By exerting mechanical force it is possible to unfold/refold RNA molecules one at a time. In a small range of forces, an RNA molecule can hop between the folded and the unfolded state with force-dependent kinetic rates. Here, we introduce a mesoscopic model to analyze the hopping kinetics of RNA hairpins in an optical tweezers setup. The model includes different elements of the experimental setup (beads, handles and RNA sequence) and limitations of the instrument (time lag of the force-feedback mechanism and finite bandwidth of data acquisition). We investigated the influence of the instrument on the measured hopping rates. Results from the model are in good agreement with the experiments reported in the companion article (1). The comparison between theory and experiments allowed us to infer the values of the intrinsic molecular rates of the RNA hairpin alone and to search for the optimal experimental conditions to do the measurements. We conclude that long handles and soft laser traps represent the best conditions to extract rate estimates that are closest to the intrinsic molecular rates. The methodology and rationale presented here can be applied to other experimental setups and other molecules.
Deep Dive into Force unfolding kinetics of RNA using optical tweezers. II. Modeling experiments.
By exerting mechanical force it is possible to unfold/refold RNA molecules one at a time. In a small range of forces, an RNA molecule can hop between the folded and the unfolded state with force-dependent kinetic rates. Here, we introduce a mesoscopic model to analyze the hopping kinetics of RNA hairpins in an optical tweezers setup. The model includes different elements of the experimental setup (beads, handles and RNA sequence) and limitations of the instrument (time lag of the force-feedback mechanism and finite bandwidth of data acquisition). We investigated the influence of the instrument on the measured hopping rates. Results from the model are in good agreement with the experiments reported in the companion article (1). The comparison between theory and experiments allowed us to infer the values of the intrinsic molecular rates of the RNA hairpin alone and to search for the optimal experimental conditions to do the measurements. We conclude that long handles and soft laser traps
Recently develop ed single-molecule techniques (2) have been used to exert force on individual mo lecu les, such as nu cleic acids (3,4,5,6,7) and p roteins (8,9,10). These techniques make it possible to test the mechanical response of biomolecules which can be used to obtain information about their structure and stability. M oreover, the study of the kinetics, p athway s, and mechan isms of biochemical reactions is p articularly suited to single-molecule methods where individual molecular trajectories can be followed (11,12,13).
Optical tweezers have been used to study folding/unfolding (F-U) of RNA hairpins (14,15,16,17). The experimental setup consists of the RNA molecule flanked by doublestranded DNA/RNA handles; the entire molecule is tethered between two polystyrene beads via affinity interactions. The handles are poly mer sp acers required to screen interactions between the RNA molecule and the beads and to p revent direct contact of the beads. One of the beads is held in the op tical trap; the other bead is controlled by a piezoelectric actuator to app ly mechanical force to the ends of the RNA molecule. In hopping experiments a given constraint, i.e. a fixed force or a fixed extension, is applied to the exp erimental system while both the force and the extension of the molecule are monitored as a function of time. Close to the transition force (around 10-20 pN for RNA or DNA hairpins at room temperature (13,14)), a hairpin molecule can transit between the folded (F) and the unfolded (U) states, as indicated by the change in the molecu lar extension: the longer extension represents the unfolded single-stranded conformation; the shorter one to the folded hairpin. From the lifetimes of the single RNA molecule in each of the two states, we can obtain the rates of the F-U reaction (1,14). Both the unfolding and folding rate constants are force-dependent following the Kramers-Bell theory (17,18,19). From their ratio the force dep endent equilibriu m constant for the F-U reaction can be obtained.
In order to obtain accurate information about the molecule under study it is important to understand the influence of the experimental setup, including the handles and the trapped bead, on the measurements. In a recent simulation, Hy eon and Thirumalai (20) examined the relationship between the amplitude of the F-U transition signal and the magnitude of its fluctuations at various handle lengths. On the other hand, experimental results have shown that the F-U kinetics was dependent on the trap stiffness (21). Several questions then arise: how different is the measured rate from the intrinsic molecular rate, i.e., the F-U rate of the RNA in the absence of handles and beads? What are the optimal working conditions to obtain the intrinsic molecular rates? To address such questions, we previously p roposed a model (22), which considered the effect of the trapp ed bead and the handles on a two-state RNA folding mechanism. In this work, we further advance our simulation by incorporating a mesoscopic model introduced by Cocco et al. (23) that takes into account the sequence-dep endent foldin g en ergy. We have then ap p lied this model to a simple hairpin, P5ab (14). We investigate how the measured rates vary with the characteristics of the exp erimental setup and how much they differ from the intrinsic molecu lar rates of the individual RNA molecule. In a comp anion p ap er, we have also measured the F-U kinetics of the RNA hairpin by op tical tweezers (1). The theoretical and experimental results agree well.
The organization of the paper is as follows. In Sec. 2 we introduce the model for the experimental setup and describe its thermodynamic properties. We also analyze the characteristic timescales of the sy stem. In Sec. 3 we d iscuss the influen ce of the different elements of the experimental setup on the kinetic rates. Limitations of the instrument which affect the measured F-U rates, such as the force-feedback time lag and the data acquisition bandwidth, are also consid ered. B ased on the various timescales of the different dynamical processes described in Sec. 2, we develop a kinetic model for the RNA hairpin and a numerical algorithm used to simulate the hopping dynamics in Sec. 4. In Sec. 5 we carry out a detailed analysis of the dependence of the kinetic rates on the characteristics of the exp eriment (such as the len gth of the handles and the stiffness of the trap ), and compare our simulation results with the exp erimentally measured F-U rates. A search of the best fit between theory and exp eriments allows us to predict the value of the intrinsic molecular F-U rate of the RNA molecule. Finally , we discuss what are the optimal experimental conditions to minimize the effect of the instrument and to obtain the intrinsic molecu lar rates.
Hopping experiments (1) were done with a single RNA hairpin P5ab, a derivative of the L-21 Tetrahymena ribozyme. The kinetics of this RNA with 1.1 Kbp handles had been
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