Exploring hysteresis and energy dissipation in single-molecule force spectroscopy

NMR structures and magnetic force spectroscopy studies of small molecules binding to models of an RNA CAG repeat expansion

RNA repeat expansions fold into stable structures and cause microsatellite diseases such as Huntington's disease (HD), myotonic dystrophy type 1 (DM1), and spinocerebellar ataxias (SCAs). The trinucleotide expansion of r(CAG), or r(CAG)exp, causes both HD and SCA3, and the RNA's toxicity has been traced to its translation into polyglutamine (polyQ; HD) as well as aberrant pre-mRNA alternative splicing (SCA3 and HD). Previously, a small molecule, 1, was discovered that binds to r(CAG)exp and rescues aberrant pre-mRNA splicing in patient-derived fibroblasts by freeing proteins bound to the repeats. Here, we report the structures of single r(CAG) repeat motif (5'CAG/3'GAC where the underlined adenosines form a 1×1 nucleotide internal loop) in complex with 1 and two other small molecules via nuclear magnetic resonance (NMR) spectroscopy combined with simulated annealing. Compound 2 was designed based on the structure of 1 bound to the RNA while 3 was selected as a diverse chemical scaffold. The three complexes, although adopting different 3D binding pockets upon ligand binding, are stabilized by a combination of stacking interactions with the internal loop's closing GC base pairs, hydrogen bonds, and van der Waals interactions. Molecular dynamics (MD) simulations performed with NMR-derived restraints show that the RNA is stretched and bent upon ligand binding with significant changes in propeller-twist and opening. Compound 3 has a distinct mode of binding by insertion into the helix, displacing one of the loop nucleotides into the major groove and affording a rod-like shape binding pocket. In contrast, 1 and 2 are groove binders. A series of single molecule magnetic force spectroscopy studies provide a mechanistic explanation for how bioactive compounds might rescue disease-associated cellular phenotypes.

Single-Molecule Force Spectroscopy of Membrane Protein Folding

Single-molecule force spectroscopy is a unique method that can probe the structural changes of single proteins at a high spatiotemporal resolution while mechanically manipulating them over a wide force range. Here, we review the current understanding of membrane protein folding learned by using the force spectroscopy approach. Membrane protein folding in lipid bilayers is one of the most complex biological processes in which diverse lipid molecules and chaperone proteins are intricately involved. The approach of single protein forced unfolding in lipid bilayers has produced important findings and insights into membrane protein folding. This review provides an overview of the forced unfolding approach, including recent achievements and technical advances. Progress in the methods can reveal more interesting cases of membrane protein folding and clarify general mechanisms and principles.

Hysteresis loop area scaling exponents in DNA unzipping by a periodic force: A Langevin dynamics simulation study

Using Langevin dynamics simulations, we study the hysteresis in unzipping of longer double-stranded DNA chains whose ends are subjected to a time-dependent periodic force with frequency ω and amplitude G keeping the other end fixed. We find that the area of the hysteresis loop, A_{loop}, scales as 1/ω at higher frequencies, whereas it scales as (G-G_{c})^{α}ω^{β} with exponents α=1 and β=1.25 in the low-frequency regime. These values are same as the exponents obtained in Monte Carlo simulation studies of a directed self-avoiding walk model of a homopolymer DNA [R. Kapri, Phys. Rev. E 90, 062719 (2014)10.1103/PhysRevE.90.062719], and the block copolymer DNA [R. K. Yadav and R. Kapri, Phys. Rev. E 103, 012413 (2021)2470-004510.1103/PhysRevE.103.012413] on a square lattice, and differs from the values reported earlier using Langevin dynamics simulation studies on a much shorter DNA hairpins.

Unzipping of a double-stranded block copolymer DNA by a periodic force

Using Monte Carlo simulations, we study the hysteresis in unzipping of a double-stranded block copolymer DNA with -A_{n}B_{n}- repeat units. Here A and B represent two different types of base pairs having two and three bonds, respectively, and 2n represents the number of such base pairs in a unit. The end of the DNA are subjected to a time-dependent periodic force with frequency (ω) and amplitude (g_{0}) keeping the other end fixed. We find that the equilibrium force-temperature phase diagram for the static force is independent of the DNA sequence. For a periodic force case, the results are found to be dependent on the block copolymer DNA sequence and on the base pair type on which the periodic force is acting. We observe hysteresis loops of various shapes and sizes and obtain the scaling of loop area both at low- and high-frequency regimes.

Simulated Force Quench Dynamics Shows GB1 Protein Is Not a Two State Folder

Single molecule force spectroscopy is a useful technique for investigating mechanically induced protein unfolding and refolding under reduced forces by monitoring the end-to-end distance of the protein. The data is often interpreted via a "two-state" model based on the assumption that the end-to-end distance alone is a good reaction coordinate and the thermodynamic behavior is then ascribed to the free energy as a function of this one reaction coordinate. In this paper, we determined the free energy surface (PMF) of GB1 protein from atomistic simulations in explicit solvent under different applied forces as a function of two collective variables (the end-to-end-distance, and the fraction of native contacts ρ). The calculated 2-d free energy surfaces exhibited several distinct states, or basins, mostly visible along the ρ coordinate. Brownian dynamics (BD) simulations on the smoothed free energy surface show that the protein visits a metastable molten globule state and is thus a three state folder, not the two state folder inferred using the end-to-end distance as the sole reaction coordinate. This study lends support to recent experiments that suggest that GB1 is not a two-state folder.

Colloidal transport and diffusion over a tilted periodic potential: dynamics of individual particles

A tilted two-layer colloidal system is constructed for the study of force-assisted barrier-crossing dynamics over a periodic potential. The periodic potential is provided by the bottom layer colloidal spheres forming a fixed crystalline pattern on a glass substrate. The corrugated surface of the bottom colloidal crystal provides a gravitational potential field for the top layer diffusing particles. By tilting the sample at an angle θ with respect to the vertical (gravity) direction, a tangential component of the gravitational force F is applied to the diffusing particles. The measured mean drift velocity v(F, Eb) and diffusion coefficient D(F, Eb) of the particles as a function of F and energy barrier height Eb agree well with the exact results of the one-dimensional drift velocity (R. L. Stratonovich, Radiotekh. Elektron, 1958, 3, 497) and diffusion coefficient (P. Reimann, et al., Phys. Rev. Lett., 2001, 87, 010602 and P. Reimann, et al., Phys. Rev. E, 2002, 65, 031104). Based on these exact results, we show analytically and verify experimentally that there exists a scaling region, in which v(F, Eb) and D(F, Eb) both scale as ν'(F)exp[-E(F)/kBT], where the Arrhenius pre-factor ν'(F) and effective barrier height E(F) are both modified by F. The experiment demonstrates the applications of this model system in evaluating different scaling forms of ν'(F) and E(F) and their accuracy, in order to extract useful information about the external potential, such as the intrinsic barrier height Eb.

Unzipping DNA by a periodic force: hysteresis loop area and its scaling

Using Monte Carlo simulations, we study the hysteresis in the unzipping of double-stranded DNA whose ends are subjected to a time-dependent periodic force with frequency (ω) and amplitude (G). For the static force, i.e., ω→0, the DNA is in equilibrium with no hysteresis. On increasing ω, the area of the hysteresis loop initially increases and becomes maximum at frequency ω*(G), which depends on the force amplitude G. If the frequency is increased further, we find that for lower amplitudes the loop area decreases monotonically to zero, but for higher amplitudes it has an oscillatory component. The height of subsequent peaks decreases, and finally the loop area becomes zero at very high frequencies. The number of peaks depends on the length of the DNA. We give a simple analysis to estimate the frequencies at which maxima and minima occur in the loop area. We find that the area of the hysteresis loop scales as 1/ω in the high-frequency regime, whereas it scales as G(α)ω(β) with exponents α=1 and β=5/4 at low frequencies. The values of the exponents α and β are different from the exponents reported earlier based on the hysteresis of small hairpins.

Kinetics and energetics of biomolecular folding and binding

The ability of biomolecules to fold and to bind to other molecules is fundamental to virtually every living process. Advanced experimental techniques can now reveal how single biomolecules fold or bind against mechanical force, with the force serving as both the regulator and the probe of folding and binding transitions. Here, we present analytical expressions suitable for fitting the major experimental outputs from such experiments to enable their analysis and interpretation. The fit yields the key determinants of the folding and binding processes: the intrinsic on-rate and the location and height of the activation barrier.

Predicted rupture force of a single molecular bond becomes rate independent at ultralow loading rates

We present for the first time a theoretical model of studying the saturation of the rupture force of a single molecular bond that causes the rupture force to be rate independent under an ultralow loading rate. This saturation will obviously bring challenges to understanding the rupture behavior of the molecular bond using conventional methods. This intriguing feature implies that the molecular bond has a nonzero strength at a vanishing loading rate. We find that the saturation behavior is caused by bond rebinding when the loading rate is lower than a limiting value depending on the loading stiffness.

Hysteresis and nonequilibrium work theorem for DNA unzipping

We study by using Monte Carlo simulations the hysteresis in unzipping and rezipping of a double stranded DNA (dsDNA) by pulling its strands in opposite directions in the fixed force ensemble. The force is increased at a constant rate from an initial value g(0) to some maximum value g(m) that lies above the phase boundary and then decreased back again to g(0). We observed hysteresis during a complete cycle of unzipping and rezipping. We obtained probability distributions of work performed over a cycle of unzipping and rezipping for various pulling rates. The mean of the distribution is found to be close (the difference being within 10%, except for very fast pulling) to the area of the hysteresis loop. We extract the equilibrium force versus separation isotherm by using the work theorem on repeated nonequilibrium force measurements. Our method is capable of reproducing the equilibrium and the nonequilibrium force-separation isotherms for the spontaneous rezipping of dsDNA.

Evolution of the potential energy landscape with static pulling force for two model proteins

The energy landscape is analyzed for off-lattice bead models of protein L and protein G as a function of a static pulling force. Two different pairs of attachment points (pulling directions) are compared in each case, namely, residues 1/56 and 10/32. For the terminal residue pulling direction 1/56, the distinct global minimum structures are all extended, aside from the compact geometry that correlates with zero force. The helical turns finally disappear at the highest pulling forces considered. For the 10/32 pulling direction, the changes are more complicated, with a variety of competing arrangements for beads outside the region where the force is directly applied. These alternatives produce frustrated energy landscapes, with low-lying minima separated by high barriers. The calculated folding pathways in the absence of force are in good agreement with previous work. The N-terminal hairpin folds first for protein L and the C-terminal hairpin for protein G, which exhibits an intermediate. However, for a relatively low static force, where the global minimum retains its structure, the folding mechanisms change, sometimes dramatically, depending on the protein and the attachment points. The scaling relations predicted by catastrophe theory are found to hold in the limit of short path lengths.

Collapse dynamics of single proteins extended by force

Single-molecule force spectroscopy has opened up new approaches to the study of protein dynamics. For example, an extended protein folding after an abrupt quench in the pulling force was shown to follow variable collapse trajectories marked by well-defined stages that departed from the expected two-state folding behavior that is commonly observed in bulk. Here, we explain these observations by developing a simple approach that models the free energy of a mechanically extended protein as a combination of an entropic elasticity term and a short-range potential representing enthalpic hydrophobic interactions. The resulting free energy of the molecule shows a force-dependent energy barrier of magnitude, DeltaE =epsilon(F - F(c))(3/2), separating the enthalpic and entropic minima that vanishes at a critical force F(c). By solving the Langevin equation under conditions of a force quench, we generate folding trajectories corresponding to the diffusional collapse of an extended polypeptide. The predicted trajectories reproduce the different stages of collapse, as well as the magnitude and time course of the collapse trajectories observed experimentally in ubiquitin and I27 protein monomers. Our observations validate the force-clamp technique as a powerful approach to determining the free-energy landscape of proteins collapsing and folding from extended states.