingcontributions to recover ∆ G . According to (9), we have:∆ G = ∆ G − ∆ W m − ∆ W b − ∆ W h . (14)The term ∆ W m is calculated using (10). In order to model the ssDNA elasticresponse (i.e. f ( x ss ) in (10)), we use the WLC model (4) with a persistence length P equal to 1.35 nm and an interphosphate distance d b equal to 0.59 nm/base [37], so that L c = (2 n bases + 4) × .
59 nm/base ≈
26 nm. On the other hand, the term ∆ W h + ∆ W b is calculated using the effective stiffness method (13) with k Feff = 0 . ± .
002 pN/nm(obtained by a linear fit of the FDCs, see inset in 2(a)).In table 1 we report the values we obtained for ∆ G , ∆ G , as well as theaforementioned stretching contributions.Results for ∆ G are in very good agreement with the theoretical ones obtained usingthe nearest-neighbour model for DNA either using Mfold parameters (∆ G = 51 k B T )[43] or the ones derived from unzipping experiments (∆ G = 48 k B T ) [44]. In what follows, we first discuss the characteristics of long handles in subsection 4.2.1,explaining why sometimes the effective stiffness method can be applied, while othertimes it cannot. To illustrate the two distinct situations, we first present in section4.2.2 a scenario based on the CD4 RNA hairpin, where the effective stiffness method etermination of folding free energies in single-molecule experiments G [ k B T ] ∆ W m [ k B T ] ∆ W h + ∆ W b [ k B T ] ∆ G [ k B T ]DNA short 295 ± ± ± ± ± ± ± ± Table 1. Fluctuation theorem and stretching contributions for CD4 DNAhairpin (short handles) and CD4 RNA hairpin (long handles). (DNAshort, first row)
Reported energies for the integration range [ λ , λ ]=[20, 80] nmcorresponding to a force range ( f min , f max ) = (13 ,
17) pN. (RNA long, secondrow)
Reported energies for the integration range [ λ , λ ]=[30, 85] nm correspondingto a force range ( f min , f max ) = (18 ,
22) pN. Error bars obtained after averaging theresults over four (DNA short) and five (RNA long) molecules at two pulling speeds,respectively. is applicable with long handles, just as with short handles (Sec. 4.1). Secondly,the development of a general approach for long handles, beyond the effective stiffnessapproximation, is covered in section 5 and exemplified with the CD4L12 RNA hairpin.
Long handles, ∼
500 bp each, typicallyrepresent a bigger challenge than their short counterpart because they are significantlysofter. This implies that long handles stiffness features a noticeable force dependence k h = k h ( f ), especially in the lower range of forces experimentally accessible with LOT.Moreover, the magnitude of k h is now lower and typically comparable to the trapstiffness, k h ∼ k b . Thus, since k Feff (cid:39) ( k − + k − ) − , the term k h significantly contributesto k Feff . This, together with the clear force dependence of k h , implies in turn that theeffective stiffness is not constant but depends on force: k Feff = k Feff ( f ). Consequently, uponcalculating stretching contributions, the terms ∆ W b , ∆ W h need to be evaluated morecarefully. At closer inspection, however, the use of long handles does not invalidate per se the effective stiffness approximation (13). The validity of (13) relies on the assumptionthat k Feff is constant over the integration range [ λ , λ ]. Indeed, in many situations, suchas with CD4 RNA hairpin, the actual integration range occurs at forces high enough sothat k h (cid:29) k b and k Feff can be taken as constant. Whenever this assumption does nothold another approach must be used. There are two typical scenarios. On the one hand,if the integration range is large (e.g. for molecules featuring a pronounced hysteresis),the force-dependence of the stiffness k Feff = k Feff ( f ) cannot be neglected (note that evenif k Feff changes marginally from pN to pN, the overall change on the whole integrationrange can be significant). On the other hand, if we reach low enough forces (e.g. byusing a molecule that refolds at very low forces), the effective stiffness also exhibits forcedependence. Indeed at low forces k h (cid:28) k b , hence k Feff ∼ k h , and as k h = k h ( f ) is steepat low f , so will k Feff be.Provided that the handle stiffness k h ( f ) and the force stiffness of the trap k b ( f )are known with a good precision, the integrals in (12) can in principle be carriedout easily, irrelevantly of k Feff being non-constant. This corresponds however to anidealized scenario which rarely occurs in practice. To begin with, the elastic properties etermination of folding free energies in single-molecule experiments k b canhave a very significant energetic impact. For instance, a modest deviation of 5% from k b = 0 .
08 pN/nm to k b = 0 .
075 pN/nm, results in a change of a dozen k B T in ∆ W b when integrated between 2 an 12 pN. Changes in the value of k b and even non-linearforce corrections in k b ( f ) do inevitably occur in LOT, not only on a day-to-day basis(depending on the laser focusing, alignment, power, intensity or temperature) but alsowithin the same day on a molecule-to-molecule basis, since the beads used for performingexperiments can usually slightly vary in size, and the trap stiffness directly depends onthis. A slight force dependence in k b ( f ) also occurs if the optical plane of the bead shiftswith force. Hence, we see that k h ( f ) and k b ( f ) are usually not characterized preciselyenough for the integrals in (12) to be computed reliably.To address the aforementioned issues, we will introduce in section 5 a novelmethodology to retrieve the optimal stiffness profile k b ( f ) and k h ( f ) directly from theFDCs obtained in pulling experiments with LOT. Before doing so, let us however showan example where long handles and the effective stiffness approximation go in pair: theCD4 RNA hairpin. The effectivestiffness method can be applied to the CD4 RNA hairpin which is a molecule showingnearly reversible folding-unfolding kinetics at the accessible pulling speeds [17, 39]. Themolecule has the same sequence as hairpin CD4-DNA presented in Section 4.1 butreplacing thymines by uracils (top panel of figure 3(a)). In the present case, the RNAhairpin is inserted between two ∼
500 bases-long hybrid RNA/DNA handles [45]. Thus,the molecular construct is formed by the RNA hairpin plus the two long hybrid handles.Pulling experiments were performed analogously as described in section 4.1.Due to the narrowness of the region in the FDCs (figure 3(a), bottom panel) wherefolding-unfolding events of CD4 RNA take place, the effective stiffness k Feff remainsfairly constant over the force range experimentally probed. This linearity of the FDCsis evidenced in the inset of figure 3(a) and justifies the use of the effective stiffnessapproximation. By fitting the FDCs slopes in the highlighted region, we obtain a valuefor k Feff equal to 0 . ± . λ , λ ] = [30, 85] nm, which correspondsto the force interval ( f min , f max ) = (18 ,
22) pN. As we did in section 4.1, the F and Rwork distributions are calculated for two pulling speeds (2 and 20 pN/s) and are shownin figure 3(b). Note that the crossing point between both distributions corresponds tothe work value equal to ∆ G . The CFT (7) is satisfied for CD4 with long handles, ascan be seen in the inset of figure 3(b). We can thus subtract from the obtained ∆ G etermination of folding free energies in single-molecule experiments F o r ce [ p N ] Distance [nm]
G G G G G G G G G G GCCCCCCCCCC A A A A A A A A AAAUUUUUUUUUU A A F o r ce [ p N ] Distance [nm] k effF k effU (λ ,f min ) (λ ,f max ) W [k B T] P F ( W ) , P R ( - W ) W = ¢G (a) (b) W [k B T] l og P F ( W ) P R ( − W ) ((
20 pN/s2 pN/s
20 pN/s2 pN/s slope = 0.91(6)
330 340320 3500-1010
Figure 3. Free-energy recovery CD4 RNA hairpin with long handles(a).
Sequence of CD4 RNA (top panel). FDCs and integration range for the work W (bottom panel). Visual evidence of the linearity of the FDCs in the integrationrange (inset) plus linear fits to the folded (solid line in the inset) and the unfoldedbranches (dashed line in the inset). (b). Forward (solid lines) and reverse (dashedlines) work distributions for two different pulling speeds calculated in the integrationrange indicated in (a) panel. Crossing points between work distributions are tagged assolid points. The CFT verification is shown as inset. Error bars have been obtainedusing the Bootstrap method. the stretching contributions ∆ W h + ∆ W b using the effective stiffness method, along theexact same lines as in 4.1. As a last step, the term ∆ W m in (9) is calculated using theWLC model (4) with P = 0 .
75 nm and an interphosphate distance d b equal to 0.665nm/base, so that L c ≈
29 nm, higher than for the CD4 DNA molecule.We report in table 1 the values we obtained for ∆ G , ∆ G , as well as for thestretching contributions. The measured value for ∆ G (70 ± k B T ) is compatible withthe previous single-molecule measurements obtained in LOT assays at 100mM Tris HClpH 8 and 1 M NaCl (∆ G ≈ k B T ) [39] and with the Mfold prediction (∆ G = 68 k B T ) [43]. We conclude that the effective stiffness approximation is valid for determiningfolding free energies from irreversible work measurements if the integration range isnarrow enough so that FDCs along the folded branch have constant slope in such range(i.e. the effective stiffness k Feff can be taken as constant).
5. Beyond the effective stiffness method
In the previous sections we introduced the effective stiffness method, testing its reliabilityin addressing the analysis of both short and long handles. We also gave evidence that itsvalidity is limited to the case of a linear elastic response and that when this conditionis not fulfilled a more general methodology becomes necessary. This is the subjectcovered by section 5.1 where we present a novel technique going beyond the effectivestiffness approximation. Then, in section 5.2 we present an application of this method etermination of folding free energies in single-molecule experiments
As can be seen in (1) the force-extension profile λ ( f ) depends on x b and x h , and theseare, by definition, related to the stiffness through: x i ( f ) = (cid:90) f k − i ( f (cid:48) ) df (cid:48) , dx i df = 1 k i ( f ) for i = b, h . (15)This hints at the fact that FDCs (i.e. the λ ( f ) profile) might allow us to retrievethe stiffness profiles needed to estimate the elastic energy contributions from bead andhandles in (12). To realize this in practice, we must assume the elastic response of thetrap and the one of the handles can be parametrised by some reasonable physical model.Starting with the handles, we will assume that the extensible WLC model (ext-WLC)is a good description. k h ( f ) = k ext-WLCh ( f ; { P, d b , Y } ) , x h ( f ) = x ext-WLCh ( f ; { P, d b , Y } ) , (16)where we introduced the usual WLC elastic parameters (i.e. persistence length P , Youngmodulus, Y , and monomer length d b ). Then, we can either model the trap stiffness asconstant, or as a linear function of force: k b ( f ) = k b , + αf , x b ( f ) = 1 α log (cid:18) αk b, f (cid:19) , (17)where α quantifies the linear dependence and k b , is the stiffness at zero force ( x b ( f ) isobtained by integrating as in (15)).Note that we can rewrite (1) as: λ ( f ) = x h ( f ) + x b ( f ) + x d ( f ) δ N + x ss ( f ) δ U + λ , (18)where we used a delta-Kronecker-like notation ( δ N ( U ) = 1 if the molecule is in the Native(Unfolded) state and zero otherwise) and explicitly introduced the offset λ , whichaccounts for the fact that the molecular extension is always measured with respect tothe micropipette. If we now rewrite the explicit dependence with respect to our modelparameters, (18) becomes: λ ( f ) = x h ( f ; { P, d b , Y } ) + x b ( f ; { α, k b , } ) + x d ( f ) δ N + x ss ( f ) δ U + λ ≡ M ( f ; { P, d b , Y, α, k b , , λ } ) , (19)where we have denoted M as the overall model underpinning the λ ( f ) response. As(1) illustrates, the knowledge of a handful of physical parameters fully determines theFDC for the N and U branches. The key idea behind our methodology is that the inverseimplication is also true: knowing λ ( f ) and given M we can extract P, d b , Y, α, k b , etermination of folding free energies in single-molecule experiments k h ( f ) and k b ( f ) for all f using the models in (16) and (17) eventually obtaining the stretchingcontributions through numerical integration of (12). Crucially, this can be done withoutany a priori knowledge of the parameters of the experimental setup.In practice, however, the fitting procedure requires a FDC featuring enoughcurvature to be able to constrain the model, and even so, the number of parametersto fit is too large for a 2-dimensional curve, so that some additional considerations mustbe taken into account. Firstly, reasonable bounds/priors on the allowed values for theparameters must be set. Secondly, it is convenient to assume that certain parametersplay a minor role in the overall FDC shape (such as Y ) or are characterized well enough(e.g. the monomer length for dsDNA) to be fixed at some nominal value and notfitted. Thirdly, computing the handles extension x h = x h ( f ) using the extensible WLCcan be slow and numerically inaccurate as it normally requires to perform a numericalinversion of f = f ( x h ). To address this, we introduce in Appendix A a formula toexplicitly invert the WLC which can then be used in (19). Fourthly, to get as manypoints as possible to constrain the fit, we have aligned all the FDCs in the starting pointso they share an identical λ offset (i.e. ’const’ in (1)(a,b)). After all these steps, fitting λ ( f ) = M ( f ; { P, d b , Y, α, k b, , λ } ) is affordable.In the following section we will show a concrete examples of the FDCs fittingprocedure and its application to extract the stretching contributions. The effective stiffness method may work well when the range of force integration is nottoo large. This condition is met in molecules exhibiting mild hysteresis. For moleculesshowing large hysteresis in pulling cycles the limits of integration f min and f max are faraway and the effective stiffness k Feff cannot be considered constant anymore. Here wepresent results for an RNA molecule (CD4L12) falling in this category and present ageneral procedure to extract the free energy of formation. CD4L12 shares the samestem than the previously discussed CD4 RNA in section (4.2.2), but with the originaltetraloop replaced by a dodecaloop (i.e. 12-loop bases), see sequence in figure 4(a). Alarge loop yields a larger entropic barrier for refolding and large hysteresis in the FDC.Pulling experiments were performed as described in section (4.1), with a pulling speedof 100 nm/s and 300 nm/s and a buffer containing 4 mM MgCl, 50 mM NaCl, and 10mM Tris. The values of P = 0 .
75 nm and d b = 0 .
665 nm were used to describe theelastic properties of the ssRNA for this buffer [39].As can be seen in figure 4(b), CD4L12 behaves as a two-state system being eitherfolded or unfolded along the FDCs. As expected, pulling cycles feature large hysteresis,with a maximal difference of nearly 20 pN between the lowest folding and largestunfolding force rips. In order to compute the work needed for the CFT (7), we mustintegrate the area under the FDC within a large force range with a very low f min . It etermination of folding free energies in single-molecule experiments G G G G G G G G G GGCCCCCCCCCC A A A A A A A A A A AUUUUUUUUUU A A A A A AAA AA (a) (b)
Force [pN] k h [ p N / n m ] Distance, λ [nm] F o r ce [ p N ] × -2 Force [pN] k e ff F [ p N / n m ] (c) Figure 4. Free-energy recovery CD4L12 RNA hairpin. (a).
Sequence andsecondary structure of CD4L12 RNA (b).
Aligned FDCs folding (red) and unfolding(blue) for a given molecule pulled at 100 nm/s.
Inset : Effective stiffness profilemeasured along the folded branch. (c).
Stiffness profile of the hybrid DNA-RNAhandles which form the molecular construct used with CD4L12 and CD4 RNA [39].Data points have been obtained using the high frequency power-spectrum methoddescribed in [31]. The red line is a fit of the extensible WLC model, yielding P = 20 ± Y = 200 ±
14 pN ( d b was not fitted but fixed to the interphosphatedistance of A-form RNA, d b = 0 .
27 nm [45]). is clear that in this case the constant stiffness approximation described in section 4.2.1does not apply, as shown in the inset of figure 4(b) where k Feff markedly changes withforce. To estimate the stretching contributions we follow the previous subsection 5.1and (19) to obtain ∆ W b , ∆ W h , and, from (14), the value of ∆ G .In order to carry out the fit prescribed by (19), we need to introduce some furtherassumptions to simplify the problem. Regarding the hybrid DNA-RNA handles, weuse the value of the interphosphate distance d b = 0 .
27 nm of A-form RNA and Youngmodulus Y = 200 pN obtained by fitting the stiffness of the handles profile (figure 4(c)).While changes in d b only moderately affect the overall curvature of k h (but they impactthe overall contour length, an effect already captured by fitting λ ), changes in Y donot. Hence fixing these two values gives a better constrained model. For the persistencelength of the handles P it is convenient to fit the deviation ∆ P (in %) with respect toa plausible expected nominal value P , i.e. P eff = P (1 + ∆ P ), that we take from thefit in figure (4) as P = 20 nm. Lastly, we also include the number of nucleotides n released in the transition between the folded and the unfolded branches as an extra freeparameter of the model. We are thus eventually left with 5 free parameters which wefit (18,19) using a standard non-linear least square regression (Levenberg-Marquardt): λ ( f ) = M ( f ) = M ( f ; { k b,0 , α, ∆ P, λ , n ) } ) . (20) etermination of folding free energies in single-molecule experiments (a) (b) Figure 5. Fitting the folded and unfolded branches of CD4L12 RNA hairpin.(a).
Solid blue line is an example of curve fitting based on (20). Data points usedfor the fit are the black diamonds. They are obtained by smoothing and filteringthe gray dots, which are themselves obtained by aggregating the unfolding FDCs ofdifferent pulling cycles from figure 4b (b).
Example of forward and reverse (solidand dashed lines) work distributions for the same molecule pulled at 100 nm/s. Dueto the large hysteresis, work distributions do not overlap.
Inset:
Illustration of the matching method to retrieve ∆ G by imposing continuity between P F ( W ) (light green)and P R ( − W ) e ( W − ∆ G ) /k B T (dark green) in log-normal scale. Solid grey line is the fittedGaussian, see [46] for details. An example of such fitting procedure is shown on figure 5(a). As can be seen, theagreement between the experimental points and the reconstructed curve is remarkable.Furthermore, all the values obtained from the fit dovetail with prior expectations.Firstly, the value of n matches with the expected number of released nucleotides (i.e.52). Secondly, the zero-force trap stiffness k b falls in expected range [31]. Thirdly, theforce-dependence parameter α of the trap stiffness is of the same order of magnitude thanvalues already reported in the literature for similar LOT settings [31]. Fourthly, ∆ P issmall so P is reasonably close to the assigned nominal value P . Another good genericindicator is the very low error on the fitted parameters, hinting at a well-constrainedmodel; a fact that is further confirmed by the observation that in the correlation matrixof the fit, most off-diagonal entries are near-zero (details not shown). We must finallystress that the choice of free parameters in (20) is convenient for the considered situationbut is by no means customary. In a context where the trap would be well characterizedand the handles would not, we may have for instance fixed k b but fitted d b . Equation(19) can be adapted at will, depending on the requirement.With the fitted values of α , k b,0 and ∆ P in hand and our assumptions for Y and d b (legitimated retrospectively by the agreement of the fit in figure 5), we are now in aposition to precisely establish the profiles of k h ( f ) an k b ( f ) through the use of equations(15), (16) and (17). We can now quantify the terms ∆ W b and ∆ W h using (12) and ∆ G using the FT. These numbers together with (14) allow us to extract ∆ G . etermination of folding free energies in single-molecule experiments P F ( W ) and P R ( − W ) obtained from theFDC (figure 4(b)). The very pronounced hysteresis and the large value of the averagedissipated work in a pulling cycle (about 60 k B T ) is such that F and R work distributionslie far apart without overlapping. Previous methods based on the overlapping of F andR work distributions are not applicable and an alternative approach must be used, suchas the Bennett acceptance ratio [47] and the “matching method”. This last methodconsists in finding the optimal ∆ G value so that P F ( W ) is the analytical continuationof P R ( − W ) e ( W − ∆ G ) /k B T . This procedure is graphically illustrated in the inset of figure5(b) and further explained in [46]. Results obtained for different molecules are shown intable 2. We note that the values of ∆ G obtained with the two methods yield compatibleresults ( matching being systematically 3-5 k B T lower than Bennett ). Our estimatedvalue ∆ G = 67 ± k B T is not far from the Mfold prediction (∆ G = 63 k B T ) showingthe reliability of the approach.∆ G Bennet [ k B T ] ∆ G Matching [ k B T ] ∆ W m [ k B T ] ∆ W b + ∆ W h [ k B T ] ∆ G [ k B T ]1045 ± ± ± ± ± ± ± ± ± ± ± ± Mean: 67 ± B T Table 2. Fluctuation theorem and stretching contributions for CD4L12RNA hairpin with long handles.
Overview of the values of ∆ G , the stretchingcorrections, and the final ∆ G estimate for 6 different molecules. All values are givenin k B T . ∆ G Bennet and ∆ G Matching provide two ways to extract ∆ G using the CFT.The value of ∆ G is obtained through (9) using the value of the Bennett estimate.The last line corresponds the only experimental setting in which the pulling speed is300 nm/s, all the other results were obtained at 100nm/s. We want to stress the sensitivity of the value of ∆ G on the accurate estimationof the stretching contributions which, being one order of magnitude larger, can leadto inconsistent results. Had we used a methodology assuming ’average’ or ’standard’stretching contributions, we would have obtained erroneous numbers. Consider forinstance subtracting the average value (cid:104) ∆ W b + ∆ W h + ∆ W m (cid:105) = 898 k B T derived fromtable 2 to the highest and the lowest estimates of ∆ G shown in the same table: itresults in two widely off values ∆ G = 1107 −
898 = 209 k B T and ∆ G = 863 −
898 = − k B T . Therefore a tailored molecule-to-molecule estimation of the stretchingcontribution is absolutely essential for molecules like CD4L12 where the effective stiffnessapproximation cannot be used. etermination of folding free energies in single-molecule experiments
6. Conclusions
We have presented a brief tutorial on the approaches commonly used to extract foldingfree energies of single molecules pulled with optical tweezers in unzipping assays. Arecurrent issue in these calculations is the large magnitude of the stretching contributionsto the full free-energy difference measured in a pulling experiment using the CFT. Suchcontributions arise from the experimental setup and include the optical trap, the elasticstretching of the handles used in the molecular construct and the extension release ofthe unfolded polymer. A great simplification in the analysis of these correction termscan be be performed when the effective stiffness of the experimental system can beapproximated as constant, as we saw in section 4. In this so-called effective stiffnessapproximation a single parameter k Feff suffices to quantify the stretching contributionsof handles and trap. We exemplified this case in the study of a DNA hairpin in section4.1. For long handles the stiffness of the handles turns out to be comparable to that ofthe trap and a force dependent k Feff is apparent. In this case, as we showed in section4.2.2, one can still use the effective stiffness approximation if the range of integrationto evaluate the work is narrow enough. This is possible if the pulling curves are nottoo irreversible and forward and reverse work distributions overlap. In contrast, forstrong irreversible pulling experiments one needs to accurately characterize all elasticcontributions from the experimental setup. Here we have introduced a novel method(section 5) based on least-squares fitting of the elastic response of the folded and unfoldedbranches. It relies on adapting the elastic parameters extracted from the literature(inter-monomer distance, persistence length, Young modulus) to the experimental dataas well as accurately retrieving the stiffness of the optical trap using the same data.One problem that remains open is the magnitude of the statistical error committedin the estimation of ∆ G . In fact, ∆ G is the difference of two large numbers (∆ G and the stretching contributions) each with a large error and extracted from the sameexperimental FDC data. How to combine the errors from these two large quantitiesremains largely unclear as they are not really uncorrelated. A rule of thumb insingle-molecule experiments is that the largest errors come from molecule to moleculeexperimental variability. It is then recommended to first extract ∆ G values for differentmolecules by subtracting elastic contributions from ∆ G on a single-molecule basis, andthen derive the mean value of ∆ G and the corresponding statistical error.The large contribution of the stretching term (14) to the full free energy ∆ G makesthe prediction of the (comparably small) value of ∆ G a difficult task. This situation isreminiscent of the enthalpy-entropy compensation problem in biochemistry [48, 49]. Inthis case free-energy differences of intra an intermolecular weak interactions (e.g. folding,binding, allostery, enzymatic reactions and so on) are typically one order of magnitudesmaller than entropies and enthalpies, i.e. ∆ G = ∆ H − T ∆ S with ∆ G (cid:28) ∆ H, T ∆ S . Inthis regard, enthalpy-entropy compensation in biochemistry appears to be similar to the∆ G -stretching compensation in force spectroscopy. The analogy is not pure coincidenceas the stretching contributions are essentially also of entropic nature and much larger etermination of folding free energies in single-molecule experiments G .The methodology we have described should be generally useful and applicable toforce spectroscopy studies of single-molecule constructs whenever elastic contributionsare present. Applications go beyond the case of measuring folding free energies suchas extracting molecular free-energy landscapes [30] measure ligand binding energies[50], protein-protein and RNA-protein interactions and characterizing heterogeneousmolecular ensembles [51]. Acknowledgements.
We acknowledge financial support from Grants Proseqo(FP7 EU program) FIS2016-80458-P (Spanish Research Council) and Icrea Academiaprizes 2013 and 2018 (Catalan Government).
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The inextensible WLC model described in (4) gives a very direct way to compute f = f ( x ), but it is not straightforward to use it to retrieve x = x ( f ). Althoughnumerical inversion using Mathematica and other software is possible (e.g. as in [52])it is useful to have explicit inversion formulae. Hence let us now quickly show that(4) can be easily inverted to express z := x/L c as a function of f . We first define thenormalized quantity ˜ f = (4 P/k b T ) f . We can then re-write (4) as ˜ f = (1 − z ) − − z .By multiplying both sides of the previous by (1 − z ) and by moving all terms to thesame side, we obtain:0 = z + a z + a z + a with a = − − ˜ f , a = 32 + ˜ f , a = − ˜ f z as a function of f simply maps to finding theroots of a cubic polynomial – a problem solved since the 15th century. The approachtaken here is the canonical one [53, 54]. We start defining the following intermediatequantities: R := 9 a a − a − a Q := 3 a − a D for cubic equations: D := Q + R (A.3)If D >
0, there is only one real solution to (A.1), and we have to define the followingintermediate quantities to express the answer: T := (cid:113) R + √ D S := (cid:113) R − √ D (A.4)(since D >
0, we also have that √ D is real, and thus there is indeed at least onereal cubic root for T and S ). The desired inverse value z ∗ = z ( f ) is then finally obtainedas: z ∗ = − a + S + T (A.5)If D <
0, there are three real roots to the cubic equation. These roots can beobtained by re-using the quantities S and T defined above, but doing so requires usingcomplex number algebra – which may not be handy. Instead, we also can define thefollowing intermediate quantity: θ := arccos (cid:32) R (cid:112) − Q (cid:33) (A.6)From which the three real roots z , z , z can be obtained directly as: z i = 2 (cid:112) − Q cos (cid:18) θ + θ i (cid:19) − a with θ = 0 , θ = 2 π, θ = 4 π (A.7) etermination of folding free energies in single-molecule experiments , z = x/L c and a property of the inextensible WLC is that the extension x is alwayssmaller than the contour length L c . Using trigonometric standard formula and the factthat 2 √− Q > z − z > z − z ≥ θ (which must belong to [0 , π ] by definition of the arccosine), which implies that z is the smallest of all the roots. Moreover, we note that all the roots must be positive,since we see in (4) that ∀ z < f ( z ) < z is the smallest of them, it therefore has to be the onewe are looking for, in [0 , z = z ∗ = z ( f ) when D <
0. The previous resultalso covers the D = 0 situation, because we then have from (A.6), θ = 0, and so we arein the limiting case z = z .Let us finally note that in the case of the extensible WLC, the key difference withthe inextensible case is the replacement L c → L c (1 + f /Y ) with Y the Young Modulus,i.e. the contour length is now force dependent. It can be shown that this implies thefollowing relationship between the two models: x extW LC ( f ) = x inextW LC ( f ) (1 + f /Yf /Y