Pulling Direction as a Reaction Coordinate for the Mechanical Unfolding of Single Molecules

Abstract

The folding and unfolding kinetics of single molecules, such as proteins or nucleic acids, can be explored by mechanical pulling experiments. Determining intrinsic kinetic information, at zero stretching force, usually requires an extrapolation by fitting a theoretical model. Here, we apply a recent theoretical approach describing molecular rupture in the presence of force to unfolding kinetic data obtained from coarse-grained simulations of ubiquitin. Unfolding rates calculated from simulations over a broad range of stretching forces, for different pulling directions, reveal a remarkable "turnover" from a force-independent process at low force to a force-dependent process at high force, akin to the "roll-over" in unfolding rates sometimes seen in studies using chemical denaturant. While such a turnover in rates is unexpected in one dimension, we demonstrate that it can occur for dynamics in just two dimensions. We relate the turnover to the quality of the pulling direction as a reaction coordinate for the intrinsic folding mechanism. A novel pulling direction, designed to be the most relevant to the intrinsic folding pathway, results in the smallest turnover. Our results are in accord with protein engineering experiments and simulations which indicate that the unfolding mechanism at high force can differ from the intrinsic mechanism. The apparent similarity between extrapolated and intrinsic rates in experiments, unexpected for different unfolding barriers, can be explained if the turnover occurs at low forces.

Theory, analysis, and interpretation of single-molecule force spectroscopy experiments

“…5. Such behavior has already been observed in proteinunfolding simulations (36), and should be found experimentally as further advances in single-molecule experimental techniques allow both lower-and higher-force regimes to be explored. In fact, in experiments, so-called catch-bonds have been reported in which force initially slows down rupture, before accelerating it at high force (37,38).…”

Coordinate-dependent diffusion in protein folding

“…In particular, it would be difficult to even resolve the unfolded and folded states using the instantaneous value of such a coordinate (43,44). Even though for any coordinate r the decay of the correlation function hrðtÞrð0Þi at long times t reflects the slowest relaxing modes, for poor coordinates the amplitude of the slow decay will be small.…”

Towards design principles for determining the mechanical stability of proteins

“…Single molecule AFM experiments allow Δx U and ΔG* to be quantified, (given an estimate for the exponential pre-factor, A), uncovering features of the underlying energy landscape of proteins 52-55 . 10 While the Bell model is most frequently employed to extract information on the unfolding energy landscape of a protein using AFM, it should be noted that a number of alternative models have now been proposed in the literature [56][57][58][59][60][61][62][63][64] . We refer the reader to this literature for further information.…”

“…These studies have demonstrated that proteins can be ranked according to their secondary structure content and arrangement -where mostly 50 alpha-helical proteins are mechanically weaker (low F U ) than those predominantly composed of beta-sheet structures (higher F U ) 44,46 . The importance of the arrangement of the secondary structure in relation to the direction of the pulling force has been demonstrated, where for example the shearing apart of two beta 70 strands requires a greater force than "un-zipping" them sequentially 56,[66][67][68][69] . Indeed, an early molecular dynamics study on the I27 protein identified a 'mechanical clamp' region within the secondary structure which involved two neighbouring betastrands 70 .…”