# Optimization of Optical Trapping and Laser Interferometry in Biological Cells
## 1 Introduction  
Since optical trapping (OT) was invented by [PERSON] et al. in the late 20 th century [1], it has found applications in biology, physics, material sciences, and engineering [2; 3]. Nanometer- to micrometer-sized objects, whose refractive indices are slightly larger than the surrounding media, are trapped by a laser tightly focused to the diffraction limit with a high numerical aperture (NA) objective lens (Figure 1a). Along with the progress of techniques to control the focal position of the laser, multiple particles are now simultaneously manipulated in three-dimensional space in a specimen. For instance, biological cells can be aligned in order to create custom-designed tissues with a bottom-up approach [4; 5]. The position of the trapped spherical particles can be measured at precisions less than nm and at a bandwidth up to 100 kHz by using the technique referred to as back-focal-plane laser interferometry (LI) [6]. As shown in Figure 1b, the probe laser is irradiated to the particle, and the interference of the scattered and unscattered lasers is detected with a quadrant photodiode (QPD) that is placed at the back-focal plane (BFP) of the condenser lens. The QPD signal then provides the particle position \((x,y)\) relative to the symmetry axis (\(z\) axis) of the laser once an appropriate calibration of the QPD signal for the probe position is performed. The high spatio-temporal resolution and low-invasive nature of the techniques (OT and LI) are particularly useful for investigating the dynamics of mesoscale (nm - \(\mu\)m) objects in vitro, e.g., biological macromolecules (proteins, nucleic acids, and polysaccharides) and organelles as their assemblies [7; 8]. For instance, OT and LI have been performed using probe particles attached to biopolymers. Thereby, the mechanics (e.g., force-extension relation) and dynamic functions (e.g., unidirectional stepping movements) of the attached molecule have been investigated. Until now, such precise execution of OT and LI that can be used for exact scientific investigation has been possible in homogeneous specimens in vitro that are prepared in a pure and quiescent buffer [7]. On the other hand, it has been challenging to conduct similar experiments in vivo in living cells. One of the reasons for this is that the optical properties are heterogeneous in cells and different between cells [9,10]. Propagation of the laser in a heterogeneous media gives rise to optical aberrations [11] that affect OT and LI [12,13]. Therefore, the strength of OT and the position sensitivity of LI show extra dispersions and systematic errors unless careful calibrations are conducted for every particle and occasion in living cells. Second, the living cell interior is highly crowded with various biomacromolecules, and therefore highly viscoelastic [14], whereas cells with high metabolic activities exhibit vigorous cytoplasmic fluctuations [15]. OT in such viscoelastic and active situations requires a strong laser power, which would induce photo-thermal and photochemical reactions and eventually lead to the damage of cells [16, 17, 18]. A probe particle in a weak optical trap, on the other hand, readily goes out of the laser focus. OT and LI in living cells must then fail. In order to achieve precise and less invasive LI in living cells, we have recently introduced 3D-feedback tracking of a probe particle that is performed in conjunction with in-situ calibration of the QPD signal [15]. The piezo-mechanical stage, on which a culture of living cells is placed, was smoothly repositioned via the feedback control to maintain the probe particle in the laser focus. The precise LI was then conducted while the probe particle is stably tracked in a weak probe laser (\(<\)0.5 mW). Although there are a variety of caveats for executing the techniques in biological cells, they have been only partially presented in our previous publication because of the space limitations [15].