Folding and unfolding of a triple-branch DNA molecule with four conformational states
Sandra Engel, Anna Alemany, Nuria Forns, Philipp Maass, Felix Ritort
aa r X i v : . [ phy s i c s . b i o - ph ] J un Folding and unfolding of a triple-branch DNAmolecule with four conformational states
Sandra Engel a ∗ , Anna Alemany b , c , Nuria Forns b , c ,Philipp Maass a † , and Felix Ritort b , c ‡ a Fachbereich Physik, Universit¨at Osnabr¨uck,Barbarastr. 7, 49076 Osnabr¨uck, Germany b Departament de F´ısica Fonamental, Facultat de F´ısica,Universitat de Barcelona, Diagonal 647, 08028 Barcelona, Spain c CIBER-BBN Networking center on Bioengineering,Biomaterials and Nanomedicine, Spain
Abstract
Single-molecule experiments provide new insights into biological processeshitherto not accessible by measurements performed on bulk systems. We reporton a study of the kinetics of a triple-branch DNA molecule with four conforma-tional states by pulling experiments with optical tweezers and theoretical mod-elling. Three distinct force rips associated with di ff erent transitions between theconformational states are observed in the folding and unfolding trajectories. Byapplying transition rate theory to a free energy model of the molecule, probabilitydistributions for the first rupture forces of the di ff erent transitions are calculated.Good agreement of the theoretical predictions with the experimental findings isachieved. Furthermore, due to our specific design of the molecule, we found auseful method to identify permanently frayed molecules by estimating the numberof opened basepairs from the measured force jump values. Keywords:
Nonequilibrium systems, single-molecule experiments,optical tweezers, DNA
PACS:
In recent years, single-molecule experiments became of great importance in biophysi-cal research since progress in nano- and microscale manufacturing technologies facili-tated the design of scientific instruments with su ffi cient sensitivity and precision to en-able the controlled manipulation of individual molecules (for reviews, see, for example,[22, 10]). In contrast to the traditionally used bulk assays, where individual biomolec-ular dynamics can get masked, single-molecule experiments provide new insights into ∗ [email protected] † [email protected], phone: 0049-541-969-3460, fax: 0049-541-969-2351, ‡ [email protected], phone: 0034-934035869, fax: 0034-934021149, olding and unfolding of a triple - branch DNA molecule with four conformational states ff ers a powerful new tool in molecular andcellular biophysics allowing for the exploration of processes occuring inside the cell atan unprecedented level. To instance just a few of the recently investigated biochem-ical processes: the transport of matter through pores or channels [19, 20, 1], interac-tions between DNA and proteins [13] or DNA and RNA [36], the motion of single-molecular motors [2, 4, 32], DNA transcription and replication [34, 15], virus infection[26, 8], DNA condensation [23] and ATP generation [35]. In addition, the structureof biological networks [30] and the viscoelastic and rheological properties of the DNA[27, 31, 6, 29] have been studied.An important class of single-molecule experiments are performed with optical tweez-ers. By means of an optical trap generated by a focused laser beam, this useful tech-nique renders it possible to exert forces on micron sized objects, achieving sub-piconewtonand sub-nanometer resolution in force and extension, respectively. Accordingly, onecan study force-induced folding-unfolding dynamics and in this way get insight intocorresponding processes in the cell and typical bond forces. Of particular interest isthe unfolding of DNA molecules, where the hydrogen bonds between the complemen-tary base pairs (bps) are disrupted. This so-called unzipping is connected to the DNAreplication mechanism.An interesting field of biophysical studies is the investigation of junctions in moleculessince they present manifold ways to interact with other substances, for instance cations.Three-way junctions are especially interesting because metal ions such as magnesiumcan bind to them and alter the tertiary structure. Here a first step of such a study ispresented where we investigate a molecule with a three-way junction alone, withoutcation binding.Many of the single-molecule experiments so far focused on molecules with a rel-atively simple free energy lanscape (FEL) exhibiting just two states, a folded and anunfolded one, or including an additional misfolding state, leading to di ff erent kineticpathways. In this work we will consider a richer situation, where metastable states asintermediates occur during the folding-unfolding route. In this context, we will addressthe following key questions:(i) Can a corresponding molecule with such intermediate states be designed on thebasis of a suitable model for a FEL?(ii) Is it possible to observe the intermediate states by perfoming pulling experimentswith optical tweezers?(iii) Can phenomenological Bell-Evans kinetic models be applied to describe thefolding-unfolding processes including intermediate states? In particular, whenvalidating the kinetic theory against the experimental results, how do the firstrupture force distributions compare with the ones predicted by the theory?A further important aspect that we looked at in some detail is the heterogenityof molecular folding-unfolding behaviour that we observed in the experiments. Bymeasuring force-distance curves (FDCs) of several molecules we classified them into olding and unfolding of a triple - branch DNA molecule with four conformational states ff erent reproducible patterns. This leads to a useful method to identify irreversiblemolecular fraying, a phenomenon which is often observed in single-molecule studies. Based on Mfold folding predictions [25, 37] and taking FEL considerations into ac-count (see sec. 5), we designed and synthesised a DNA molecule which is composedof three parts and hence referred to as triple-branch molecule. It consists of a stemas introduced in [21] with 21 bps and two nearly identical hairpin branches whichare formed of 16 bps and a loop with four bases, thus comprising a total number of114 bases, see fig. 1. To avoid misfolding, the second hairpin branch di ff ers at twopositions from the first one. hairpin branch 1 hairpin branch 2stem Total: 114bs
Figure 1: Structure of the triple-branch DNA molecule.The triple-branch molecule is inserted between two identical short double-strandedDNA (dsDNA) handles of 29 bps each [9], leading to a total number of 172 bases,corresponding to a total contour length in the unfolded state of about 100 nm. Each ofthese polymer spacers is chemically linked to a bead. One of the beads is retained withthe help of a pipette via air suction, the other one is optically trapped in a laser focus[28]. On the 5 ′ end of the DNA molecule, biotin is attached to enable a connection with olding and unfolding of a triple - branch DNA molecule with four conformational states
4a streptavidin-coated bead (SA bead) whose diameter is 1 . µ m. Biotin is a vitaminwhich establishes a strong linkage to the proteins avidin and streptavidin. The 3 ′ end ismodified with the antigen digoxigenin, able to interact with an antidigoxigenin-coatedbead (AD bead). The latter has a diameter of 3 . µ m. Figure 2 shows the di ff erentcomponents of the molecular construct, that is to say the triple-branch DNA molecule,the handles and the beads, captured in optical trap and micropipette, respectively. Notethat it is not a true-to-scale representation.The pulling experiments are carried out with a miniaturised dual-beam laser opticaltweezers apparatus [11] at room temperature ( ≃
25 °C) and at salt concentration of 1M NaCl aqueous bu ff er with neutral pH (7.5) stabilised by Tris HCl and 1 M EDTA.The dual-beam optical tweezers collect data at 4 kHz and can operate with a feedbackrate of 1 kHz. Spatial resolution constitutes 0 . ≃ µ m. Forces up to 100 pN can be achieved, whereas the force resolution is 0 .
05 pN.
DNA triple-branch molecule
SAbead
Optical trap Micropipette
Handle 1 Handle 2
StemHairpinbranches
Loop Loop molecular extension
AD bead absolute distance
Figure 2: Sketch of the experimental setup. In our experiments we measure the relativedistance X rather than its absolute value.Optical tweezer pulling experiments permit the measurement of the force f as wellas the total distance X between the centre of the optical trap and the tip of the mi-cropipette, see fig. 2. In the experiment we vary the trap-pipette distance X ( t ) witha constant speed v = dX / dt in the range of 45 nm / s to 200 nm / s, which correspondsto a constant average loading rate r of 3 . / s and 13 . / s in between rip events,respectively. The experiment consists of loading cycles which in turn are divided intoan unfolding part (during ”pulling”) and a folding part (during ”pushing”). The load-ing cycles are repeated as long as the tether connection is unbroken. Otherwise a newconnection has to be established, possibly a new molecule must be searched and linkedto a new bead. In sec. 4 we will discuss di ff erent patterns found in the measured curvesand present a detailed analysis of two representative molecules. We chose them amongseven molecules exhibiting the first and among five molecules featuring the secondpattern. For each molecule we recorded, on average, approximately 50 cycles. Thepulling speeds, ranging from 45 to 200 nm / s, influence the experimental results onlyweekly due to a logarithmic dependence of the first rupture force with the speed. Wefound compatible data for sets of similar molecules. In the theoretical analysis of thedata in sec. 5, we concentrate on the largest set of 82 loading cycles for the molecule olding and unfolding of a triple - branch DNA molecule with four conformational states / s. The study of the above mentioned second molecule comprises55 cycles. Based on the design of the triple-branch molecule, we have to distinguish between fourconformational states (see fig. 3):1. a completely folded molecule.2. a completely unfolded stem with the hairpin branches still folded.3. stem and either hairpin branch 1 or 2 are completely unfolded.4. a completely unfolded molecule. d ′ x ( n, f ) = u l ( f ) + d ′ n = 21 x ( n, f ) = d d x ( n, f ) = u l ( f ) x ( n, f ) = u l ( f ) + d d d n = 37 n = 53 l = 2 nd l = 2 nd + n loop d l = 2 nd + 2 n loop dn = 0 Figure 3: The four stable or metastable states of the triple-branch molecule: 1 - foldedmolecule, 2 - unfolded stem, 3 - stem and one hairpin branch unfolded, 4 - unfoldedmolecule. The values of x ( n , f ) refer to the end-to-end distance given in eq. (2) withthe number n of opened bps corresponding to the molecular construct shown in fig. 1and the contour length l according to eq. (3).Figure 4(a) displays a typical unfolding route and fig. 4(b) a typical unfolding andfolding trajectory in form of a FDC, which in fact records the evolution of the force as afunction of time and relative trap position X . With rising X , the force f first increasesalmost linearly according to an elastic response of the DNA handles, which consist ofdsDNA and are stable over the whole range of forces where the unfolding / foldingof the triple-branch molecule takes place. The overstretching transition of the linkers,typically at 65 pN, lies much above the forces we explore. At a first rupture force f a sudden decrease (”jump”) ∆ f occurs, which is caused by the unfolding of the stem.This unfolding goes along with an abrupt change in the length of the molecule whensingle-stranded DNA (ssDNA) is released. As a consequence, the bead in the optical In our experiments we measure the relative distance between trap and pipette, X , rather than the absolutevalue. The force f exerted on the molecular construct leads to a displacement f / k b of the bead in the opticaltrap, where k b = .
08 pN / nm is the rigidity of the trap. Hence, the relative distance X is related to the relativemolecular extension x m (see fig. 2) by x m = X − f / k b . olding and unfolding of a triple - branch DNA molecule with four conformational states
11 12 13 14 15 16 17 18 100 150 200 250 f o r ce [ p N ] relative distance X [nm]
14 16 18 20 150 200 250 (a) f o r ce [ p N ] relative distance X [nm] foldingunfolding f f f f , rf f , rf f , rf (b) Figure 4: Force as a function of change in the trap-pipette distance (a) for one typicalunfolding trajectory for the first investigated molecule and (b) for one typical unfoldingand folding trajectory for the second investigated triple-branch molecule. Indicated arethe three first rupture forces belonging to the transitions 1 ( f ), 2 ( f ) and 3 ( f ) and therespective refolding forces, labelled f i , rf . The inset in (a) shows four further unfoldingcurves for the first investigated molecule.trap moves towards the centre of the trap, visible as the force drops. Following thejump ∆ f , there is again a linear increase up to the next force rip at a first rupture force f , where one of the hairpin branches unfolds, which in turn leads to the force jump ∆ f . Eventually, the second hairpin branch unzips at a first rupture force f with a jump ∆ f . The linear regime following this last force rip corresponds to the stretching of thewhole molecular construct including handles and the already unfolded triple-branchmolecule.Upon decreasing X from the completely unfolded state 4, the force first followsclosely the corresponding unfolding part of the trajectory. However, a backward tran-sition does not occur at ( f − ∆ f ) ≃ . . ± .
8) pN, cp. fig. 4(b), hence manifesting a hysteresis e ff ect. Moreover, an in-vestigation of a larger number of folding trajectories reveals that the bases of the hairpinbranches do not always pair conjointly in well-defined events during a short time inter- olding and unfolding of a triple - branch DNA molecule with four conformational states . ± .
8) pN instead of ( f − ∆ f ) ≃ . . ± .
8) pN,which is again considerably lower than ( f − ∆ f ) ≃ . n of broken bonds as state variable and to calculate a FEL as function of this variable(see sec. 5). The refolding of the hairpin branches in the folding trajectories exhibitless sharp transitions, see fig. 4(b). During folding, in particular at the beginning in theunfolded state, a huge number of secondary structures can be found which implies thatthe kinetic pathways are less predefined and accordingly, the transitions get smearedout. With respect to a theoretical treatment, moreover, a description in terms of thesimple state variable n becomes unlikely to be su ffi cient. In a refined analysis, manymore configurations should have to be included as relevant states in a coarse-graineddescription [17]. Such refined analysis, however, goes beyond the scope of this workamd we therefore concentrate on the unfolding process in the following.There are plenty of possible ways to analyse the FDCs in order to find out the firstrupture forces and the force jumps. In our procedure we arranged the normalised data,i.e. the relative distance X and the force f , in windows of a certain size of data points.For all consecutive windows we then calculated the slope of the considered data points,the span, i.e. the maximum distance between the lowest and the highest force value,and the mean of the relative distance X j as a moving average. Transitions betweenthe conformational states take place where the slope is minimal and the span maximalunder the condition that an appropriate number of contiguous windows is connected.Having found the X j of the three force rips, slope and axis intercept are calculated bylinear regression for each conformational state. One can now easily calculate the firstrupture forces as the intersection points with the four fitted lines and extract the forcejump values, as exemplified in fig. 5 for both molecules whose unfolding trajectorieswere depicted in figs. 4(a) and 4(b), respectively. Since the data acquisition rate is
12 13 14 15 16 17 18 19 20 150 200 250 f o r ce [ p N ] relative distance X [nm] (a)
11 12 13 14 15 16 17 18 19 100 150 200 250 f o r ce [ p N ] relative distance X [nm] (b) Figure 5: The first rupture forces and force jump values of the unfolding trajectoriesshown in fig. 4 are extracted as indicated here from the intersection of the fitted greylines and the transitions (vertical lines). In part (a) the procedure is shown for themolecule used in the theoretical analysis in sec. 5 and in (b) for the molecule withpermanent fraying behaviour. olding and unfolding of a triple - branch DNA molecule with four conformational states f i as well asthe jumps ∆ f i are subject to stochastic fluctuations, as can be seen in fig. 4(a) wherewe show four unfolding trajectories belonging to di ff erent pulling cycles of the samemolecule in the inset. An analysis revealed that the fluctuations of the force jumps ∆ f i are about ten times smaller than the fluctuations of the f i . Accordingly, in sec. 5, wewill disregard the fluctuations in the ∆ f i and use only their averages ∆ f i that will bediscussed in more detail in the following section. ff erent unfolding patterns related to the number ofopened bps Applying the above mentioned procedure to analyse the experimental data of severalmolecules, we found two predominating patterns in the unfolding trajectories whichare reflected in the distributions of the first rupture forces f i as follows. In the firstpattern, see fig. 6(a), these distributions have a similar shape for all three force rips. Thehistograms indicate the existence of one maximum slightly below 17 pN. The meanvalues for the three first rupture forces are f = (17 . ± .
9) pN, f = (16 . ± .
6) pNand f = (16 . ± .
7) pN. In contrast, in a second pattern we detected a strikinglylower value for the first rupture force of the first rip f = (14 . ± .
8) pN, as depicted infig. 6(b), whereas the other two rip forces lie basically in the same range of 16 to 18 pN.As in the former case, the second rip tends to have a slightly smaller first rupture force, f = (16 . ± .
8) pN, than the third rip, f = (17 . ± .
5) pN.The small f observed in the second molecule suggests the occurrence of permanentmolecular fraying. Obviously less force is needed to unfold the stem than typically, cp.fig. 4(a) with 4(b), since some bps at its basis are partly or completely melted. In otherwords, during folding, this molecule does not reach an entirely folded state but somebps of the stem next to the handles remain irreversibly and permanently open. Thisphenomenon has been observed previously in several pulling experiments [11, 33]. Apossible reason for irreversible fraying is the formation of reactive oxidative speciesdue to the impact of the laser light of the optical trap leading to a degradation of theDNA bases [12]. The so generated singlet oxygens are known to oxidise certain nucleicacids, such as guanine and thymine, irreversibly. This could explain our observationthat once a molecule shows fraying it does not change back again to normal behaviour.Due to the fact that we work with polystyrene microspheres which are more prone tophotodamage than the DNA bases themselves, their wide ranging interaction with thebases might be reduced replacing polystyrene by silica beads, which exhibit consid-erably minor irreversible oxidative damage. It would be very interesting to carry outsuch experiments.The insets in figs. 6(a) and 6(b) depict the corresponding force jump distributionsof both molecules. While the first molecule possesses a large first force jump valueof ∆ f = (1 . ± .
2) pN and two smaller force jumps at the second and third rip of ∆ f = (0 . ± .
2) pN and ∆ f = (1 . ± .
2) pN, the frayed molecule features threeforce jumps of approximately the same value, i.e. ∆ f = (0 . ± .
05) pN, ∆ f = (0 . ± .
06) pN and ∆ f = (0 . ± .
04) pN, respectively. This illustrates clearly theinfluence of irreversible fraying in the latter case since, roughly estimated, the samenumber of bps is expected to open in all three rips which should not be the case in anentirely folded molecule. olding and unfolding of a triple - branch DNA molecule with four conformational states r e l a t i v e f r e q u e n c y first rupture force [pN] first ripsecond ripthird rip 0 0.1 0.2 0.3 0.5 1 1.5force jump [pN] (a) r e l a t i v e f r e q u e n c y first rupture force [pN] first ripsecond ripthird rip 0 0.1 0.2 0.3 0.8 0.9 1force jump [pN] (b) Figure 6: Histograms of the three first rupture forces during unfolding for two repre-sentative triple-branch molecules: (a) the one used in the theoretical analysis in sec. 5and (b) the one exhibiting permanent fraying behaviour. Note that the average forcevalue of the first rip has decreased in (b) as compared to (a). The insets show thecorresponding histograms for the force jump values.During a force rip, the relative distance X is constant and thus ∆ X =
0. Thereforethe change in the relative molecular extension, ∆ x m = ∆ X − ∆ f / k e ff = ∆ x ( n , f ) , (1)is only related to the force jump ∆ f and the combined sti ff ness of bead and handles k e ff , given by 1 / k e ff = / k b + / k h , where k b is the trap sti ff ness and k h the rigidityof the handles, respectively. With the help of a linear least squares fit, the e ff ectivesti ff ness k e ff of the molecular construct is extracted from the average slope of the FDCsand amounts to (0 . ± . / nm. Note that, as expected, k e ff is smaller than k b ( ≃ .
08 pN / nm).For a certain force value f and assuming an elastic model for the released ssDNA,the number n of opened bps is related univocally to the equilibrium end-to-end distanceof the DNA molecule x ( n , f ), whereas its change ∆ x ( n , f ), in turn, equals ∆ x m . Usingthis relation, it is now possible to estimate the change in the number ∆ n i of bps which olding and unfolding of a triple - branch DNA molecule with four conformational states x ( n , f ) can be de-composed into two parts, cp. fig. 3. The first part, the elongation u l ( f ) of the meanend-to-end distance of the ssDNA along the force direction, accounts for the ideal elas-tic response of the ssDNA, where l is the contour length. The second part contains thecontribution of the diameter of stem and hairpin branches, respectively. Acccordingly, x ( n , f ) = u l ( f ) + , n = + +
16 bps d , n = +
16 bps d ′ , n =
21 bps , (2)where d ≃ d ′ , when the stem is unfolded,depends on the orientation of the branches. We set d ′ = d as a working value .Regarding the contour length l , which depends, amongst others (cp. sec. 5), on thenumber n of opened bps, and considering again solely the configurations of the fourconformational states, one gets l = nd + , n = n loop d , n = +
16 bps2 n loop d , n = + +
16 bps , (3)where the interphosphate distance d is taken to be 0 .
59 nm / base and n loop = f i the corresponding number of openedbps n i is calculated separately by considering the di ff erences of l between the states.Di ff erent types of models can be used to calculate u l ( f ). Prominent examples bor-rowed from polymer physics are the freely jointed chain (FJC) and the worm-like chain(WLC) model. According to ref. [27], the FJC model includes an extra term, leadingto the expression u l ( f ) = l + fY ! " coth b fk B T ! − k B Tb f . (4)Here Y denotes the Young modulus, b is the Kuhn length, k B the Boltzmann constantand T the temperature. Typical values of the Kuhn length and the Young modulus underworking conditions of T ≃
25 °C and 1 M NaCl concentration are b = .
42 nm and Y =
812 pN [27] or, as published recently, b = .
15 nm and Y = ∞ [11], respectively.In the WLC model [3], the force f ( u l ), due to an elongation u l , is given by f ( u l ) = k B TP "
14 (1 − u l / l ) − + u l l , (5)and to obtain u l ( f ), this equation has to be inverted. Based on the WLC model, wetested the influence of the persistence length P in a typical range of 1.0 to 1 . Due to this simplification, the ∆ n of the first rip is likely to be slightly underestimated and the secondrip’s ∆ n overestimated. However, it will not a ff ect the change in the total number ∆ n tot of opened bps sincewe consider the change of x ( n , f ), and the d ′ contributions will cancel each other out. olding and unfolding of a triple - branch DNA molecule with four conformational states molecule 1 molecule 2parameter change in no. of opened bps [bps]model b & P [nm], Y [pN] ∆ n ∆ n ∆ n ∆ n tot ∆ n ∆ n ∆ n ∆ n tot b = . Y =
812 [27] 18 (2) 14 (2) 15 (2) 46 (3) 13 (1) 15 (1) 15 (1) 42 (2)FJC b = . Y = ∞ [11] 19 (3) 15 (2) 16 (2) 50 (3) 14 (1) 16 (1) 16 (1) 46 (2) P = . P = . P = . Table 1: Overview over the change in the number of opened bps for di ff erent modelsand parameters for the two representative molecules. The numbers in brackets are thestandard deviations.with P = . ∆ n , depending on themodel around 13 or 14 bps, so that 7 or 8 bps are not closed after the folding processis completed. Performing single-molecule experiments without knowing the exact in-fluence of permanently frayed bps can lead to misinterpreted results. Checking theappropriate parameters for the polymer models with the help of the change in the num-ber of opened bps of the hairpin branches, one can estimate the number of irreversiblyfrayed bps at the basis of the stem . The kinetics of the unfolding process can be described on a coarse-grained level basedon a Gibbs free energy G ( n , f ) as a function of the number n of sequentially openedbps for an applied force f . For small forces, including f =
0, the FEL is expectedto have a shape as displayed in fig. 7(a). In general, G ( n ,
0) increases monotonouslywith n . However, local minima occur at the metastable states 2, 3 and 4 because thereis an increase of entropy associated with the release of additional degrees of freedomwhen the stem-hairpin-junction and end-loops of the hairpin branches are opened. Withrising force the FEL is expected to get tilted, so that the energies of the metastable statesare lowered. With a knowledge of G ( n , f ) we can apply standard transition rate theoryand write for the transition rate from state i to i + Γ i , i + ( f ) = γ i γ i , i + ( f ) , (6)where γ i is an attempt rate and γ i , i + ( f ) is the Boltzmann factor corresponding to theactivation barrier ∆ G i , i + ( f ) that has to be surmounted, γ i , i + ( f ) = exp − ∆ G i , i + ( f ) k B T ! . (7)When considering only sequential configurations in the evaluation of the FEL, dif-ferent structures compatible with a given n can occur once the stem is completely un-folded. These refer to di ff erent possibilities of breaking the bps in the hairpin branches We like to note that the checking of the change in the number of opened bps can be, in principle, alsoapplied to non-permanent, reversible molecular fraying. olding and unfolding of a triple - branch DNA molecule with four conformational states n G ( n, f ) ∆ G , ( f ) γ , ( f ) (a) G ( n , f ) no. of opened bps n [bps]FEL at f=0FEL at f=16.21 pN (b) Figure 7: (a) Sketch of the expected FEL for the triple-branch molecule at zero forceas a function of the number n of opened bps. (b) FEL calculated from eqs. (8) to (11)at two di ff erent forces in units of the thermal energy k B T .1 and 2. In order to point out this ”degeneration”, a new parameter α is introduced,leading to the FEL G ( n , α, f ). At a given n , we must calculate the (restricted) partitionsum over the configurations α to get G ( n , f ). In order to find G ( n , α, f ), we considerthe following decomposition, G ( n , α, f ) = G form ( n , α ) + G ssstr ( n , α, f ) − f ∆ x ss l , (8)where G form ( n , α ) is the free energy of formation of the configuration ( n , α ), G ssstr ( n , α, f )is the strain energy of the unfolded ssDNA and f ∆ x ss l is a Legendre term (for the defi-nition of ∆ x ss l see eq. (10) below).The free energy of formation G form ( n , α ) is written as G form ( n , α ) = X all bps g µ,µ + + G junc ( n ) + G (1)loop ( n ) + G (2)loop ( n ) . (9)The first term refers to the nearest neighbour model developed in [5, 7], which specifiesthe interaction g µ,µ + between a base pair µ and the directly adjacent one µ +
1. It wasshown to provide reasonable agreement with experiments [16, 21, 33]. For example,applying this model onto a sequence 5 ′ -TCCAG. . . -3 ′ and its complementary part 3 ′ -AGGTC. . . -5 ′ , the stack energy reads G stack = g TC / AG + g CC / GG + g CA / GT + g AG / TC + . . . The most recent values of g µ,µ + lie in the range of -2.37 to − .
84 kcal / mol at 25°C[11]. The terms G junc ( n ), G (1)loop ( n ) and G (2)loop ( n ) in eq. (9) describe the free energy re-duction due to the release of the stem-hairpin-junction and end-loops and are estimatedfrom [25, 37] as G (1)loop = G (2)loop = .
58 kcal / mol and G junc = .
90 kcal / mol.The strain energy G ssstr ( n , α, f ) of the unfolded single-stranded part [21] with contourlength l = l ( n , α ) can be calculated from the work needed to stretch the unpairedbases. We like to remind the reader that we denote the elongation of the mean end-to-end distance of the ssDNA in force direction by u l ( f ). In what follows we chose In previous publications of some of the authors this contour length was denoted by l n ,α to emphasize thedependence on n (and, in addition, α here). This dependence is caused by the change of the contour lengthin the transitions. For easier reading we suppress to give it explicitely in the following. Further details aboutthe contour length were already discussed in sec. 4. olding and unfolding of a triple - branch DNA molecule with four conformational states u l ( f ) is monotonously increasing with f , it has an inverse f l ( u ) = u − l ( f ),which is the force that is exerted by a ssDNA chain with contour length l , if its meanend-to-end distance is elongated by u . Accordingly, setting ∆ x ss l = u l ( f ), we can write G ssstr ( n , α, f ) = ∆ x ss l Z du ′ f l ( u ′ ) = f ∆ x ss l − f Z d f ′ u l ( f ′ ) . (10)Finally, we computed G ( n , f ) by G ( n , f ) = − k B T ln X α exp − G ( n , α, f ) k B T ! . (11)In fig. 7(b) the FEL is depicted for f = f = .
21 pN. It exhibits thebehaviour anticipated in fig. 7(a): for zero force it has minima at the stable / metastablestates and it becomes tilted with rising force. When approaching the force regimewhere the rips occur in fig. 4(a), the levels of the minima become comparable. We wantto point out that the DNA sequences shown in fig. 1 have been designed deliberately toyield the multiple-state structure seen in fig. 7(b). This gives us some confidence in themodel underlying the construction of the G ( n , α, f ) in eq. (8).Based on the FEL we can easily calculate the transition probability W ( f i | f i − ) forthe first rupture force f i in the i th transition if the first rupture force was f i − in the( i − W ( f i | f i − ) = γ i r γ i , i + ( f ) exp h − γ i r Z ff ∗ i − d f ′ γ i , i + ( f ′ ) i , (12)where f ∗ i = f i − ∆ f i (and f = f ∗ = r was the loading rate, see sec. 2, and γ i and γ i , i + ( f ) were defined in eq. (6). The activation energy ∆ G i , i + ( f ) appearing in eq. (7)was calculated, as indicated in fig. 7(a), from the G ( n , f ) by determining the energy G min i ( f ) of the local minimum belonging to state i and the saddle point energy G saddle i , i + ( f )of the i th transition between the i th and ( i + ∆ G i , i + ( f ) = G saddle i , i + ( f ) − G min i ( f ).The attempt rate γ i was used as the only fitting parameter.For the joint probability density of the three first rupture forces we then obtain Ψ ( f , f , f ) = W ( f | W ( f | f ) W ( f | f ) , (13)which allows us to calculate the distributions shown in fig. 6(a).Figure 8 displays the histograms for the three rips from fig. 6(a) in comparisonwith the distributions calculated from our theory. In view of the available statistics (82cycles, see sec. 2), the agreement is quite satisfactory. An important class of biophysical studies is the investigation of junctions in moleculessince they present manifold ways to interact with other substances, for instance cations.Three-way junctions are especially interesting because metal ions such as magnesium[14, 18] can bind to them and alter the tertiary structure. Here a first step of such a olding and unfolding of a triple - branch DNA molecule with four conformational states r e l a t i v e f r e q u e n c y experimenttheory
16 17 experimenttheory
14 16 18 experimenttheory
Figure 8: Comparison of the first rupture force distributions from fig. 6(a) with thetheory for the first ( f ), the second ( f ) and the third ( f ) transition, see left, middle andright panel, respectively. The best fit was obtained for attempt rates γ = Hz, γ = . Hz and γ = . Hz.study is presented where we investigate a molecule with a three-way junction alone,without in vivo relevant substances.One of our aims of this work was to study whether the construction and kineticsof more complex DNA molecules with richer folding-unfolding behaviour can be de-scribed by proper extensions of theories developed successfully for two-state systemsso far. Our results show that this is indeed possible, at least for the unfolding trajec-tories. A triple-branch molecule has been specifically designed to produce a four-statesystem based on a model for the free energy landscape. This design was successfuland we were able to prove the existence of these states by the emergence of associatedforce rips in pulling experiments. The first rupture forces have been systematicallyrecorded in these pulling experiments and their distributions have been calculated. Atransition rate theory based on the free energy landscape was successful in describingthese distributions.Two patterns have been found in the measured unfolding trajectories, one indicat-ing the anticipated unfolding behaviour and the other one pointing to the occurrenceof irreversible molecular fraying. This characterisation was possible by connecting theextracted force jump values to the change in the number of opened bps at each transi-tion. For this estimation we tested the validity of two polymer models for the elasticresponse of ssDNA (FJC and WLC) and di ff erent sets of parameters in order to find thebest agreement with the expected values. This analysis is useful to compare the elasticproperties measured in DNA unzipping experiments with those obtained by stretchingssDNA polymers [11].One class of molecules required a smaller force than anticipated to unfold the stemsince some bps at its basis are partly or completely melted due to photodamaging.Permanent molecular fraying is an usually undesired, but frequent e ff ect in single- olding and unfolding of a triple - branch DNA molecule with four conformational states
Acknowledgements
S. E. thanks the Deutscher Akademischer Austauschdienst (DAAD) for providing fi-nancial support (FREE MOVER and PROMOS) for stays at the Small Biosystems Labin Barcelona where the experiments have been performed. A. A. is supported by grantAP2007-00995. F. R. is supported by the grants FIS2007-3454, Icrea Academia 2008and HFSP (RGP55-2008).
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