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 freeenergy 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. T he energy landscape perspective has provided a conceptual framework to describe how RNA (1) and proteins (2-4) fold. Some of the key theoretical predictions (5, 6), have been confirmed by experiments (7). More refined comparisons require mapping the full folding landscape of biomolecules. Advances in laser optical tweezer (LOT) experiments have been used to obtain free-energy profiles as a function of the extension of biomolecules under tension (7-12).The usefulness of the LOT technique, however, hinges on the assumption that information about the fluctuating biomolecule can be accurately recovered from the raw experimental data, namely the time-dependent changes in the positions of the beads in the optical traps, attached to the biomolecule by doublestranded DNA (dsDNA) handles (Fig. 1). Thus, we only have access to the intrinsic folding landscape of the biomolecule (in the absence of handles and beads) indirectly through the bead-bead separation along the force direction. Many extraneous factors, such as fluctuations of the handles (13, 14), rotation of the beads, and the varying applied tension due to finite trap stiffness, can distort the intrinsic folding landscape. Moreover, the detectors and electronic systems used in the data collection have finite response times, leading to filtering of high-frequency components in the signal (15). Ad hoc attempts have been made to account for handle effects based on experimental estimates of stretched DNA properties, using techniques similar to image deconvolution (8,11,16). Theory has been used to extract free-energy information from nonequilibrium pullin...

On artifacts in single-molecule force spectroscopy

“…However, if one can measure the elastic properties of the linker, and thereby find the function V ðq − xÞ, then one can estimate G M ðxÞ by "deconvolving" or "inverting" the transform given in Eq. 2 (26,33). Interestingly, for a harmonic linker, this problem is the same as finding G A ðqÞ in a constant pulling velocity experiment from the free energy along the trap separation obtained using Jarzynski's identity (13).…”

Nonlinear irreversible thermodynamics of single-molecule experiments

“…The probability distribution associated to these fluctuations, although may take a stationary shape, is bimodal and it cannot be taken as an equilibrium distribution compatible with thermodynamic equilibrium. According to the experimental and simulation results reported in [3], for tensions around 15pN, the intrinsic free-energy of the RNA molecule is bistable as a function of the elongation x. A series expansion of F x around the critical value of the elongation x = x c + x, yields the Landau-De Gennes-like model…”

“…The effective free-energy is reported in experiments and simulations from which the values of the parameters and important biological information can be obtained [2,3,[5][6][7]26]. The numerical solutions of Eq.…”

“…Many mesoscopic systems of great interest ranging from biology, such as proteins, RNA or DNA molecules [1][2][3][4][5][6][7][8][9][10], to nano-engeneering, like rotaxanes and catenanes [11][12][13], have similar dynamic behaviors due to their intrinsic properties and structure, as well as to their strong coupling with the surroundings. In particular, when they are subject to stressing conditions, like stretching forces or radiative interaction, the dynamics of these systems may present critical oscillations and stochastic resonance phenomena.…”

“…In order to highlight the ideas and simplify the presentation, we apply our formalism to analyze single RNA macromolecule stretching experiments which provide information on its energy landscape [2][3][4][5][6][7][8][9]16]. Specifically, the existence of a loop in the force-extension curve (FEC) implies that there is a range of tensions for which two stable configurations of the system are possible.…”