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Precisely quantifying the energetics that drive the folding of membrane proteins into a lipid bilayer remains challenging. More than 15 years ago, atomic force microscopy (AFM) emerged as a powerful tool to mechanically extract individual membrane proteins from a lipid bilayer. Concurrently, fluctuation theorems, such as the Jarzynski equality, were applied to deduce equilibrium free energies (ΔG0) from non-equilibrium single-molecule force spectroscopy records. The combination of these two advances in single-molecule studies deduced the free-energy of the model membrane protein bacteriorhodopsin in its native lipid bilayer. To elucidate this free-energy landscape at a higher resolution, we applied two recent developments. First, as an input to the reconstruction, we used force-extension curves acquired with a 100-fold higher time resolution and 10-fold higher force precision than traditional AFM studies of membrane proteins. Next, by using an inverse Weierstrass transform and the Jarzynski equality, we removed the free energy associated with the force probe and determined the molecular free-energy landscape of the molecule under study, bacteriorhodopsin. The resulting landscape yielded an average unfolding free energy per amino acid (aa) of 1.0 ± 0.1 kcal/mol, in agreement with past single-molecule studies. Moreover, on a smaller spatial scale, this high-resolution landscape also agreed with an equilibrium measurement of a particular three-aa transition in bacteriorhodopsin that yielded 2.7 kcal/mol/aa, an unexpectedly high value. Hence, while average unfolding ΔG0 per aa is a useful metric, the derived high-resolution landscape details significant local variation from the mean. More generally, we demonstrated that, as anticipated, the inverse Weierstrass transform is an efficient means to reconstruct free-energy landscapes from AFM data.
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Experiments on Probing the Configuration Space of Post-Buckled Panels
Abstract This paper describes a primarily experimental study in which a nonlinear structural component (a slender, mechanically buckled panel) is subject to probing. That is, equilibrium configurations are explored when a specific location on the panel is subject to the application of a (variable) displacement constraint and characterized by a corresponding probe force. This probe force (in this study located at the center of the rectangular panels) is measured using a load cell and the resulting shape(s), taken up by the panel, measured using digital image correlation (DIC). Although the probe is only applied at a single location, this arrangement supplies considerable information about the changing equilibrium landscape including revealing co-existing equilibrium configurations using large perturbations and associated hysteresis phenomena. In addition, monitoring the probing force, and specifically when it drops to zero, provides a window into “free” equilibria that would otherwise be unstable and unobservable. Finally, it is shown that the probed equilibrium configurations provide the “landscape” within which any dynamically induced trajectories evolve including snap-through oscillations.
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- Award ID(s):
- 1926672
- PAR ID:
- 10233568
- Date Published:
- Journal Name:
- Journal of Applied Mechanics
- Volume:
- 87
- Issue:
- 12
- ISSN:
- 0021-8936
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1115/1.4048197
