Parameters Derived from a Single Force Curve
To quantitatively characterize the specific interaction between a pair of molecules, a set of distinct parameters is derived from the force curves. An additional parameter, determined on the basis of the total number of performed measurements, is an unbinding probability that is related to the number of receptors present on the substrate surface or in the cell membrane.
Figure 5.18 Example of the force curve recorded for the interaction of an antigen-antibody pair, showing a typical jump related to single molecule interaction. Reprinted with permission from .
The pull-off force and force histogram
The unbinding force needed to separate two interacting molecules is delivered from the analysis of the retraction part of the force curve. It should be pointed out here that the AFM can measure only a so-called pull-off force (Fig. 5.19).
The pull-off force (called also rupture or detachment force) is defined as a difference between the force Fb, corresponding to the free cantilever position (when the interacting force is negligible) and the maximum value of the force Fa:
The pull-off force value measured for two surfaces, bearing the complementary proteins that interact in a specific way, is usually a superposition of two components: (i) discrete, short- range one, dominating within the binding sites that is related to the strength of a single molecular complex (referred here as specific forces), and (ii) that originating from long-range, distance-dependent forces dominating outside of the binding site (referred here as non-specific forces). The non-specific forces vary in response to the properties of the environment surrounding both molecules of interest. Very often, it is difficult to separate between the specific and non-specific interactions since the strength of the latter one can be comparable. For example, the determined non-specific forces  ranged from 60 pN to about 400 pN (!), which was significant in comparison with the measured specific interaction forces (240 ± 160 pN for ConA- ASA, 370 ± 110 pN for ConA-CaY and 180 ± 130 pN for PAP- aPAP pairs). For analogous protein embedded in the plasma membrane, the non-specific force was 60 ± 30 pN for ConA-PC3 cells. Such a low value can be explained by the overall interactions present on the cell surface that are not involved in the molecular recognition phenomenon.
Figure 5.19 Force histograms showing only one peak attributed to the specific interaction between mannose-type glycans present on the surface of prostate (PC-3) cells probed with AFM tip functionalized with lectin ConA. The solid line is a Gaussian fit used for determination of the unbinding force. Reprinted with permission from .
The force histogram is created using the bin size reflecting the minimum value of detected forces (see Chapter 3 for more details). Shape of the histogram is characteristic for the studied pair of molecules and depends on the number of molecules present within the contact area. To quantify the unbinding force, many force curves must be collected and analyzed using histograms that can reveal either only one peak (Fig. 5.19) or multiple peaks (Fig. 5.20).
The single force maximum can be directly attributed to a rupture of a single molecular complex unbinding. The multiple peaks can correspond to the case of simultaneous rupture of more complexes (Fig. 5.20).
Figure 5.20 Force histograms showing multiple peaks observed when surface of melanoma cells (WM35) is probed with ConA- modified tip. The solid line is a Gaussian fit used for determination of the unbinding force. Reprinted with permission from .
The most probable unbinding force is usually determined by fitting the Gaussian functions to the maxima present in the histogram of the measured rupture force. Unbinding force is calculated as a center of the fitted Gauss distribution and the corresponding error is a standard deviation determined from the half width of the peak at its half maximum height. Such fitting procedure gives the position of the maximum with reasonable reliability but it does not describe fully the shape of force histogram which is determined by the stochastic nature of the unbinding process. It can be analytically described by the probability density function:
where a, b, and c are parameters fitted to the unbinding force histogram obtained at the given loading rate rf. The probability density function is determined by the two parameters describing rupture of a single molecular complex: the energy barrier xb and the dissociation rate constant k0. The fit delivers the estimates of those parameters. They are calculated according to
The dissociation rate constant and the energy barrier position determined in such way were in a reasonable agreement with values were obtained using two independent experiments [46, 48], namely, using the QCM and the AFM working in a dynamic force spectroscopy mode (DSF). The dissociation rate constant of 0.137 ± 0.029 s-1 obtained from the fit is placed between values 0.095 ± 0.002 s-1 (from QCM measurements) and 0.170 ± 0.060 s-1 (from AFM measurements). The fitted position of the energy barrier was in good agreement with the reported value (0.23 ± 0.01 nm versus 0.229 ± 0.004 nm).