^{1,a)}, George C. Schatz

^{1}and Mark A. Ratner

^{1}

### Abstract

The thermodynamic driving force in the folding of a class of oligorotaxanes is elucidated by means of molecular dynamics simulations of equilibrium isometric single-molecule force spectroscopy by atomic force microscopy experiments. The oligorotaxanes consist of cyclobis(paraquat--phenylene) rings threaded onto an oligomer of 1,5-dioxynaphthalenes linked by polyethers. The simulations are performed in a high dielectric medium using MM3 as the force field. The resulting force versus extension isotherms show a mechanically unstable region in which the molecule unfolds and, for selected extensions, blinks in the force measurements between a high-force and a low-force regime. From the force versus extension data the molecular potential of mean force is reconstructed using the weighted histogram analysis method and decomposed into energetic and entropic contributions. The simulations indicate that the folding of the oligorotaxanes is energetically favored but entropically penalized, with the energetic contributions overcoming the entropy penalty and effectively driving the folding. In addition, an analogy between the single-molecule folding/unfolding events driven by the atomic force microscope (AFM) tip and the thermodynamic theory of first-order phase transitions is discussed. General conditions (on the molecule and the AFM cantilever) for the emergence of mechanical instabilities and blinks in the force measurements in equilibrium isometric pulling experiments are also presented. In particular, it is shown that the mechanical stability properties observed during the extension are intimately related to the fluctuations in the force measurements.

The authors thank Professor J. Fraser Stoddart, Dr. Wally F. Paxton, and Dr. Subhadeep Basu for insightful remarks and the AFOSR-MURI program of the DOD, NSF (Grant No. CHE-083832), and the Network for Computational Nanoscience for support of this work. I.F. thanks Dr. Gemma C. Solomon and Dr. Neil Snider for discussions on an earlier version of this manuscript.

I. INTRODUCTION

II. THEORETICAL METHODOLOGY

A. Pulling simulations

B. Reconstructing the PMF using WHAM

C. Energy-entropy decomposition of the PMF

III. RESULTS AND DISCUSSION

A. Phenomenology of the pulling experiments

1. The approach to equilibrium

2. Reversible unfolding, mechanical instability, and blinks in the force measurements

3. The PMF and the thermodynamic driving force in the folding

4. Dependence of the PMF on the number of threaded rings

B. Interpretation of the pulling experiments

1. Analogy with first-order phase transitions

2. Dependence of the isotherms on the cantilever spring constant

3. Disappearance of the mechanical instability in the soft-spring limit

4. Emergence of the mechanical instability in the stiff-spring limit

5. Relation between force fluctuations and mechanical stability

6. Origin of the blinking in the force measurements

7. Final remarks

IV. CONCLUSIONS

### Key Topics

- Force measurement
- 31.0
- Atomic force microscopy
- 12.0
- Free energy
- 10.0
- Helmholtz free energy
- 10.0
- Entropy
- 9.0

## Figures

Structure of the [3]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.

Structure of the [3]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.

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 [3]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.

The approach to equilibrium in the pulling simulations. The figure shows force vs extension profiles for the [3]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 [3]rotaxane under equilibrium conditions (, ). Typical structures (labels 1–7) observed during the extension are shown in Fig. 5.

Time dependence of the force and the molecular end-to-end distance during the pulling of the [3]rotaxane under equilibrium conditions (, ). Typical structures (labels 1–7) observed during the extension are shown in Fig. 5.

Snapshots of the [3]rotaxane during its extension. The numerical labels shown here are employed in Figs. 4, 6, and 9.

Snapshots of the [3]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 [3]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.

Changes in the Helmholtz free energy of the [3]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 [3]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.

Time dependence and probability density distribution of the radius of gyration and the end-to-end molecular extension when the [3]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.

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 [3]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.

Potential and entropic contributions to the PMF [Eq. (18)] of the [3]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 [3]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.

Probability density distribution of the radius of gyration for the [3]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.

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.

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 [3]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 .

Left panel: dependence of the isotherms on the cantilever spring constant during the extension of the [3]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 [3]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 .

Dependence of the instability of the isotherms on the cantilever spring constant for the [3]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 [3]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).

Standard deviation in the force measurements as a function of for three different cantilever spring constants during the pulling of the [3]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 .

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 [3]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 [3]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).

The upper panels show the probability density distribution of the force measurements (in ) during the extension of the [3]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 [3]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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