DNA calorimetric force spectroscopy at single base pair resolution

We used a temperature-jump optical trap (Fig. 1a and Sec. 1, Methods) to unzip a 3593bp ( ≈ 3.6 kbp) DNA hairpin ending in a GAAA tetraloop. Pulling experiments were conducted at temperatures 7–42 °C at 1M NaCl in 10 mM Tris-HCl buffer (pH 7.5). In Fig. 1b and Supplementary Figs. 1, 2, we show the measured force-distance curves (FDCs). They exhibit a sawtooth pattern upon increasing the trap-pipette distance, λ, until the hairpin unzips completely and the elastic response of the released single-stranded DNA (ssDNA) is measured (rightmost part of Fig. 2b). Upon increasing T, the hairpin unzips at progressively lower forces (horizontal dashed lines in Fig. 1b) from  ~20pN at 7 °C to  ~14pN at 42 °C. This indicates that the DNA stability decreases with T, and the energy parameters of the NN model are temperature dependent: the higher the temperature, the lower the hairpin’s free energy of hybridization. Moreover, the molecular extension of the ssDNA at a given force increases with T, yielding a total of  ~ +500 nm between 7^∘ and 42 °C (Fig. 1b, horizontal grey double arrow).

T-dependence of the ssDNA elasticity

Deriving the full NNBP parameters in unzipping experiments requires measuring the T-dependent ssDNA elasticity. To do this, we extracted the force versus the hairpin’s molecular extension (x) curve (hereafter referred to as FEC) by using the relation

with x being shown as a green vertical line in Fig. 1a. Here, x_b and 2x_h are the bead displacement relative to the trap’s center and the handles extension, respectively (grey vertical lines in Fig. 1a). Notice that all extensions are f and T dependent. To determine the term x_b + 2x_h in Eq. (1), we have used the effective stiffness method⁵¹ with x_b + 2x_h = f/k_eff and x = λ − f/k_eff (Supplementary Methods, Sec. 2) where k_eff is obtained from a linear fit to the first FDC slope, i.e., when the hairpin is fully folded (Supplementary Figs. 3 and 4). This gives the FEC at each T (Supplementary Fig. 5).

From Fig. 1a, x = x_ss + x_d, where x_ss is the ssDNA extension and x_d is the projection of the helix diameter (d = 2 nm⁵²) on the pulling axis, which is described by Eq. (7) (Methods). To model the ssDNA elasticity, we used the inextensible WLC model^53,54 (Sec. 2, Methods). In this model, the extension x_ss at a given force is proportional to the number of released bases n, \({x}_{{{{\rm{ss}}}}}(f,T)=n{x}_{{{{\rm{ss}}}}}^{(1)}(f,T)\) where x⁽¹⁾ is the extension per base. For the fully folded hairpin, n = 0 and x = x_d, whereas for the fully unzipped hairpin, \(x=2{x}_{{{{\rm{ss}}}}}=2(N+L/2){x}_{{{{\rm{ss}}}}}^{(1)}\) with N the number of base pairs in the stem and L the loop size. At a given T, we measured \({x}_{{{{\rm{ss}}}}}^{(1)}(f,T)\) by fitting the WLC in Eq. (6) (Methods) to the FEC after the last force rip, i.e., where the hairpin is fully unfolded and n = 2(N + L/2) (Supplementary Fig. 3). Applying this procedure to the FECs at all T, gives the temperature-dependent persistence length (l_p) and interphosphate distance (d_b) of the ssDNA (Fig. 2a and Supplementary Table 3). As T increases, l_p (blue squares), varies from \({l}_{p}^{280{{{\rm{K}}}}}=0.74(7)\) nm to \({l}_{p}^{315{{{\rm{K}}}}}=0.88(4)\) nm (≈ +30%). A linear fit to the data gives the slope 6(1) ⋅ 10⁻³ nm/K (blue line). Moreover, the interphosphate distance, d_b, (orange circles) shows a weak linear T-dependence (≈ +5%) of slope 4(1) ⋅ 10⁻⁴ nm/K (orange line). Similar behavior has been observed for shorter ssDNAs of 20–40 nucleotides⁵⁵ and polypeptide chains³⁵. The observed increase in x_ss with T is predicted in Debye-Huckel theory due to the entropy of the cloud of counterions. Upon increasing temperature, the screening of the phosphates repulsion is reduced, and l_p increases. Our results confirm this trend with l_p increasing with temperature tenfold relative to d_b, which remains almost constant in the studied temperature range.

Derivation of the NNBP entropies

To derive the entropies of the different NNBPs, we have decomposed the full unzipping curve into segments of variable length encompassing different regions along the FEC. Each segment is delimited by two peaks corresponding to force rips along the FEC. The peaks have been selected by looking for local maxima along the experimental signal. To do so, the FEC function derivatives have been numerically computed, and the maxima were determined by imposing df(x)/dx = 0. We notice that the magnitude of the force rip is proportional to the number of released bases in that unfolding event. To ensure the maximal signal-to-noise ratio, we discarded events within the FEC statistical errors by only accounting for force-rips with Δf ≥ 2pN. Figure 2c shows examples of segments starting and ending at a peak (colored circles). Considering all possible combinations of starting and ending peaks, a set of K segments has been obtained for each T. Note that peaks are not necessarily consecutive, as different segments can overlap, making the analysis more robust. In a single unzipping curve, the typical number of peaks is N_p ~ 30 − 35 (see Fig. 2c). No further selection step is applied to the segments for analysis, resulting in a total number of segments K = N_p(N_p − 1)/2 ~ 400 − 600 per temperature. Notice that such a large number of segments, each being a different sequence, permits measuring many sequences in a single unzipping experiment. In this regard, the unzipping approach is high throughout compared to oligo hybridization experiments, where the measurement of melting curves must be repeated for every oligo sequence.

The entropy of hybridization, ΔS_0,k(T), of each segment k is given by

with Δx_k the extension of segment k (Sec. 3, “Methods”). Equation (2) is analogous to the Clausius-Clapeyron equation^20,35,53 in classical thermodynamics. Here, f and x are the equivalent quantities of pressure and volume in hydrostatic systems. The r.h.s. of Eq. (2) depends on the average unzipping force of segment k measured at different temperatures, f_m,k(T), according to the equal area Maxwell construction for segment k (colored horizontal dashed lines in Fig. 2c). f_m,k(T) varies linearly with T, all segments showing the same slope—0.165(3)pN/K within statistical errors (Fig. 2b and Supplementary Table 3). The integral in Eq. (2) accounts for the work needed to stretch the ssDNA and orient the molecule along the pulling axis between zero force and f_m,k(T) (Supplementary Fig. 6A and Table 3). Equation (2) applied to the full FEC gives the hairpin total entropy of hybridization (Supplementary Fig. 6B).

To apply Eq. (2) for a given segment k, we must identify the DNA sequence limited by the initial and final peaks. A WLC curve passing through a peak at (x, f) gives the number n of unzipped bases at that peak (dashed-grey lines in segment Δx_k). The initial and final values n_A and n_B (orange segment in Fig. 2c) identify the DNA sequence of that segment. Let k = 1, 2, …, K enumerate the different segments. The entropy of segment k at zero force and temperature T in the NN model is given by the sum of the individual entropies of all adjacent NNBPs within that segment,

where the sum runs over the ten independent NNBP parameters labeled by the index i, and Δs_i is the entropy of motif i with multiplicity c_k,i, i.e., the number of times motif i appears in segment k. The entropy ΔS_0,k(T) in the l.h.s of Eq. (3) is measured using Eq. (2), and c_k,i for each motif is obtained from the segment sequence. A stochastic gradient descent algorithm has been designed to solve the system of K non-homogeneous linear equations in Eq. (3) and derive the Δs_i parameters at each T (Sec. 4, Methods). The results for the T-dependent DNA NNBP entropies Δs_i are shown in Fig. 2d and reported in Supplementary Table 4. To determine the statistical errors of Δs_i(T), we have propagated the values of the elastic parameters at the limits of their confidence interval. Minimum and maximum values for Δs_i(T) set the confidence intervals with the corresponding mean (Supplementary Methods, Sec. 5). The same method has been applied to all NNBP thermodynamic parameters.

NNBP free energies and enthalpies

From the previously derived NNBP entropies Δs_i(T), we can also derive the NNBP enthalpies, Δh_i(T), from the relation,

with Δg_i(T) the free energies of the different motifs. The Δg_i(T) values are derived via a Monte Carlo optimization based on the prediction of the theoretical FDC according to the NN model^28,29,40. The equilibrium FDC is obtained by calculating the equilibrium force at different values of the trap position given a set of energies and elastic parameters. At each step, λ (Eq. (1)) is increased by a fixed Δλ = 3 nm. The contribution of each element of the molecular construct (optical trap, dsDNA handles, ssDNA, and hairpin diameter) is computed. For a given λ and number of unzipped base pairs n, the corresponding applied force, f, is retrieved by inverting Eq. (1). This calculation is repeated for all values of n. For a given λ, the total energy of the molecular system has two competing contributions: the positive stretching free energy, ΔG_el(n, f), acting as an externally applied work to unfold the hairpin, and the negative hybridization free energy, ΔG₀(n), which keeps the hairpin folded. The equilibrium values of n^* and f^* are determined by finding the absolute minimum over n of ΔG(n, f) for that λ. At each force rip, it is fulfilled that ΔG_el(Δn^*, f) = ΔG₀(Δn^*) rendering the equilibrium FDC a sawtooth pattern made of a sequence of gentle slopes separated by force rips (Supplementary Methods, Sec. 4). The derivation of the equilibrium FDC for each set of energies is the step in an optimization Monte Carlo algorithm used to minimize the error between the theoretical and the experimental FDCs in Fig. 1b (Sec. 5, “Methods”). This procedure gives the eight NNBP independent free energies. The other two parameters (GC/CG and TA/AT) are obtained from the circular symmetry relations^38,39,40.

Fits are shown in Fig. 3a for three selected temperatures, and the Δg_i(T) are shown in Fig. 3b (see also Supplementary Table 5). Results (blue circles) agree with the unified oligonucleotide (UO) dataset (black line) and the energy parameters obtained by Huguet et al. in ref. ⁴⁰ (grey line). In this reference, unzipping experiments at room temperature (298K) were combined with melting temperature data of oligo hybridization over the vastly available literature. Overall agreement is observed, except for some motifs such as AC/TG and GA/CT where the UO energies are lower. The ten NNBP enthalpies were obtained from Eq. (4) at each T and are shown in Fig. 2d (see also Supplementary Table 6). The agreement between the Δg_i(T) values in Fig. 3b with previous measurements under the assumption Δc_p,i = 0 for all motifs^40,46, underlines the strong compensation between the temperature-dependent enthalpies and entropies shown in Fig. 2d that mask the finite Δc_p,i’s.

NNBPs heat capacity changes

The temperature-dependent NNBP entropies and enthalpies permitted us to derive the heat capacity changes Δc_p,i for all motifs by using the relations,

where T_m,i is the melting temperature of motif i where Δg_i(T_m,i) = 0, and Δs_m,i and Δh_m,i are the entropy and enthalpy at T_m,i, fulfilling Δh_m,i = T_m,iΔs_m,i. To derive the ten Δc_p,i, we fit the NNBP entropies to the equation \({A}_{i}+\Delta {c}_{p,i}\log (T)\), being \({A}_{i}=\Delta {s}_{m,i}-\Delta {c}_{p,i}\log ({T}_{m,i})\). The results are shown in Fig. 4a and Table 1 (column 1). From the Δc_p,i, we combined Eqs. (5) with Eq. (4) to fit the experimental values of Δg_i(T) (blue dashed lines in Fig. 3b) to obtain T_m,i. From the T_m,i, we retrieve Δs_m,i and Δh_m,i from Eqs. (5a), (5b) (red and blue dashed lines in Fig. 2d). The fitting procedure is described in Supplementary Methods, Sec. 6. Results for T_m,i, Δs_m,i and Δh_m,i are shown in Fig. 4b and Table 1. Notice the high T_m values of the individual motifs, a consequence of the high enthalpies of the NN motifs.