Force Spectroscopy: Cantilever Calibration

To translate the deflection of the cantilever, , to units of force, , it is necessary to determine the spring constant of the cantilever, (i.e., ). There are several techniques for calibrating tips and theoretical methods that provide an approximation of (Sader, 1995). Determination of using empirical methods involves taking measurements of cantilever deflection with application of a constant known force (Senden and Ducker, 1994) or measurements of the cantilever's resonant frequency (Hutter and Bechhoefer, 1993). The method we use for calibrating cantilevers is based on Hutter and Bechhoefer (1993) and is briefly outlined below.

For ligand-receptor force measurements we use triangle-shaped, unsharpened, gold-coated silicon-nitride cantilever tips that have spring constants ranging from 10 mN/m to 50 mN/m. The cantilever tip can be treated as a simple harmonic oscillator whose power spectrum of thermal fluctuation can be used to derive the spring constant. In brief, the cantilever is raised several microns from the surface and its natural frequency of vibration (resonant frequency) is monitored for 2-3 seconds. Since each vibration mode of the cantilever receives the thermal energy commensurate with one degree of freedom, k BT/2, the measured variance of the deflection áx 2 ñ can be used to calculate the spring constant, i.e., 1/2 k BT = 1/2 k c áx 2 ñ, where k B and T are Boltzmann’s constant and temperature, respectively. To separate deflections belonging to the basic (and predominant) mode of vibration from other deflections or noise in the recording system, the power spectral density of the temperature-induced deflection is determined, and only the spectral component corresponding to the basal mode of vibration is used to estimate the spring constant. Using this approach, the spring constants of cantilevers can be calibrated in either air or solution. The calculated spring constant k C can then be used to calculate rupture force, F, by F = k cC DV , where DV is the change in voltage detected by the photodiode just prior to and immediately after the rupture event and C is a calibration constant that relates deflection and photodiode voltage. C is determined from the deflection of the cantilever pressed against a rigid surface, such as the bottom of a plastic petri dish.

3-Cantilever Functionalization