Two-State Folding Energy Determination Based on Transition Points in Nonequilibrium Single-Molecule Experiments

Abstract

Many small protein domains or nucleic acid structures undergo two-state unfolding-refolding transitions during mechanical stretching using single-molecule techniques. Here, by applying the Jarzynski equality (JE), we analytically express the folding energy of a molecule as a function of the experimentally measured transition points ξ* obtained with two typical time-varying mechanical constraints: the force constraints F(t) and the position constraints R(t) of a Hookian spring attached to one end of the molecule. Compared to previous applications of JE based on the integration of accurately measured force-extension curves of a tether that typically contains the molecule of interest and handles, our approach just needs to accurately measure a single data point. In the case of the F(t) process, the calculation is handle-independent. The broad applications of the theory are demonstrated by measuring the folding energies of a DNA hairpin, a DNA G-quadruplex, and the titin I27 domain based on transition forces using magnetic tweezers.

Structural–elastic determination of the force-dependent transition rate of biomolecules

“…k u ( F ) and k r ( F ) curves cross at F c ∼ 8.9 pN, predicting that at this force the unzipped and zipped states have equal probabilities, which is close to the value determined by constant force equilibrium measurements reported in our previous paper on the same DNA within ∼1 pN (Fig. S4 in ref. 35 ).…”

Unexpected Low Mechanical Stability of Titin I27 Domain at Physiologically Relevant Temperature

“…The tug-of-war between the two factors determines the overall temperature dependent thermal stability of I27. Here we note that the absolute value of Δ G 0 ( T ) determined from the current study (∼−3 to −5 kcal mol –1 depending on temperature) based on equilibrium transitions is significantly smaller than those reported in previous AFM studies (∼ −11 to −12 kcal mol –1 depending on temperature) based on nonequilibrium work done during the pulling process estimated by Jarzynski equality. ,− It has been known that Jarzynski equality tends to overestimate the free energy change when the experiments are far from the equilibrium . Therefore, it is likely that the previous AFM studies overestimated the free energy change since the experiments were carried out at conditions far away from equilibrium indicated by a much larger unfolding forces (>100 pN) than the critical force (3–5 pN depending temperature) where unfolding and refolding have equal probabilities.…”

“…The force-dependent folding energy at different temperatures can be determined from the equilibrium probability ratio of the unfolded ( p u ) and folded ( p 0 ) states by the equation From it the temperature-dependent zero-force folding energy can be calculated by Δ G 0 ( T ) = Δ G ( F , T ) – ΔΦ­( F ), where Here x 0 ( f ) and x u ( f ) are the force–extension curves of I27 in the folded and unfolded states that have been well characterized (Text S4). , We performed the equilibrium unfolding and refolding dynamics measurements at temperatures of 23 °C, 25 °C, 27 °C, 29 °C, 31 °C, 35 °C, and 37 °C and analyzed the temperature dependence of Δ G 0 ( T ) (Figure ). The resulting temperature dependent Δ G 0 ( T ) can be fitted with a linear function of temperature, from which the associated entropic change Δ S 0 = −0.13 ± 0.01 kcal mol –1 K –1 and enthalpic change Δ H 0 = −44.1 ± 3.7 kcal mol –1 during I27 folding are determined based on Δ G 0 ( T ) = Δ H 0 – T Δ S 0 .…”

Characterization of G-Quadruplexes Folding/Unfolding Dynamics and Interactions with Proteins from Single-Molecule Force Spectroscopy

“…In force-ramp experiments, the folding rate k f of G4s can be obtained by measuring the time-evolution of the refolding probability p fold ( t ) and fitting p fold ( t ) with a single exponential function p fold ( t ) = p st [1 − exp(− k f t )] ( Figure 3 D,E) [ 50 ]. The free energy can be also estimated using non-equilibrium theorems such as Jarzynski equality [ 40 , 50 , 99 , 100 ]; however, such an approach tends to overestimate the unfolding free energy, particularly when the experiments are far from the equilibrium.…”