From mechanical folding trajectories to intrinsic energy landscapes of biopolymers

Abstract

In single-molecule laser optical tweezer (LOT) pulling experiments, a protein or RNA is juxtaposed between DNA handles that are attached to beads in optical traps. The LOT generates folding trajectories under force in terms of time-dependent changes in the distance between the beads. How to construct the full intrinsic folding landscape (without the handles and beads) from the measured time series is a major unsolved problem. By using rigorous theoretical methods--which account for fluctuations of the DNA handles, rotation of the optical beads, variations in applied tension due to finite trap stiffness, as well as environmental noise and limited bandwidth of the apparatus--we provide a tractable method to derive intrinsic free-energy profiles. We validate the method by showing that the exactly calculable intrinsic free-energy profile for a generalized Rouse model, which mimics the two-state behavior in nucleic acid hairpins, can be accurately extracted from simulated time series in a LOT setup regardless of the stiffness of the handles. We next apply the approach to trajectories from coarse-grained LOT molecular simulations of a coiled-coil protein based on the GCN4 leucine zipper and obtain a free-energy landscape that is in quantitative agreement with simulations performed without the beads and handles. Finally, we extract the intrinsic free-energy landscape from experimental LOT measurements for the leucine zipper.

Fig. 1.

Dual-beam optical tweezer setup for studying the equilibrium folding landscape of a single protein molecule under force.

Fig. 2.

GRM hairpin in an optical tweezer setup. First row shows the exact end-to-end distributions along for each component type in the system: (A) GRM, (B) dsDNA handle, (C) polystyrene bead. Handle, bead, and trap parameters are listed in Table S1 (GRM column). (Upper) Probabilities projected onto cylindrical coordinates . (Lower) Projection onto z alone. (D) Result for the total system end-to-end distribution _tot derived by convolving the component probabilities and accounting for the optical traps. (E–G) Construction of the original GRM distribution starting from _tot. (E) _tot (purple) and (blue) as a function of z on the bottom axis, measured relative to , the average extension for each distribution. For _tot, the upper axis shows the z range translated into the corresponding trap forces F. After removing the trap effects, is the distribution for constant force F₀ = 11.9 pN. (F) , describing the total probability at F₀ of fluctuations resulting from both handles and the rotation of the beads. (G) Constructed solution for (solid line), obtained by numerically inverting the convolution . Exact analytical result for is shown as a dashed line; z_N is the position of the N peak.

Fig. 3.

Effects of handle characteristics on the free-energy profile of the GRM in a LOT setup. (A) Total system free energy ℱ_tot = −k_BT ln _tot for fixed L = 100 nm, and varying ratios l_p/L. All of the other parameters are in Table S1 (GRM column). The exact analytical free energy at F₀ =11.9 pN (dashed line) for the GRM alone, , is shown for comparison. (B) For each ℱ_tot in A, the construction of at F₀, together with the exact answer (dashed line). (C) For system parameters matching the experiment (Table S1), the variance of the point-spread function broken down into the individual handle, bead, and linker contributions. The fraction for each component is shown as a function of varying handle elastic modulus γ.

Fig. 4.

Intrinsic characteristics of the LZ26 leucine zipper at constant F₀, derived from SOP simulations in the absence of handles/beads. (A) LZ26 free energy at F₀ = 12.3 pN vs. end-to-end extension z. (Right) Representative protein configurations from the four wells (N, I1, I2, U), with asparagine residues colored blue. (B) Average fraction of native contacts between the two alpha-helical strands of LZ26 (the “zipper bonds”) as a function of z. (Left) Lists of the a and d residues in the heptads making up the amino acid sequence for each LZ26 strand, placed according to their position along the zipper. Asparagines (N) are highlighted in blue. (C) For the residues listed in B, the residue contact energies used in the SOP simulation [rescaled BT (30) values].

Fig. 5.

(A and B) A trajectory fragment and the probability distribution from SOP simulations of the LZ26 leucine zipper at constant force F₀ = 12.3 pN in the absence of handles/beads. (C and D) A trajectory fragment and the total system distribution _tot at z_trap = 503 nm. C shows both the total extension z_tot(t) (purple) and the protein extension z_p(t) (gray). Triangles mark times when the protein makes a transition between states, and the arrows point to two enlarged portions of the trajectories. In all cases the z-axis origin is z_I1, the peak location of the I1 intermediate state. (E–G) Leucine zipper free-energy profiles extracted from time series (third row = simulation, fourth row = experiment). First column shows the total system end-to-end distribution _tot and the corresponding at constant force F₀ = 12.3 pN. In the experimental case F₀ = 12.3 ± 0.9 pN is the midpoint force at which the I1 and U states are equally likely. For _tot, z_trap = 503 nm (simulation), 1553 ± 1 nm (experiment). Force scales at the top are the range of trap forces for _tot. Second column shows the computed intrinsic protein free-energy profiles compared with the total system profile, ℱ_tot (shifted upward for clarity). (F) SOP simulations for the protein alone at constant F₀ provide a reference landscape, drawn as a dashed line. (H) Dotted curve is the reconstructed at the midpoint force F₀ = 12.1 ± 0.9 pN, from a second, independent experimental trajectory, with z_trap = 1547 ± 1 nm. curves have a median uncertainty of 0.4 k_BT over the plotted range (see SI Text for error analysis).