Parallel, linear, and subnanometric 3D tracking of microparticles with Stereo Darkfield Interferometry

Abstract

While crucial for force spectroscopists and microbiologists, three-dimensional (3D) particle tracking suffers from either poor precision, complex calibration, or the need of expensive hardware, preventing its massive adoption. We introduce a new technique, based on a simple piece of cardboard inserted in the objective focal plane, that enables simple 3D tracking of dilute microparticles while offering subnanometer frame-to-frame precision in all directions. Its linearity alleviates calibration procedures, while the interferometric pattern enhances precision. We illustrate its utility in single-molecule force spectroscopy and single-algae motility analysis. As with any technique based on back focal plane engineering, it may be directly embedded in a commercial objective, providing a means to convert any preexisting optical setup in a 3D tracking system. Thanks to its precision, its simplicity, and its versatility, we envision that the technique has the potential to enhance the spreading of high-precision and high-throughput 3D tracking.

Fig. 1. Description and optical characterization of SDI.

(A) The red and blue colors denote, respectively, parallel incident light and light diffused by the scatterer (same wavelengths). (B) Schematic view of the setup in the absence of scatterer. For the sake of clarity, the objective and the 4f setup are represented as a unique lens. The incoming parallel light is blocked, ensuring dark field. (C) When a scatterer is present, the light goes through the slits and creates a PSF consisting of two interference patterns. (D). Vertical stack of the SDI images (objective, 100×) as a function of the defocus (axial position z) of the tracked object. The transverse distance ϵ_x between the two spots is proportional to the defocus (fig. S2). (E) Typical transverse density profile of an SDI pattern. One fringe and the envelope are being fitted by Gaussians (Gaus.). (F) Theoretical number of photons by frame and information per photon for each profile shown in (E). Interferences allow us to increase the number of photons while keeping a good precision. Thin black lines join points with equal theoretical precision σ (values in millipixels). (G) SD of the measured axial position of a microsphere as a function of light intensity (objective, 100×). This is compared with the theoretical Cramér-Rao bounds computed from the experimental profile drawn in d and its envelope. The maximal light intensity is constrained by the camera’s well depth (here, 30,000 electrons per pixel). (H) Distribution of the inferred 3D-positions, at maximal light intensity, of stuck microspheres. A total of 1280 frames are analyzed. No averaging is performed. The mechanical and thermal drifts are subtracted to assess the optical noise of the setup (see Materials and Methods).

Fig. 2. Application of SDI to nanometric force spectroscopy.

(A) Schematic representation of the SDI setup attached to magnetic tweezers. (B) In this picture, the bead is tracked with SDI. Its z position is directly related to the number of bases hybridized in the hairpin. The magnets are designed so that both light sources go through the gap between them. (C) Typical field of view. Each pair of fringes, materialized by two green boxes, corresponds to one magnetic bead. Scale bar, 10 μm. (D) The hybridization of an 8-bp oligonucleotide causes a shortening of the DNA molecule of typically 1 nm since dsDNA is shorter than ssDNA above 5 pN. (E) The 1-nm steps caused by the oligonucleotide hybridization are measured by tracking the position of the magnetic microsphere with the SDI. Figure S6 shows that the binding rate increases linearly with the concentration of oligonucleotide. (F) Two DNA substrates are tested with the same oligonucleotide: In the configuration on the left, there is no free base when the oligonucleotide hybridizes. The analysis of the kinetic parameters of the two-level systems described above allows accessing the stacking free energy by comparing k_off and k_on in both configurations. k_on being equal for both configurations (see fig. S7), k_off contains all the information about the free energy of stacking. (G) Description of the helicase stepping experiment. As the helicase Upf1 unwinds a base pair of the hairpin, the measured extension increases by twice the length of a sDNA base, that is, roughly 0.9 nm at 9 pN. ADP, adenosine diphosphate. (H) While unwinding the dsDNA recursively, Upf1 displays discrete steps (top inset). At the stalling position, Upf1 displays a ratchet-like behavior, going forth and back by steps of 1 bp (bottom inset).

Fig. 3. Application of SDI to single-cell tracking and direct embedding in a commercial objective.

(A) Schematic description of the modified objective (Olympus, achromatic 20×).The SDI slits brought in contact with two prisms of opposite angles are added in the Fourier plane of the objective. (B) Picture of the slits and of the mechanics allowing their insertion in the objective. Photo credit: Vincent Croquette, Laboratoire de Physique de l’Ecole Normale Supérieure. (C) Picture of the disassembled objective. Photo credit: Vincent Croquette, Laboratoire de Physique de l’Ecole Normale Supérieure. (D) Schematic representation of the tracking experiment: Algae C. reinhardtii are inserted in a flow cell containing tris-acetate-phosphate buffer, and their movement is tracked. (E) Image of an alga obtained with the integrated objective. White bar, 10 μm. (F) Horizontal profile of light intensity corresponding to the image e. The lateral shift between the two interference profiles allows measuring the axial position z. (G) Dependency of the distance between the interference fringes on the position of the focus for an alga fixed on the surface. (H) A 3D trajectory obtained thanks to the SDI modified objective. Acquisition frequency, 10 Hz. Each point corresponds to one frame. Colors represent z. (I) Distribution of the z position of the algae over 3314 positions taken from 117 individual trajectories. (J) Distribution of the vertical angles, while an alga crosses the middle of the flow cell (z between 60 and 90 μm); 332 crossing events and 117 trajectories.