Reconstructing folding energy landscapes from splitting probability analysis of single-molecule trajectories

Abstract

Structural self-assembly in biopolymers, such as proteins and nucleic acids, involves a diffusive search for the minimumenergy state in a conformational free-energy landscape. The likelihood of folding proceeding to completion, as a function of the reaction coordinate used to monitor the transition, can be described by the splitting probability, p fold (x). P fold encodes information about the underlying energy landscape, and it is often used to judge the quality of the reaction coordinate. Here, we show how p fold can be used to reconstruct energy landscapes from single-molecule folding trajectories, using force spectroscopy measurements of single DNA hairpins. Calculating p fold (x) directly from trajectories of the molecular extension measured for hairpins fluctuating in equilibrium between folded and unfolded states, we inverted the result expected from diffusion over a 1D energy landscape to obtain the implied landscape profile. The results agreed well with the landscapes reconstructed by established methods, but, remarkably, without the need to deconvolve instrumental effects on the landscape, such as tether compliance. The same approach was also applied to hairpins with multistate folding pathways. The relative insensitivity of the method to the instrumental compliance was confirmed by simulations of folding measured with different tether stiffnesses. This work confirms that the molecular extension is a good reaction coordinate for these measurements, and validates a powerful yet simple method for reconstructing landscapes from single-molecule trajectories.single-molecule biophysics | force spectroscopy | nucleic acid folding | protein folding | optical tweezers S tructure formation by biological polymers like proteins and nucleic acids, an essential process linked to biological function, is typically described by energy landscape theory, in which folding is viewed as a diffusive search through the conformational space of the molecule for the minimum-energy structure (1, 2). This search takes place on an energy landscape, with the surface describing the energy of the molecule as a function of all possible conformations. Because of the many conformational degrees of freedom in even a small biopolymer, folding landscapes are inherently multidimensional, forming a hypersurface. Experimental measurements, however, typically follow some observable that is used to monitor the progress of the reaction. As a result, the full energy hypersurface is projected onto a 1D profile along the chosen reaction coordinate. There is great interest in measuring such energy landscape profiles directly, because they provide a fundamental basis for understanding folding phenomena.Recent advances in single-molecule approaches have provided powerful tools for measuring landscapes. Most notably, 1D landscape profiles can be reconstructed in several ways from force spectroscopy measurements, wherein tension is applied to a molecule and the resulting changes in the molecular extension, the reaction coordinate, are measured (3). Lands...

Direct measurement of sequence-dependent transition path times and conformational diffusion in DNA duplex formation

“…Furthermore, landscapes reconstructed by Eq. 3 were found to agree well with those reconstructed from the inverse Boltzmann transform (which makes no assumptions about D), suggesting not only that the correct barrier height and curvature are being reliably recovered (within experimental error) but also that any effects from the small sequence dependence of D do not noticeably alter the landscape reconstruction (44). To reduce noise in the numerical differentiation, p fold (x) was first smoothed in a 1-nm window with a boxcar filter.…”

“…An interesting implication of the sequence dependence of D is that D should be position dependent along the duplex, leading to changes in the shape of the transition paths (51). Such position dependence has been previously sought but not reliably observed (44,63); it may soon be detectable through analysis of transition path shapes.…”

“…The shape of the energy barrier for each hairpin was reconstructed from the extension trajectories using the committor, p fold , as described previously (44). Briefly, p fold (x) was calculated empirically from the trajectory at each value of the extension, and then the landscape was reconstructed using the following:…”

Communication: Transition-path velocity as an experimental measure of barrier crossing dynamics

“…[1][2][3][4][5][6][7][8][9][10] For example, when the reactant is a protein or DNA in its folded state and the product is the same molecule in the unfolded state, the time evolution of the protein's extension x(t) (i.e., the distance between the termini of the molecular chain) during individual folding and unfolding events has been observed using single-molecule force spectroscopy. 5 These studies have attracted considerable theoretical attention.…”

Enhanced stochastic fluctuations to measure steep adhesive energy landscapes

“…Despite this limitation, the demonstrated method may be appropriate to investigate the effects of surface roughness and hydropathicity, as well as biological interactions, including ligand-receptor binding and the folding landscapes of proteins and nucleic acids with steep energy wells. This capability would greatly benefit determination of detailed energy landscapes, from which we could calculate the kinetic rates (70) and landscape roughness (71), and predict the structural stability of macromolecules.…”