^{1}and Michael Urbakh

^{1,a)}

### Abstract

We propose an analytical approach to describe the active rebinding and force hysteresis observed in single-molecule pulling experiments. We derive equations for dependences of the measured quantities on the properties of molecular potential, effective stiffness of the pulling spring, and the pulling velocity. The calculations predict that the energy dissipated per an unbinding-rebinding cycle strongly increases with the steepness of the molecular potential and with decreasing the spring stiffness. A comparison of analytical results with Langevin simulations shows that the scaling relations for the barrier heights and most probable forces are more accurate in the case of active rebinding than for unbinding. Our consideration demonstrates that simultaneous analysis of probability density functions for unbinding and rebinding forces improves essentially the accuracy of retrieval information on intrinsic parameters of the molecular complex from the force measurements.

I. THE MODEL

II. RESULTS AND DISCUSSION

III. CONCLUSIONS

### Key Topics

- Adhesion
- 16.0
- Langevin equation
- 8.0
- Free energy
- 5.0
- General molecular properties
- 3.0
- Probability density functions
- 3.0

## Figures

(a) A schematic presentation of the rebinding and unbinding measurements. [(b)–(e)] Hysteresis in the force-extension curves calculated with Eq. (1) for the Morse [(b) and (c)] and Lennard-Jones [(d) and (e)] potentials and for two values of the spring constant [(b) and (d)] and [(c) and (e)]. Black and gray (blue and red online) curves show force traces for unbinding and rebinding processes, respectively. Parameter values: , , , , , , and .

(a) A schematic presentation of the rebinding and unbinding measurements. [(b)–(e)] Hysteresis in the force-extension curves calculated with Eq. (1) for the Morse [(b) and (c)] and Lennard-Jones [(d) and (e)] potentials and for two values of the spring constant [(b) and (d)] and [(c) and (e)]. Black and gray (blue and red online) curves show force traces for unbinding and rebinding processes, respectively. Parameter values: , , , , , , and .

The total potential experienced by the pulled molecule for three different positions of the cantilever, (a), (b), and (c). Panels (a) and (c) give the potential at the critical positions and , for which the barriers for rebinding and unbinding vanish, respectively. Panel (d) presents the positions of the local minima and maximum of the potential as functions of . Variations in the bound and free states and the maximum are shown by dotted, dashed, and solid curves, respectively. For clarity we use the logarithmic scale to present variation in and with . Parameter values: , , , and .

The total potential experienced by the pulled molecule for three different positions of the cantilever, (a), (b), and (c). Panels (a) and (c) give the potential at the critical positions and , for which the barriers for rebinding and unbinding vanish, respectively. Panel (d) presents the positions of the local minima and maximum of the potential as functions of . Variations in the bound and free states and the maximum are shown by dotted, dashed, and solid curves, respectively. For clarity we use the logarithmic scale to present variation in and with . Parameter values: , , , and .

(a) Barrier heights for the unbinding and rebinding as a function of the applied force, and (b) PDFs for unbinding and rebinding forces. Circles and squares and the corresponding curves show the results for the unbinding and rebinding processes, respectively. (a) Solid curve presents results of numerical calculations of ; dashed and dotted curves show predictions of analytical scaling Eqs. (5) with given by Eq. (6), and the scaling relation for the unbinding barrier with the fitting parameter , respectively. (b) Symbols and curves show results of numerical calculations of and and the corresponding analytical results [Eq. (7)]. For unbinding we used . The inset demonstrates considerable difference between analytical PDFs for unbinding obtained with (solid line) and without (dashed line) introduction of the fitting parameter . , other parameters as in Figs. 1(a) and 1(b).

(a) Barrier heights for the unbinding and rebinding as a function of the applied force, and (b) PDFs for unbinding and rebinding forces. Circles and squares and the corresponding curves show the results for the unbinding and rebinding processes, respectively. (a) Solid curve presents results of numerical calculations of ; dashed and dotted curves show predictions of analytical scaling Eqs. (5) with given by Eq. (6), and the scaling relation for the unbinding barrier with the fitting parameter , respectively. (b) Symbols and curves show results of numerical calculations of and and the corresponding analytical results [Eq. (7)]. For unbinding we used . The inset demonstrates considerable difference between analytical PDFs for unbinding obtained with (solid line) and without (dashed line) introduction of the fitting parameter . , other parameters as in Figs. 1(a) and 1(b).

Pre-exponential factors entering the rate equation (4) for unbinding (a) and rebinding (b) as functions of the applied force. Solid and dashed curves present results of numerical calculations and analytical scaling relations (5). Parameters as in Fig. 2.

Pre-exponential factors entering the rate equation (4) for unbinding (a) and rebinding (b) as functions of the applied force. Solid and dashed curves present results of numerical calculations and analytical scaling relations (5). Parameters as in Fig. 2.

The most probable unbinding and rebinding forces vs the pulling velocity. Analytical and numerical results for unbinding are presented by black curves and symbols (blue online), while those for rebinding are shown by gray curves and symbols (red online). Calculations have been done for the Morse potential and two values of the spring constants: (solid lines and circles and squares) and (dashed lines and triangles). In analytical calculations for and 1 N/m we used and , respectively. Other parameters as in Fig. 1.

The most probable unbinding and rebinding forces vs the pulling velocity. Analytical and numerical results for unbinding are presented by black curves and symbols (blue online), while those for rebinding are shown by gray curves and symbols (red online). Calculations have been done for the Morse potential and two values of the spring constants: (solid lines and circles and squares) and (dashed lines and triangles). In analytical calculations for and 1 N/m we used and , respectively. Other parameters as in Fig. 1.

[(a) and (b)] PDFs of the dissipated energy calculated for the Morse potential and two values of the spring constant, (a) and (b). Circles, squares, and triangles show PDFs for , 50, and 300 nm/s, respectively. The curves are simple connectors between the data. (c) The mean dissipated energy vs pulling velocity calculated for (circles and solid curve) and (squares and dashed curve). Symbols and curves present results of Langevin simulations and analytical calculations with Eqs. (6) and (11). Parameters as in Fig. 3, and for the description of unbinding we used the fitting parameters and for and 0.2 N/m, respectively.

[(a) and (b)] PDFs of the dissipated energy calculated for the Morse potential and two values of the spring constant, (a) and (b). Circles, squares, and triangles show PDFs for , 50, and 300 nm/s, respectively. The curves are simple connectors between the data. (c) The mean dissipated energy vs pulling velocity calculated for (circles and solid curve) and (squares and dashed curve). Symbols and curves present results of Langevin simulations and analytical calculations with Eqs. (6) and (11). Parameters as in Fig. 3, and for the description of unbinding we used the fitting parameters and for and 0.2 N/m, respectively.

Article metrics loading...

Full text loading...

Commenting has been disabled for this content