# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 4 Experimental Results and Discussions  
### LI Sensitivity Depending on Optical Properties  
As explained in previous sections, the LI sensitivity depends on various material parameters and experimental conditions in a complicated manner. Furthermore, the factor \(G\) in Equation (3) changes its value gradually day by day, even if one tries to keep the same experimental conditions as much as possible. Calibration of the LI sensitivity is therefore required for every experiment, even when in vitro samples are measured. When a probe particle is dispersed in a homogeneous solvent with known viscosity, the LI sensitivity and the stiffness of OT can be obtained by observing thermal fluctuations of the probe particle by utilizing the Fluctuation-dissipation theorem [24]. The position sensitivity of LI and the stiffness of OT do not change if the reference solvent used for the calibration and the material of interest have almost the same refractive index. Since dilute samples have their refractive indices similar to the solvents, this calibration procedure has been widely used for many LI experiments [23]. Using the calibration procedure, LI sensitivities were measured in 0-90% glycerol solutions with three types of probe particles (melamine: \(n_{\text{p}}=1.68\), latex: \(n_{\text{p}}=1.58\), and silica: \(n_{\text{p}}=1.43\)). The factor \(G\) (\(a\), \(w_{0}\), \(\lambda\)) in Equation (3) is affected by optical and electronical conditions other than refractive indices(\(n\) and \(n_{\text{p}}\)), and some of them are hard to control, as explained in previous sections. The contribution of the factor \(G\) to the LI sensitivity was therefore removed from the LI sensitivity by normalizing as \(A\equiv S/S_{40}\) where \(S_{40}\) is the sensitivity measured in the same setup in 40 wt% glycerol solution. Results of the LI sensitivity are shown in Figure 4a as a function of refractive indices of surrounding media \(n\). The refractive indices of glycerol solutions were obtained from Ref. [31]. The concentration of the solutions were determined in the range where the refractive index is pertinent to that of eukaryotic cells used in this study (\(n\sim 1.38\pm 0.02\) for isotonic cytoplasm) [25; 26; 27; 28; 29; 30; 9]. We compared the experimental results to the theoretical predictions based on geometrical optics (Equation (4)) and the Rayleigh scattering (Equation (5)) approximations. In our experimental conditions, Equations (4) and (5) provide similar predictions, and the overall trend of the normalized LI sensitivity \(A\) conforms to these formulae. These results imply that the simple formulae can be extended to apply to the experimental condition \(a\sim\lambda\), as long as \(n\) and \(n_{\text{p}}\) are chosen for LI experiments feasible in living cells. As we see in Figure 4a, the normalized LI sensitivity strongly depends on the refractive index \(n\) of surrounding solutions when silica particles are used, but the dependence is modest for melamine particles. This indicates that the LI sensitivity in heterogeneous living cells could be more stable if melamine particles are used as a probe. On the basis of the observation above, the sensitivity \(f(n,\,n_{\text{p}})\) which is not affected by the uncontrollable factor \(G\) can be estimated as \(Af_{\text{R}}(n_{40},\,n_{\text{p}})\) where \(n_{40}\) represents the refractive index of 40% glycerol solution. As it is shown in Figure 4b, all \(f(n,\,n_{\text{p}})\) estimated from experimental results are collapsed to a single line that is proportional to \(\Delta n\equiv n_{\text{p}}-n\), and they agree well to theoretical prediction \(f_{\text{R}}(n,\,n_{\text{p}})\). Note that OT and LI experiments, including ours, are possible at conditions where \(n_{\text{p}}\) is slightly larger than \(n\) since the scattering force prevents OT at too large \(\Delta n\)[1]. In this case,\(f(n,\,n_{\text{p}})\) is simply proportional to \(\Delta n\) and does not largely depend on the regimes for approximations, which is consistent to our results.