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Dynamic force spectroscopy (DFS) makes it possible to investigate specific interactions between two molecules such as ligand-receptor pairs at the single-molecule level. In the DFS method based on the Bell-Evans model, the unbinding force applied to a molecular bond is increased at a constant rate, and the force required to rupture the molecular bond is measured. By analyzing the relationship between the modal rupture force and the logarithm of the loading rate, microscopic potential barrier landscapes and the lifetimes of bonds can be obtained. However, the results obtained, for example, in the case of streptavidin/biotin complexes, have differed among previous studies and some results have been inconsistent with theoretical predictions. In this study, using an atomic force microscopy technique that enables the precise analysis of molecular interactions on the basis of DFS, we investigated the effect of the sampling rate on DFS analysis. The shape of rupture force histograms, for example, was significantly deformed at a sampling rate of 1 kHz in comparison with that of histograms obtained at 100 kHz, indicating the fundamental importance of ensuring suitable experimental conditions for further advances in the DFS method.

The various interactions between a pair of functional molecules, for example, DNA, ligand-receptor and antigen-antibody systems, play essential roles in biological processes and molecular devices based on molecular recognition properties [

Dynamic force spectroscopy (DFS) is a technique that enables the investigation of specific interactions between two molecules at the single-molecule level [_{b}, where _{b} is the potential barrier position. Therefore, the rupture probability depends on the magnitude of the applied force. Using the Bell-Evans model [_{b} and the bond lifetime are obtained. There are no other techniques that enable the analysis of chemical reaction processes at the single-molecule level, and DFS has been successfully used to analyze the molecular dynamics of systems such as ligand-receptor pairs and DNA molecules [

Several modifications of the original Bell-Evans model [

In this paper, we further analyze the experimental conditions; we investigated the effect of the sampling rate on DFS measurement using an avidin-biotin complex. In DFS analysis based on the Bell-Evans model [^{5} pN/s [

These points must be considered before discussing the details of molecular interactions. In fact, although such effects have not been examined in detail, we found that the sampling rate of 1 kHz has a critical effect on rupture-force measurement. For a low loading rate, in particular, the shape of rupture-force histograms obtained at a 1 kHz sampling rate was significantly deformed in comparison with that of histograms obtained at 100 kHz, indicating the fundamental importance of suitable experimental conditions when employing the DFS method.

First, we explain the methodology that we developed and the details of sample preparation. _{b}. Since the lifetime of a molecular bond depends on the potential barrier height, the rupture probability depends on the magnitude of the applied force [

where _{(}_{f}_{)}, ^{*}, _{b}, _{B},

According to ^{*} linearly depends on the logarithm of the loading rate _{0} (d

where _{off(0)} is the lifetime of the molecular bond.

To enable the use of

As shown in ^{*} [

Force curves were measured at a high sampling rate (100 kHz) for various constant loading rates, and then these force curves were numerically converted to those at lower sampling rates of 10 kHz and 1 kHz. Namely, every 10 and every 100 data were picked up from the full series of the original data obtained for the 100 kHz measurement and treated on the basis of the sampling theorem to provide the data for 10 kHz and 1 kHz, respectively. The method of resampling that we carried out is based on the sampling theorem (Nyquist’s theorem). The data obtained at 100 kHz is sufficient to produce sampling rates of 10 kHz and 1 kHz. To reduce the sampling rate, for example, from 100 kHz to 10 kHz, the data obtained at 100 kHz was changed after the application of a low-pass filter with a frequency of 5 kHz. The resampling process is considered sufficient to reduce the amount of data to that actually observed at a small sampling rate.

As described in the introduction, in this study, we focused on the sampling rate used in force measurement. This is because, as shown in

A gold-coated cantilever was immersed in a solution of 8-amino, 1-octanethiol hydrochloride molecules (1 mM in ethanol) for 48 h to form a closely packed SAM with amino groups on the surface. After rinsing with ethanol, the cantilever was immersed in a solution of biotin-PEG3400-COO-NHS molecules (Shearwater Polymers, 0.1 mM in ethanol) for 20 h to fix a biotin (biotin-PEG) molecule onto the probe apex. Finally, the cantilever was rinsed with ethanol. To investigate the effect of the sampling rate, Bio-Levers (Bio-Lever, Olympus, 0.006 N/m and 0.03 N/m, rectangular shape) and microcantilevers (OMCL-RC800PB, Olympus, 0.06 N/m, rectangular shape) were prepared. These cantilevers were chosen because their stiffness is typical of cantilevers widely used in DFS experiments using an AFM. The soft cantilever of 0.006 N/m stiffness is suitable for obtaining a weak force in a small-loading rate measurement, while the cantilevers of 0.06 N/m and 0.03 N/m stiffness are suitable for measuring a stronger force a large-loading rate measurement.

To avoid multiple-bonding events, the density of the target molecules in the SAM was reduced and free biotin molecules were introduced so that the probability of bonding became 5–10% for each tip-sample approach. Only one peak becomes dominant after this treatment [

To clarify and remove the effect of noise on measurement when the signal level is low, the sampling rate should be 5–10 times larger than the cutoff frequencies indicated by arrows in

^{3} pN/s (4(a)-I to 4(a)-III) and 1.9 × 10^{2} pN/s (4(b)-I to 4(b)-III), where the force curves for 10 kHz and 1 kHz were numerically obtained, as described above, from the data obtained at a sampling rate of 100 kHz. The method of obtaining the rupture force is similar to that by Kasas

For the loading rate of 1.6 × 10^{3} pN/s (^{2} pN/s, the smaller number of measurement points becomes a critical issue. Although the rupture force is clearly determined for sampling rates of 10 kHz and 100 kHz, it cannot be accurately measured for the sampling rate of 1 kHz owing to the reduced number of measurement points (10 measurement points in this case).

For the loading rate of 1.9 × 10^{2} pN/s, since the magnitude of the rupture force becomes close to that of the thermal noise, we should carefully choose an appropriate sampling rate. A rupture force of about 20 pN was observed for sampling rates of 100 kHz and 10 kH. However, if the sampling rate is insufficient, the differential coefficient used to obtain the position of the rupture in the force curve cannot be determined with confidence under thermal noise. In fact, as shown in

How high should the sampling rate be necessary for measurement? In DFS, as explained concerning the procedures shown in ^{5} pN/s. A higher sampling rate is necessary for measurement at a larger loading rate. For the examples shown in ^{3} pN/s. To obtain an accurate rupture force at a loading rate of 10^{5} pN/s, it is necessary to observe the force curve obtained at a significantly higher sampling rate. For example, when the cantilever with a spring constant of 0.06 N/m is used at a sampling rate of 20 kHz, about 200 measurement points are obtained even at a loading rate of 10^{5} pN/s, enabling the accurate estimation of the rupture force.

How does the measurement inaccuracy affect the histogram? As has been discussed, the rupture force cannot be determined when the sampling rate used in the force curve measurement is not sufficiently high; this affects the analysis of the histogram.

This inaccuracy affects the final result,

There is another method by which the potential barrier position can be estimated by analyzing the shape of the rupture force distribution [

_{b} in

This result indicates that although the method used here has very high efficiency, when the potential position is estimated from the peak width, it is extremely important to measure the force curves at a sufficiently high sampling rate.

Using an atomic force microscopy technique that enables the precise analysis of molecular interactions on the basis of DFS, we investigated the effect of the sampling rate on DFS analysis. The sampling rate of 1 kHz is not sufficiently high at both large and small loading rates, resulting in a difference in the slope of the relationship between the modal rupture force and the logarithm of the loading rate, which is used to obtain potential barrier positions and bond lifetimes. The potential barrier position can be estimated from a single histogram when rupture forces are accurately measured. However, the shape of rupture force histograms was found to be significantly deformed at a sampling rate of 1 kHz in comparison with that of histograms obtained at 100 kHz, indicating the fundamental importance of suitable experimental conditions for further advances in the DFS method. The consideration of suitable experimental conditions, which have not been well investigated, is essential for increasing the usefulness and practicality of DFS analysis.

Schematic diagrams to explain the analysis method of Dynamic force spectroscopy (DFS). (

Schematic illustrations of the method of sample preparation and the effect of sampling rate on force curve measurement.

Power spectra of the three cantilevers with spring constants of 0.006 N/m, 0.03 N/m and 0.06 N/m, whose cutoff frequencies were estimated to be 2 kHz, 10 kHz and 5 kHz, respectively.

Force curves obtained at a sampling rate of 100 kHz for loading rates of 1.6 × 10^{3} pN/s ((a)-I to (a)-III) and 1.9 × 10^{2} pN/s ((b)-I to (b)-III), where force curves for 10 kHz and 1 kHz were numerically converted from data obtained at a sampling rate of 100 kHz.

Histograms of the rupture force obtained for sampling rates of 100 kHz, 10 kHz and 1 kHz based on a series of data shown in

Relationship between the modal rupture forces and the logarithm of the loading rate at sampling rates of 100 kHz, 10 kHz and 1 kHz, which correspond to the values derived from the histograms shown in

Potential barrier position estimated from the distribution width,

Loading rate | _{b} (100 kHz) |
_{b} (10 kHz) |
_{b} (1 kHz) |
---|---|---|---|

1.6 × 10^{3} pN/s |
0.29 nm | 0.31 nm | 0.21 nm |

6.8 × 10^{2} pN/s |
0.30 nm | 0.42 nm | 0.62 nm |

4.4 × 10^{2} pN/s |
0.57 nm | 0.51 nm | 0.54 nm |

1.9 × 10^{2} pN/s |
0.47 nm | 0.43 nm | 0.75 nm |