$$\rightleftharpoonup{xx}$$
$$\longleftharp{xx}$$,
$$\longrightharp{xx}$$,
The decrease in the resonant frequency (Δf) correlates in a linear manner with the adsorbed mass (Δm), as defined by the Sauerbrey equation.17

Here, f is the resonant frequency, Cf is a constant that depends on the geometrical and physical characteristics of the given quartz and the resonant frequency, and A is the sensor surface area.
In most applications, the adsorbed layer is not completely rigid but viscoelastic. The resulting dampening of the quartz sensor oscillation is referred to as dissipation (D). The monitored dissipation changes (ΔD) correlate with the viscoelastic properties of the bound mass18 and are defined as follows8.

Here, Edissipated is the energy lost during one oscillation period, and Estored is the total energy of the freely oscillating sensor.
To analyze and quantify the binding parameters, frequency isotherms are derived by plotting the equilibrium frequency shifts (ΔΔfe) against the protein concentrations. ΔΔfe is defined as
![figure-results-3 Static equilibrium equation, ΔΔf_e = [(Δf_t2 - Δf_tt)], mathematical formula, physics concept.](/files/ftp_upload/58224/58224eq3.jpg)
Here, Δft1 represents the beginning of the protein adsorption and Δft2 the equilibrium state. Nonlinear curve fitting can be performed by using a Hill expansion of the Langmuir equation as follows6,8.

Here, ΔΔfmax is the ΔΔfe of the protein concentration resulting in maximum (saturating) binding, Kd is the apparent dissociation constant for the protein/membrane complex, and n is the Hill coefficient.
The Hill coefficient (n) describes the cooperativity of binding. For n = 1, the Hill adsorption model is a simple Langmuir isotherm (the equal binding sites and all molecules bind independently of each other to the lipid bilayer). If n ≠ 1, a bound ligand changes the membrane binding affinity for other ligands, either increasing (n > 1, positive cooperativity) or decreasing (n < 1, negative cooperativity) the affinity.
Figure 1 shows a schematic of the experimental workflow used in our laboratory to measure the shifts in resonance and frequency during Ca2+-dependent binding and the release of AnxA2 to the lipid bilayer in the liquid phase. An exemplary recording is shown in Figure 2. Figure 2A shows the recording of the frequency curve and Figure 2B shows the dissipation shifts. The prominent drop in frequency upon the addition of the liposomes (Figure 2A [step 1]) indicates their adsorption. Because the buffer-filled vesicles are not rigid, but viscoelastic, the dissipation increases (Figure 2B [step 1]). Subsequently, the coalescing vesicles rupture. The concomitant release of the buffer inside the vesicles decreases the adsorbed mass until a stable plateau is reached (Figure 2A [step 2]). Of note, the addition of vesicles results in a high dissipation shift, while the shift in response to the bilayer is much smaller due to the rigid homogenous nature of the SLB (Figure 2B [step 2]). Step 3 in Figure 2A and 2B records the binding of AnxA2 to the lipids, which adds mass, as seen by the clear frequency shift, but does not interfere with the bilayer structure, as indicated by the only small change in dissipation. When Ca2+ is removed by the chelating agent EGTA (Figure 1 and Figure 2 [step 4]), AnxA2 dissociates from the lipid film. The frequency, as well as the dissipation recordings, shift to the levels seen with the bilayer only (compare steps 2 and 4 in Figure 2A and 2B), indicating that AnxA2 binding is totally dependent on Ca2+ and that the lipid film remains intact.
AnxA2, as do most of the annexins, depends on negatively charged lipids such as PS. This is clearly seen when POPS is absent in the lipid bilayer (Figure 3). Figure 3A shows the recording of the frequency curve and Figure 3B shows the dissipation shifts. Note that the frequency shifts to a stable baseline at -25 Hz, yet the dissipation is not altered (Figure 3B [step 2]), indicative of a proper bilayer formation. However, no changes in frequency (Figure 3A) or dissipation (Figure 3B) are observed after the addition of AnxA2 in the presence of Ca2+ (Figure 3A and 3B [step 3]) or EGTA (Figure 3A and 3B [step 4]), as AnxA2 cannot interact with the lipid film.

Figure 1: Graphical model of the experimental workflow. This workflow illustrates the vesicle absorption to the hydrophilic sensor surface (step 1), the vesicle fusion/rupture leading to the SLB formation (step 2), and the Ca2+-dependent adsorption (step 3) and EGTA-dependent desorption of AnxA2 (step 4). Please click here to view a larger version of this figure.

Figure 2: Exemplary recording. These panels show (A) the time-dependent monitoring of the 7th overtone resonance frequency and (B) the dissipation shifts of the quartz sensors during measurement. The application of the liposomes causes a rapid drop in the frequency baseline, whereas the dissipation baseline increases (step 1). The stabilization of the baselines indicates the formation of the bilayer (step 2). The AnxA2 (200 nM) adsorption (in the presence of Ca2+) onto the POPS-containing lipid bilayer adds mass without significantly changing the dissipation, indicating that the lipid film is not perturbed (step 3). The recovery of the frequency baseline upon Ca2+ chelation with EGTA indicates the total desorption of the protein (step 4). Please click here to view a larger version of this figure.

Figure 3: Negative control experiment, demonstrating that AnxA2 does not bind to SLBs in the absence of POPS. These panels show the addition of liposomes and the SLB formation (steps 1 and 2). No changes in (A) frequency or (B) dissipation are apparent after the addition of AnxA2 (step 3; 200 nM, in the presence of Ca2+) or EGTA (step 4). Please click here to view a larger version of this figure.
| Composition | ΔΔF/Hz after formation of SLBs | ΔΔD*10-6 after formation of SLB |
POPC/POPS
(80 : 20) | 26.3 ± 0.2 | 0.26 ± 0.03 |
POPC/PI(4,5)P2
(95 : 5) | 26.5 ± 0.5 | 0.31 ± 0.02 |
POPC/POPS/Chol
(60 : 20 : 20) | 29.2 ± 0.2 | 0.45 ± 0.09 |
POPC/POPS/ PI(4,5)P2/Chol
(60 : 17 : 3 : 20) | 29.6 ± 0.6 | 0.43 ± 0.10 |
POPC/DOPC/POPS/ PI(4,5)P2/Chol
(37 : 20 : 20 : 3 : 20) | 29.4 ± 0.4 | 0.39 ± 0.14 |
Table 1: Lipid composition and formation data of the SLB7.