Structure of the rotaxane. Oxygen atoms are depicted in red, the naphthalene units in black, and the polyether carbons in orange. The rings are depicted in blue with the pyridinium ions in brown.
Schematic of an isometric single-molecule force spectroscopy experiment using an AFM. In it, one end of the molecular system is attached to a surface and the other end to an AFM tip attached to a cantilever. During the pulling, the distance between the surface and the cantilever is controlled and varied at a constant speed . The deflection of the cantilever from its equilibrium position measures the instantaneous applied force on the molecule by the cantilever , where is the cantilever spring constant and is the fluctuating molecular end-to-end extension.
The approach to equilibrium in the pulling simulations. The figure shows force vs extension profiles for the rotaxane immersed in a high dielectric medium at 300 K for different pulling speeds of (a) , (b) , (c) , and (d) . The harmonic cantilever employed has a soft-spring constant of , is the distance between the surface and the cantilever, and is the instantaneous applied force. The black lines correspond to the extension process; the gray ones correspond to the contraction.
Time dependence of the force and the molecular end-to-end distance during the pulling of the rotaxane under equilibrium conditions (, ). Typical structures (labels 1–7) observed during the extension are shown in Fig. 5.
Snapshots of the rotaxane during its extension. The numerical labels shown here are employed in Figs. 4, 6, and 9.
Changes in the Helmholtz free energy of the rotaxane plus cantilever obtained from the data shown in Fig. 4. The blue line corresponds to the pulling and the red one corresponds to the contraction. The dotted lines provide an estimate of errors in the thermodynamic integration due to force fluctuations at each pulling step. The difference in the degree of convexity of regions I and III is evidenced by extrapolating the data in each region outside of its domain through fitting to quartic polynomials (black dashed lines). The labels correspond to the structures shown in Fig. 5.
Time dependence and probability density distribution of the radius of gyration and the end-to-end molecular extension when the rotaxane plus cantilever is constrained to reside in the unstable region (II) of Fig. 6 with . Typical structures encountered in this regime (labels 1a–3a) are shown in Fig. 8.
Structures observed in the unstable region of the pulling simulations. The numerical labels are employed in Fig. 7.
Potential and entropic contributions to the PMF [Eq. (18)] of the rotaxane along the end-to-end distance . The results are averages of three different pulling simulations. The error bars correspond to twice the standard deviation obtained from a bootstrapping analysis. The solid lines result from a spline interpolation of the available data points. Typical structures (labels 1–7) are shown in Fig. 5.
Probability density distribution of the radius of gyration for the rotaxane when is fixed at 7.0 Å (dots), 47 Å (open circles), and 64.9 Å (crosses). Note the bistability along when is fixed at the concave region of the PMF.
PMF as a function of the molecular extension extracted from force measurements using WHAM for rotaxanes with different numbers of threaded rings. The error bars correspond to twice the standard deviation obtained from a bootstrapping analysis.
Schematic variation in the Helmholtz potential of the molecule plus cantilever as a function of for different extensions suggested by the phenomenological observations around the region of mechanical instability. The labels I–III correspond to the different stability regions in Fig. 6.
Left panel: dependence of the isotherms on the cantilever spring constant during the extension of the rotaxane. Right panel: ratio between the thermal fluctuations and the average in the force measurements. The cantilever spring constants employed are all expressed in terms of the cantilever spring constant used in the pulling simulations .
Dependence of the instability of the isotherms on the cantilever spring constant for the rotaxane. The right panel shows the critical values of the force as a function of . Here and correspond to the values of the force when the curves exhibit a maximum and minimum, respectively. The left panel shows the extension lengths that enclose the unstable region in the isotherms, as well as the average end-to-end distance at the critical points .
Standard deviation in the force measurements as a function of for three different cantilever spring constants during the pulling of the rotaxane. In each case, the dotted line indicates the value of which sets the limit between the stable and unstable branches in the extension, see Eq. (40).
Effective potential for the molecule plus cantilever for different values of the extension . In the panels, the blue open circles correspond to , the open black circles to the PMF , and the solid red lines to the cantilever potential . The cantilever spring constant employed is the same as the one used in the pulling simulations presented in Sec. III A. Note the bistability in the effective potential for .
The upper panels show the probability density distribution of the force measurements (in ) during the extension of the rotaxane using cantilevers of varying stiffness. The force function is defined by Eq. (3). The lower panels show the associated spatial probability density distributions (in ) [Eq. (10)] for the rotaxane plus cantilever. The color code is given in the far right. The spring constants are expressed in terms of , the value employed in the pulling simulations presented in Sec. III A. Note how bistability along , and hence blinks in the force measurements, arises for soft cantilevers and decays for stiff ones in accordance with Eq. (42).
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