Elasticity, structure, and relaxation of extended proteins under force

Abstract

Force spectroscopies have emerged as a powerful and unprecedented tool to study and manipulate biomolecules directly at a molecular level. Usually, protein and DNA behavior under force is described within the framework of the worm-like chain (WLC) model for polymer elasticity. Although it has been surprisingly successful for the interpretation of experimental data, especially at high forces, the WLC model lacks structural and dynamical molecular details associated with protein relaxation under force that are key to the understanding of how force affects protein flexibility and reactivity. We use molecular dynamics simulations of ubiquitin to provide a deeper understanding of protein relaxation under force. We find that the WLC model successfully describes the simulations of ubiquitin, especially at higher forces, and we show how protein flexibility and persistence length, probed in the force regime of the experiments, are related to how specific classes of backbone dihedral angles respond to applied force. Although the WLC model is an average, backbone model, we show how the protein side chains affect the persistence length. Finally, we find that the diffusion coefficient of the protein's end-to-end distance is on the order of 10 8 nm 2 /s, is position and side-chain dependent, but is independent of the length and independent of the applied force, in contrast with other descriptions.internal diffusion | potential of mean force | protein elasticity T he development of single-molecule force spectroscopies [atomic force microscopy (AFM), and optical or magnetic tweezers] in the last two decades has opened a whole new and exciting field (1, 2). It is now possible to manipulate biomolecules directly at a molecular level and to study their behavior under force (3). It is therefore not surprising that these techniques have been applied in a broad gamut of contexts, in particular for proteins. In some cases, including enzyme catalysis (4), protein-ligand interaction (5), or folding and unfolding events (6), force is used as a probe, altering the free-energy landscape and the dynamics of the protein (7), and provides valuable kinetic and mechanistic insights. For other systems, force is of direct biological relevance [e.g., cellular adhesion (8) or muscle elasticity (9)].The elasticity of a polypeptide is typically modeled using the worm-like chain (WLC) (1, 2) model of polymer elasticity. Although this simple model from polymer physics has proven remarkably successful at describing and interpreting experimental data, it lacks molecular details associated with proteins extended under force. However, substrate flexibility to adopt the right geometry is key in many situations, including, e.g., molecular recognition (10) or enzymatic reactivity (11). Therefore, a precise understanding of how force modulates and influences protein flexibility at a single-amino acid level is essential and is not provided by the aforementioned model.From a dynamical perspective, it is still unclear how applied force affects internal diffusion ...

How force unfolding differs from chemical denaturation

“…These snapshots of the protein at representative forces suggest that in general no secondary structure is seemingly formed, despite the fact that the protein does exhibit some lateral fluctuations and local structure. The average end-to-end distances L obtained at different forces down to 30 pN, are actually well-fitted by a WLC chain force-extension profile with a contour length L c = 28:4 nm and a persistence length p = 0:39 nm (12). This persistence length value is in remarkable agreement with that obtained by AFM experiments, including some on the same protein (7,10,26), suggesting that simulations reach equilibrium.…”

“…Force-extended configurations of ubiquitin were generated following a protocol described and used previously (12). Briefly, the protein is first unfolded at a very high force.…”

“…We use simulation setups previously developed in our groups to generate chemically unfolded structures (18,19) and force-unfolded (in the range of 30-250 pN) configurations (12,20). We focus on ubiquitin, which is a 76-residue protein well-characterized experimentally, both by bulk techniques (14,21,22) and by force spectroscopies (23).…”

“…Although this approach has proved useful in probing protein folding and unfolding at a single-molecule level, the unfolded configurations generated by such techniques can be very different from the chemically unfolded ones. As an illustration of these large differences, under chemical denaturing conditions the peptide chain usually follows Flory's scaling law for a random coil in a good solvent (8,9), whereas the force-unfolded configurations follow the worm-like chain (WLC) model of polymer entropic elasticity (10)(11)(12). In these two types of experiments, protein refolding therefore occurs from two different regions of the free-energy surface, which may influence the folding mechanism and its kinetics.…”

Correlation Force Spectroscopy for Single Molecule Measurements

“…1 τ in equation ( which gives R~0 .7 μs τ , or about 10 2 times faster. It has been suggested by molecular dynamics simulation that the slow relaxation of single molecules in mechanical SMFS is an artifact due to anchoring the ends of the molecule [146]. While this is plausible until further experiments reveal more evidence, two orders of magnitude lagging in relaxation can be due to real underlying molecular events.…”