Benchmarks of Biomembrane Force Probe Spring Constant Models

Main Text

Thanks to its soft spring constant, compatibility with live cells, and user-friendly maneuverability, the biomembrane force probe (BFP) demonstrates its unique strength as a picoforce technique for characterizing single-molecule dynamics on live cells (1, 2). The recent upgrades with simultaneously fluorescence imaging (3, 4, 5) and dual BFP setup (6) have further increased the recognition of the BFP by biophysics and mechanobiology fields as a powerful nanotool to visualize molecular mechanosensing events on live cell surface (7). The BFP uses a micropipette-aspirated red blood cell (RBC) with a glass bead attached to its apex as a force transducer. The spring constant determination of this force transducer is fundamental to the BFP technique as this is the basis for the force measurement. Approximating the BFP as a Hookean spring, three analytical expressions have been published in Biophysical Journal for calculation of the BFP spring constant by Evans et al. in 1995 (8), Simson et al. in 1998 (9), and Heinrich et al. in 2007 (10). While all three expressions were derived from RBC mechanics, different simplifying assumptions were made to set up the mathematical problems and different approximations were used to solve the mathematical problems, which introduce different amount of errors. In this brief communication, we benchmarked the three published analytical expressions of the BFP spring constant. We found that their accuracies depend on BFP configurations (Fig. 1).

Figure 1.

Illustration of three BFP setup configurations. (A) The RBC is aspirated with a short tongue (R_p ≈ L_p) is shown. (B) The RBC is aspirated with a long tongue (R_p ≪ L_p) is shown. (C) The RBC is aspirated with a large micropipette (R_p ≫ 1 μm) is shown. (D) The bead is misplaced at the RBC apex, off the central axis. R_p, R₀, and R_c are the respective radii of the micropipette lumen, the spherical portion of the aspirated RBC, and the circular adhesion area between the RBC and the probe bead. L_p is the length of the RBC tongue aspirated inside the micropipette. Of note, assumptions of the all discussed theoretical analyses of the RBC deformation are valid for (A) and (B), but invalid for (C) and (D). To see this figure in color, go online.

The three analytical expressions of BFP spring constant (k_p) to be considered are

$$k_p=\frac{\pi \Delta p}{\left(\frac{1}{R_p}-\frac{1}{R_0}\right)}\frac{1}{\mathrm{l}\mathrm{n}\left(\frac{4R_0^2}{R_p{R}_c}\right)},$$ (1)

hereafter referred to as the Evans’ model (8),

$$k_p=\frac{\pi \Delta p}{\left(\frac{1}{R_p}-\frac{1}{R_0}\right)}\frac{1}{\mathrm{l}\mathrm{n}\left(\frac{4R_0^2}{R_p{R}_c}\right)-\left(1-\frac{R_p}{4R_0}-\frac{3R_p^2}{8R_0^2}+\frac{R_{\text{c}}^2}{R_0^2}\right)},$$ (2)

hereafter referred to as the Simson’s model (9), and

$$k_p=\frac{\pi \Delta p}{\left(\frac{1}{R_p}-\frac{1}{R_0}\right)}\frac{1}{\mathrm{l}\mathrm{n}\left(\frac{4R_0^2}{R_p{R}_c}\right)-\left(1+\frac{R_p^2+R_{\text{c}}^2}{4R_0^2}\right)},$$ (3)

hereafter referred to as the Heinrich’s model (10). In all three models, R_p, R₀, and R_c are the respective radii of the micropipette lumen, the spherical portion of the aspirated RBC, and the circular contact area between the RBC and the probe bead. Δp is the aspiration pressure (Fig. 1).

The calibration experiments were performed under three BFP configurations: 1) the RBC was aspirated with a short tongue (R_p ≈ L_p) (Fig. 1 A); 2) the RBC is aspirated with a long tongue (R_p ≪ L_p) (Fig. 1 B); and 3) aspirated by a large micropipette (R_p ≫ 1 μm) (Fig. 1 C). These configurations were chosen because Evans et al. (8) mentioned a caveat that Eq. 1 is accurate only when the length of the RBC tongue aspirated into the micropipette is short and close to the inner radius of micropipette. Also, the optimal micropipette size is R_p = 1 μm. Notably, several calibration experiments for the BFP spring constant used these configurations of R_p ≪ L_p and R_p > 1 μm (9, 11, 12, 13). These previous calibration experiments were done using another force transducer to exert force to the BFP. By comparison, the present study used thermal fluctuation analysis.

Calibration of the BFP spring constant was done using the previously described thermal fluctuation analysis (14), which is based on the equipartition theorem

$$\frac{1}{2}{k}_p\text{Var}\left(X\right)=\frac{1}{2}{k}_{B}T,$$ (4)

where Var(X) is the variance of the thermally excited random displacements X of the force probe, k_B is the Boltzmann constant, and T is the absolute temperature.

Due to the finite temporal resolution (limited by the camera speed), the measured displacements X_m are an average of X over the time window during which a single frame of image is acquired. This so-called “motion-blur” effect can be corrected by a scaling factor S(α) (15)

$$\text{Var}\left(X\right)=\text{Var}\left(X_{\text{m}}\right)S^{-1}\left(\alpha \right),$$ (5a)

$$S\left(\alpha \right)=\frac{2}{\alpha }-\frac{2}{{\alpha }^2}\left(1-\text{exp}\left(-\alpha \right)\right),$$ (5b)

where α is the ratio of the camera exposure time to the characteristic fluctuation time of the BFP, represented by the ratio of its spring constant to its friction coefficient. Thus, α = Ak_p where A is proportional to the camera exposure time but inversely proportional to the BFP friction coefficient. Since k_p = CΔp from Eqs. 1 and 2, it follows from substituting Eqs. 4 and 5b into Eq. 5a that

$$\text{Var}\left(X_{\text{m}}\right)=\frac{k_{B}T}{C\Delta p}\left\{\frac{2}{AC\Delta p}-\frac{2}{{\left(AC\Delta p\right)}^2}\left[1-\exp \left(-AC\Delta p\right)\right]\right\}.$$ (6)

Similar to the method as previously described (14, 15), to correct for low frequency drifts, we calculated a series of Var(X_m) values after passing the displacement raw data (Fig. 2 A, black) through a series of high-pass filters to obtain low-frequency drift free data (Fig. 2 A, red). The high-pass filters were generated with equi-ripple characteristics using the Parks-McClellan algorithm in Labview programming platform (15). The histograms of corrected displacement X_m were then fit with the Gaussian to derive the variance Var(X_m) (Fig. 2 B).

Figure 2.

Benchmarks of BFP spring constant models by the motion-blur method for the “short tongue” configuration. (A and B) BFP thermal fluctuation with the micropipette held stationary is shown. (A) Representative raw (black) and low-frequency drift corrected (red) data of measured displacements of the BFP (X_m) by tracking the RBC-bead edge over 15 s are shown. The displacement raw data were passed through a series of high-pass filters as previously described (15) to obtain the drift-free data. (B) Corresponding X_m histograms and the Gaussian fits to obtain the corrected variance Var(X_m). (C) Var(X_m) is plotted versus reciprocal suction pressure 1/Δp (symbol) and nonlinearly fitted by the motion blur model Eq. 6 (curve). The results represent mean ± SEM of n ≥ 3 measurements under the same configuration. (D–F) The motion-blur corrected variance Var(X) is calculated from (5a), (5b) using the best-fit AC and C values and plotted versus 1/k_p. k_p is the BFP spring constant calculated from either Evans’ model (D; Eq. 1), Simson’s model (E; Eq. 2) or Heinrich’s model (F; Eq. 3). The three sets of data were then fitted by linear regression (solid line) to calculate the slope, y-intercept corresponding errors (SEM) and R². To see this figure in color, go online.

We first performed the calibration under the previously described optimal condition that the RBC is aspirated with a short tongue (Fig. 1 A; R_p = 1.1 μm, L_p = 1.2 μm). The drift-free Var(X_m) was plotted vs. 1/Δp. Consistent with our previous calibration, the Var(X_m) of the force probe increased as the suction pressure decreased. Eq. 6 was nonlinearly fit to the background-subtracted Var(X_m) vs. 1/Δp data, which returns two best-fit parameters, AC and C (Fig. 2 C). The corrected Var(X) from Var(X_m) was calculated using (5a), (5b) with α = ACΔp. Next, Var(X) was plotted vs. 1/k_p calculated by either the Evans’ model (Eq. 1; Fig. 2 D), Simson’s model (Eq. 2; Fig. 2 E), or the Heinrich’s model (Eq. 3; Fig. 2 F). It is evident that all three data sets display a linear trend, as predicted by Eq. 4. The respective linear fits to the three data sets had slopes of 4.13 ± 0.24, 4.82 ± 0.28, and 5.62 ± 0.33 pN nm, respectively (Fig. 3). The first value is in excellent agreement with the k_BT value at room temperature (4.1 pN nm) (Fig. 2 D), supporting the validity of Evans’ equation. By comparison, the second (Fig. 2 E) and third (Fig. 2 F) values are slightly larger than the 4.1 pN nm value predicted by the equipartition theorem (Fig. 3), suggesting that the Simson’s and the Heinrich’s model are less accurate than Evans’ model in this BFP configuration.

Figure 3.

Benchmarks of the three BFP spring constant models. The slopes of Var(X) (mean ± SEM) versus 1/k_p plots obtained using the Evans’ model published in 1995 (blue), Simson’s model published in 1998 (green), and the Heirinch’s model published in 2007 (red) are compared for three BFP setup configurations. The broken horizontal line indicates the k_BT value (4.1 pN nm) predicted by the equipartition theorem Eq. 4. To see this figure in color, go online.

Next we repeated our calibration but performed the experiment in the second BFP configuration where the RBC was aspirated with a long tongue (Fig. 1 B; R_p = 1.1 μm, L_p = 2.9 μm). The linear fit to the data where k_p was calculated with the Evans’ and the Simson’s models returned slopes of 2.79 ± 0.13 pN nm and 3.24 ± 0.15 pN nm, much lower than the theoretical k_BT value of 4.1 pN nm. By comparison, the linear fit to the data where k_p was calculated with the Heinrich’s model returned a slope of 3.95 ± 0.19 pN nm, close to the prediction by the equipartition theorem. These results suggest that the Heinrich’s model is more accurate than the other two in this BFP configuration (Figs. 3 and S1 A).

The calibration experiment was also performed in the third configuration where the RBC was aspirated by a large micropipette (Fig. 1 C; R_p = 1.6 μm). The linear fits to the two data sets where k_p was respectively calculated with the Evans’, the Simson’s, and the Heinrich’s models returned 6.30 ± 0.38, 6.96 ± 0.42 and 8.90 ± 0.54 pN nm, respectively. All values far exceed the predicted 4.1 pN nm value (Figs. 3 and S1 B), suggesting that none of these calculations is accurate under this BFP configuration.

The above comparisons between theoretical models and experimental data indicate that the accuracy of an analytical expression for BFP spring constant depends on the configuration of the BFP setup. This finding may be explained as follows. The Evans’ model takes into account the RBC membrane elasticity, but was obtained as an approximate solution (16). The Herinch’s model is the linear approximate solution to the simpler problem of a pure fluid membrane (10). The Simpson’s model is also linearized, but is a series solution to the same simplified problem (9). The inclusion of RBC elasticity pushes the linear range of BFP extension to larger deflections than that predicted by the fluid membrane models. Nevertheless, when the assumptions used to simplify the physical problem are less valid, a “more complete” setup of the mathematical problem and a “more approximate” solution to the mathematical problem may not provide a more accurate answer to the physical problem in the real world.

The BFP technology also has its limitations. Since it was first introduced by the Evans’ lab in 1995 (8), quite a few labs have tried to set up this powerful technology but very few ultimately succeeded. This is because BFP is technically challenging, requiring highly skilled and dedicated operators. Force is converted from probe bead displacement using a BFP spring constant k_p that depends on many parameters, i.e., Δp, R_p, R₀, and R_c. All are subject to measurement errors. In our previous BFP works (2, 3, 4, 5), we used Evans’ model because we found it more accurate under our experimental configuration, which used the optimal micropipette size R_p around 1 μm to aspirate RBC with a short tongue (R_p ≈ L_p), as calibrated previously (14) and again in this paper. When the RBC aspiration tongue is longer (R_p ≪ L_p), the Heinrich’s model is more accurate and should be used. Simpson’s model represents an intermediate between Evans’ model and Heinrich’s model. Its most appropriate configuration should be L_p>R_p with an intermediate RBC tongue length. However, when RBC aspiration micropipette size is too big (R_p ≫ 1 μm), none of these models are accurate. It is therefore important not to use micropipettes with large orifice to assemble the BFP. Furthermore, it has been well-demonstrated that the BFP spring displays nonlinearity when RBC extension exceeds 200 nm (9). For a typical spring constant of k_p = 0.3 pN/nm, this corresponds to a force of 60 pN. For force measurements of larger magnitude, a higher suction pressure should be used to increase k_p to keep the RBC extension below 200 nm to maintain the BFP linearity.

While inaccuracy of the spring constant introduces uncertainty of the force value, a bigger contribution to error comes from the experimental variations in attaching the probe bead to the RBC apex. If the bead is not in that exact location, the axisymmetric assumption of the theoretical analysis of the RBC deformation is invalid even if the micropipette is fabricated perfectly (Fig. 1 D). These uncertainties outweigh that, due to the non-linear force-extension behavior of the BFP, which is relatively small. In conclusion, it is critical to configure BFP with rigorous criteria so as to ensure the assumptions underlying the spring constant model are valid (Fig. 1 A and B) and the experimental variations are avoided during the data acquisition period, thereby determining force accurately.

Author Contributions

L.J. and C.Z. designed the study. L.J. performed experiments. L.J. and C.Z. wrote the manuscript.

Editor: Simon Scheuring.