Stereo Darkfield Interferometry : a versatile localization method for subnanometer force spectroscopy of single molecules and 3D-tracking of single cells
Martin Rieu, Thibault Vieille, Gaël Radou, Raphaël Jeanneret, Nadia Ruiz, Bertrand Ducos, Jean-François Allemand, Vincent Croquette
SStereo Darkfield Interferometry : a versatile localizationmethod for subnanometer force spectroscopy of singlemolecules and 3D-tracking of single cells.
Martin Rieu , Thibault Vieille , Gaël Radou , Raphaël Jeanneret , Nadia Ruiz ,Bertrand Ducos , Jean-François Allemand , and Vincent Croquette These authors contributed equally to this work Laboratoire de physique de L’École normale supérieure de Paris, CNRS, ENS,Université PSL, Sorbonne Université, Université de Paris, 75005 Paris, France Institut de Biologie de l’Ecole Normale Supérieure (IBENS), Ecole normale supérieure,CNRS, INSERM, Université PSL, Paris, FranceAugust 25, 2020
Abstract
Super-resolutive 3D tracking, such as PSF engineering or evanescent field imaging has long beenused to track microparticles and to enhance the throughput of single molecules force spectroscopicmeasurements. However, current methods present two drawbacks. First, they lack precision comparedwith optical tweezers or AFM. Second, the dependence of their signal upon the position is complexcreating the need for a time-consuming calibration step.Here, we introduce a new optical technique that circumvents both issues and allows for a simple,versatile and efficient 3D tracking of diluted particles while offering a sub-nanometer frame-to-frameprecision in all three spatial directions. The principle is to combine stereoscopy and interferometry, suchthat the z (axial) position is measured through the distance between two interferometric fringe patterns.The linearity of this stereoscopy technique alleviates the need for lookup tables while the structuredinterferometric pattern enhances precision. On the other hand, the extended spatial footprint of this PSFmaximizes the number of photons detected per frame without the need of fancy cameras, nor the need forcomplex hardware. Hence, thanks to its simplicity and versatility, we believe that SDI (Stereo DarkfieldInterferometry) technology has the potential to significantly enhance the spreading of 3D tracking.We demonstrate the efficiency of this technique on various single-molecule measurements thanksto magnetic tweezers. In particular we demonstrate the precise quantification of two-state dynamicsinvolving axial steps as short as 1 nm. We then show that
SDI can be directly embedded in a commercialobjective providing a means to track multiple single cells in 3D .
Video-based 3D tracking of micro-particles has be-come a major tool revealing the dynamics of mi-croorganisms motility [1] and the kinetics of molec-ular motors interacting with biopolymers tethered tomicrobeads [2]. However, in spite of recent signifi-cant improvements [3] [4] [5], these techniques suf- fer from the limitation of their precision and fromthe requirement of object-specific calibrations priorto any measurement. In particular, while thesecamera-based methods present the extremely usefuladvantage of being parallelizable, they were not ableto reproduce the very precise nanometric measure-ments made with the optical tweezers (OT) or AFMthat revolutionized the understanding of molecular Present address : Depixus R (cid:13) a r X i v : . [ phy s i c s . b i o - ph ] A ug otors stepping [6][7] [8][9]. The reason behindthis difference lies in the huge number of photonsrecorded in OT or AFM in comparison with the oneobtained using camera sensors.The last decade has witnessed significant im-provements [10] of microparticles 3D tracking tech-niques, providing better axial and lateral precisionand smaller optical aberrations. However, most ofthese competing solutions have made improvementssolely for one feature. For instance, while some al-low subnanometer precision, they are not suited forparallelization [8]. Others do not allow long real-time tracking, as they rely on very high acquisitionrates [11] incompatible with the maximal transferrate between cameras and computers.Among these methods, evanescent fields basedones are precise but limited in their detection range,of the order of the wavelength of the exciting light[12][13][14]. Technologies based on the dependenceof the point spread function (PSF) with the axial po-sition [15] [16] [17] [18] [19] [20] [21] [22] allowprecise 3D-tracking in the far field but suffer fromthe complexity of their experimental implementa-tion and from their dependence upon the quantitativemeasurement of the image shape that increases theirsensibility to optical imperfections such as spheri-cal aberration or astigmatism. More importantly, theneed for lookup tables calibrated for each individ-ual emitter (or at least for sufficient positions in thefield of view) [10] limits their throughput and theirusage. Tracking of free particles variable in sizequickly passing in the field of view is then almost im-possible. A significant amelioration of ring-shapedPSF’s [16][15] was recently proposed [23], allowingnano-particles tracking in living cells, but is limitedin axial range (a few hundreds of nanometers) andprecision (5-6 nm).By comparison, stereoscopic methods [24] [25][26] are more versatile and proficient in the contextof heterogeneous and fast-moving particles. Usingtwo light sources with different incidences, they pro-duce two images coinciding when the particle is infocus. As the tracked object moves out of focus bydz, the shift between both images dx increases lin-early with dz. This linearity is a great feature ofthe method alleviating the requirement of a calibra-tion step and insuring its independence with respectto the particle size. However, the necessary reduc-tion of the numerical aperture to at least half of thepossible angles results in a larger point-spread func-tion (PSF) and thus a reduced precision. This lossof information can be in principle compensated by increasing the number of photons used to build theirPSF, but the finite well-depth (the maximum numberof photons treated by one pixel during one frame)of cameras sets a limit to this strategy. At the sametime, such a numerical aperture limits the axial mea-surement range.Here, we present a method, Stereo Darkfield In-terfetrometry (SDI) [27] that combines the linear-ity and parallelization ability of stereoscopic meth-ods with the high per-photon information content ofstructured PSF. It thus allows to optimize the use ofthe transmission rate and of the well depth of cam-eras. Furthermore, the technique enables to reach abetter compromise between the axial measurementrange and the resolution by selecting a small rangeof incident angles and by using interferometry tostructure light and thus to boost precision. Our ap-proach is designed to track in real time a large num-ber of diluted microparticles with subnanometer ax-ial and lateral precision with no limit on the dura-tion of the experiments, thus enhancing significantlythe throughput of subnanometer 3D single-particletracking. Furthermore, it allows to greatly simplifythe tracking of free microorganisms by suppressingthe need to constitute lookup tables.We illustrate the precision of the method byacquiring, for the first time with a camera-basedmethod, traces from force spectroscopic eventsshowing nanometric stepping (helicase stepping,oligonucleotide hybridization), thus paving the wayfor the multiplexing of measurements that were sofar exclusively feasible with optical tweezers oratomic force microscopes (AFM), and for this rea-son hardly parallelizable.We then illustrate the versatility of the methodby measuring dynamic properties of unicellular al-gae through the acquisition of their 3D trajectories,thus expanding the utility of our approach to a widerange of applications in biological imaging.
Figure 1 presents the principle of Stereo DarkfieldInterferometry (
SDI ). Illumination is generated bytwo superluminescent LEDs (SLED) sending colli-mated parallel beams with symmetric incidence an-gles of ± ◦ from the optical axis (Figure 1.a). Themicroscope is built around an infinitely corrected ob-jective and a field lens producing an image on a cam-era chip. An afocal system built with 4 lenses (Fig-ure S1) is used to access the image focal plane of the2bjective that is physically located inside the objec-tive mount. A black absorbant film with slits is usedas a filter in the Fourier plane, i.e. the back focalplane. In the absence of diffusing particles, no lightreaches the camera as the two illumination beamsare focused on the black region located between theslits. The Fourier filter is built with four symmetricalslits spanning the x axis. When a micron size parti-cle is placed in the objective focal plane, the lightstemming from each of the sources is diffused in acone which covers mostly the two closest slits. Theslits select two beams that are focused on the cam-era chip by a lens placed in contact with the slits(Figure 1.b). When the particle is at the focal plane,the beams originating from both slit pairs coincide.In order to separate them along the y direction, twoglass slides tilted with opposite incidences ( ± ◦ )are placed just behind each slit pairs in close con-tact with the imaging lens. Their incidence is chosento be large enough so that the two images comingfrom both pairs of slits do not coincide. Due to thedouble-slit configuration, each of the beam pairs is-sued from the same light source, as in the so-calledYoung double-slit experiment, gives rise to an in-terferometric pattern. When the object moves awayfrom the focal plane by δ z , both patterns are trans-lated in a shear mode with opposite directions alongthe x -axis (Figure 1.c). The distance between each ofthem is linearly related to the axial displacement δ z (Figure S2), whereas the average of their positions isrelated to x - y movement of the tracked object, as in[24].The structure of the interferometric fringes of SDI increases both the axial and in-plane resolutionof the bead position. Indeed, assuming a photon lim-ited noise, the optical noise σ i on the 3D localiza-tion of the bead along the i direction must verify (seeSupp. Materials S.III): σ i > g i N (cid:82) f (cid:48) ( i ) f ( i ) di (1), where N is the number of photons received dur-ing one frame, f is the profile of the interferometricpattern and g is the coefficient relating the signal dis-placement to the displacement of the tracked object.For the transverse displacement, g x is simply the ge-ometric magnification of the optical system. The ax-ial displacement g z depends on the magnification andon the angle of the incident light as described by theformula given in Supp. Materials S.IV.In practice, for real-time tracking, the numberof photons received per seconds is limited by the transfer rate of the camera and its well-depth (maxi-mum number of electrons per pixel). Increasing thespread of a Gaussian PSF results in a loss of pre-cision. However, SDI allows spreading the PSF onseveral pixels without losing precision (Figure 1.d-f): the PSF is indeed structured, increasing its spatialderivative and thus the value of the integral in thedenominator in equation 1. We show in Figure 1.gthe experimental and theoretical values of the pre-cision of
SDI : the structuration of the signal al-lows a 4-fold improvement of the spatial precisioncompared to the equivalent unstructured PSF. At themaximum light intensity allowed by the camera welldepth, the frame-to-frame error on the position of thetracked objects due to photon noise is as low as 0.2nm in y and z directions, and as 0.1 nm in the x di-rection (Figure 1.e). The difference between z and x -axis is due to a difference between the transverseand the axial magnification of the setup (see S.IV).The slightly poorer precision in y is due to the asym-metry of the SDI profile pattern (no structure in the y direction of the slits presented in this paper).The double-slit configuration creates a patternthat is localized in the space of spatial frequenciesand that can be easily filtered out using Fourier trans-form and deconvolved from low-frequency signals(see Supp. Materials S.V). That, combined with thereduced range of angles selected by the slits, reducesthe sensitivity to overall PSF deformations and al-lows increased depth of field compared to traditionalstereoscopy [24]. Thus, the z axial position can betracked up to 6 µ m with 40X and 100X objectives(Fig S2).Most interestingly, Stereo Darkfield Interefer-ometry can be directly embedded into an objec-tive and thus can be commercially distributed as a3D-tracking module for any preexisting microscope.Due to the technical complexity of planar high mag-nification objectives, such an embedding would needto be performed by the objective supplier. However,in order to demonstrate this possibility, we disas-sembled a simpler X20 objective and added the SDIslits directly in the back focal plane of the objective.We brought two prisms of opposite angles directlyin contact with the slits (Figure 3a-c). These prismsproduce a phase shift in the Fourier space that trans-lates into a position shift in the real image space,preventing the superimposition of the two interfer-ometric patterns. They play the same role as theparallel slides presented above. In this low magni-fication configuration, the precision is micrometric,while the axial depth of view is greatly enhanced3ith a conserved linear response : the distance be-tween the interferometric fringes depends linearly onthe z position over ranges of the order of 150 µ m(Figure 3.h). Residues to linearity over such a rangelay below 0 . µ m and are shown in Figure S5. Con-cerning illumination, the angles corresponding to theslits being smaller, one parallel light source is in thiscase sufficient to perform the measurement. The pa-rameters of this configuration are perfectly suited forthe 3D-tracking of microorganisms. We now illustrate the precision of our tracking tech-nique through various single-molecule force spec-troscopy measurements. These measurements areperformed with magnetic tweezers : DNA moleculesare attached between a glass surface and a micro-metric magnetic bead (Figure 2.a-b). A force is ap-plied by approaching magnets while the extensionof the DNA molecule is measured by tracking theposition of the bead with the help of
StereoscopicDarkfield Interferometry (Figure 2.c). The followingexamples are to our knowledge the first nanometricmeasurements of relevant biomolecular effects per-formed with a camera. They open the possibility toincrease importantly the throughput of precise enzy-matic measurements and to overcome the lack of par-allelization that characterizes high resolution opticaltweezers and AFM.Figure 2e shows the kinetics of hybridization ofshort oligonucleotides that were obtained by usingthe nanometric precision of
SDI . A DNA moleculewith a short single stranded (ss) segment is attachedbetween a magnetic microsphere (MyOne TM ) andthe surface and pulled at a force of ∼
13 pN. Afree 8-bp oligonucleotide complementary to the ss-DNA segment of the tethered molecule is injectedin the solution. As the oligonucleotide hybridizesto the molecule, 8 bases of the latter are convertedfrom ssDNA to dsDNA, causing a shortening of themolecule of ≈ . SDI to in-vestigate the effect of the stacking of the oligonu-cleotide with the neighbour dsDNA. For this pur-pose, we reproduced the experiment on two differ-ent substrates (Figure 2f). On one of them, thefully stacked configuration, the oligonucleotide hy-bridizes with a single stranded gap whose size is ex-actly 8 bp. On the other, the half-stacked configura-tion, the single-stranded gap is larger (14 bp) than theoligonucleotide, thus preventing stacking interactionon one of the free ends of the ss fragment. While asimilar experiment was pioneered by Whitley et al. in [29] using FRET, we here are able to perform it ina fluorescence-free setup, thus avoiding an eventualbias that could be introduced by the interaction be-tween the fluorescent label and the DNA. From thedifference in the measured hybridization kinetics be-tween the two substrates (Figure 2f.), we notably ex-tract the free energy of stacking (see Figure S12) andfind ∆ G stacking , AG = . ± . Helicases are essential enzymes that unwind dsDNAto separate the two DNA strands. They are key el-ements of DNA replication and repair. As an he-licase unwinds a DNA hairpin pulled at 9 pN andtethered between a surface and a magnetic bead, themeasured elongation increases by twice the lengthof a ssDNA base for each base pair unwound, that4 igure 1:
A schematic description of the SDI setup. The red and blue colors denote respectively parallel incident light and lightdiffused by the scatterer (same wavelengths). a. In the absence of the scatterer, the incoming parallel light is blocked, ensuringdarkfield. b. When a scatterer is present, the light goes through the slits and creates a PSF consisting of two interference patterns. c. Vertical stack of the SDI images (objective 100X) as a function of the defocus (axial position z ) of the tracked object. Thetransverse distance ε x between the two spots is proportional to the defocus. d. Typical transverse density profile of a
SDI pattern.One fringe and the envelope are being fitted by Gaussians. e. Distribution of the inferred position, at maximal light intensity, ofthe 3D position of stuck microspheres. 1280 frames are analyzed. No averaging is performed. The mechanical and thermal driftare subtracted in order to assess the optical noise of the setup (see Materials and Methods). f. Theoretical number of photonsby frame and information per photon for each profile shown in d. Interferences allow to increase the number of photons whilekeeping a good precision. Thin black lines join points with equal theoretical precision σ (values in millipixels). g. Standarddeviation of the measured axial position of a microsphere as a function of light intensity (objective 100X). This is comparedwith the theoretical Cramér-Rao bounds computed from the experimental profile drawn in d and its envelope. The maximal lightintensity is constrained by the camera’s well-depth (here 30.000 electrons per pixel). is roughly 1 nm (see Figure 2g.). Distinguishing in-dividual steps of an helicase with magnetic tweezersthus requires tracking the bead with subnanometricresolution.Upf1 is an helicase belonging to the Super-Family 1 that we studied in previous works usingmagnetic tweezers [30][31]. On Figure 2h, we showthat the resolution enhancement provided by SDI al-lows resolving its individual steps, provided that the ATP concentration is low enough (500 nM). The 30-bp hairpin is open step-by-step until the remainingunwound part becomes too short and starts to os-cillate spontaneously between its open and closedstates under the influence of the applied force andof the thermal fluctuations (starting from 272s). Atsome positions, the enzyme displays a ratchet-likebehavior with back-and-forth steps of one base pair.Away from this stalling position, the distribution of5 igure 2: a.
Schematic representation of the SDI setup attached to magnetic tweezers. b. In this picture, the bead is trackedwith
SDI . Its z -position is directly related to the number of bases hybridized in the hairpin. The magnets are designed so thatboth light sources go through the gap between them. c. A typical field of view. Each pair of fringes, materialized by two greenboxes, corresponds to one magnetic bead. Scale bar, 10 µ m. d. The hybridization of a 8-bp oligonucleotide causes a shorteningof the DNA molecule of typically 1 nm since dsDNA is shorter than ssDNA above 5 pN. e. The 1-nm steps caused by theoligonucleotide hybridization are measured by tracking the position of the magnetic microsphere with the SDI. f. Two DNA sub-strates are tested with the same oligonucleotide : in the configuration on the left, there is no free base when the oligonucleotidehybridizes. The analysis of the kinetic parameters of the two-level systems described above allows accessing the stacking freeenergy by comparing k o f f and k on in both configurations. k on being equal for both configurations (see Figure S11), k o f f containsall the information about the free energy of stacking. g. Description of the helicase stepping experiment. As the helicase Upf1unwinds a base pair of the hairpin, the measured extension increases by twice the length of a ssDNA base, that is roughly 0.9nm at 9 pN. h) While unwinding the dsDNA recursively, Upf1 displays discrete steps (top inset). At the stalling position, Upf1displays a ratchet-like behavior, going forth and back by steps of 1 bp (bottom inset). step sizes displays intermediate sizes between 1 bpand 2 bp (c.f. Figure S13). Such intermediate stepsizes were already observed in optical tweezers ex-periments [32] and interpreted as an asynchronousrelease of both stands. Stalling positions of the en-zyme are reproducible from beads to beads and canbe easily identified even at higher ATP concentra-tion. We show for example in Figure S14 a trace ofthe helicase domain of yeast-Upf1 unwinding a 30- bp hairpin at a concentration of 10 uM ATP and thecorresponding distribution blockage positions over80 single-helicase events (Figure S15). Blockagesare determined with the resolution of the base pair.6 .3 3D-micrometric-tracking of dilutesingle cells.
Usually, 3D tracking techniques of microorganisms[15] are dependent on calibration libraries consist-ing of images taken at different vertical position.Interpolating these images allows retrieving the ax-ial position of an object. The linearity of
SDI avoids the use of such lookup tables and thus greatlyreduces image processing needed for 3D-tracking,while dark field illumination facilitates automatic de-tection.Using a modified objective as described in thesection
Setup enables the 3D-tracking of free micro-particles in any commercial microscope. We showin Figure 3.f the image of the model micro-swimmer
Chlamydomonas reinhardtii acquired with the mod-ified objective. We studied the dynamics of freelyswimming algae confined in a microfluidic Hele-Shaw cell (i.e. width and length (cid:29) height) of thick-ness ≈ µ m. Filtering out short trajectories (<1.5s spent in the field of view) and cells with defi-cient vertical swimming (spanning less than 110 umin the axial direction), we analysed the statistics of117 independent trajectories and found that the dy-namic is largely dominated by the presence of con-fining walls. As illustrated by the typical track inFigure 3.i, most of the cells performed back and forthmovements between the upper and lower boundaries(see also Figure S20). The distribution of their axialposition displays peaks at 20 − µ m from the celllimit (Figure 3.j), in accordance with previous 2Dhorizontal measurements [35]. Measuring the verti-cal angle α z of each of the 117 trajectories withinthe middle of the chip (for z between 60 and 90 µ m) we found that it is symmetrically distributedaround 0 (Figure 3.k) with a clear peak at ≈ ± ◦ .This value compares reasonably well with the mostprobable outgoing angle following wall scatteringas measured in [36], which shows that after inter-acting with either wall the cells keep swimming inthe same direction until colliding with the oppositeboundary. This is consistent with the long ≈ ≈ .
5s to cross the height ofthe channel. Finally, the symmetry observed in thedistribution of angles indicates that gravity axis [38]does not play any role in this confined configuration,simply because the gravitational torque [39] felt bythe algae is too small to bias the cell swimming di-rection over such a short vertical distance.
We presented a new 3D tracking method,
Stereo-scopic Darkfield Interferometry . The darkfield al-lows for a reduction of the background noise, whilethe stereoscopic aspect allows for the linearity ofthe measurement and the interferometry ensures anAngström level resolution. By tracking the positionof a magnetic bead with higher precision, the tech-nique allows performing force spectroscopy experi-ments that were so far only possible with hardly par-allelizable methods (AFM or optical tweezers) likehelicase stepping measurements. It also enables the3D-tracking of dilute free micro-organisms over alarge range of axial positions without the need of anyfocus feedback nor any calibration. Because of thehigh information content of the PSF, the method canbe performed at relatively low frequencies that arecompatible with real-time tracking.Regarding its application to force spectroscopy,the method offers to the community working withmagnetic tweezers the means to study the discrete ki-netic mechanisms of molecular motors. We showedsome preliminary data on the helicase Upf1 but thetechnique could also be applied to the studies ofother enzymes, like topoisomerases or polymerases.For example, the addition of one base by a poly-merase results at 10 pN in the decrease of extensionof the DNA molecule by a distance of roughly 1.5 Å.Thus, our optical precision (a frame-by-frame stan-dard deviation of 2 Å in the axial direction due tophoton noise) should theoretically allow the distinc-tion of the discrete incorporation of bases by poly-merases. However, there is still an obstacle to beovercome that is related to the Brownian motion ofthe magnetic beads. Indeed, while we measuredAngström level standard deviation on the position ofbeads fixed on the glass surface, this noise increasedto the nanometer range (at 160 Hz of acquisition fre-quency) when the beads were attached to a DNAsubstrate. This difference is partly due to the mag-netic anisotropy [40] of the beads as pointed in [41]but mainly, it is due to the large increase of the ax-ial hydrodynamic drag applied on the bead causedby the proximity of the surface [42]. In our cases,the correction to the drag is close to factor 15 (seeFigure S17), and thus implies an increase of the in-tegrated Brownian motion of a factor √ (cid:39)
4. Aswe show in the Supplementary S.IX, this problemcannot be solved by using longer dsDNA handles asthe induced loss of stiffness has a worse effect onthe noise than the viscosity close to the surface. A7 igure 3:
Direct integration of a modified objective in a microscope (here Olympus IX81) in order to implement
Stereo Dark-field Interferometry in an existing setup. a. Schematic description of the modified objective (Olympus, achromatic 20X). The
SDI slits brought in contact with two prisms of opposite angles are added in the Fourier plane of the objective. b. Picture ofthe slits and of the mechanics allowing their insertion in the objective. c. Picture of the disassembled objective. d. Schematicrepresentation of the tracking experiment : algae
Chlamydonomonas reinhardtii are inserted in a flow cell containing TAP bufferand their movement is tracked. f. An image of an alga obtained with the integrated objective. White bar : 10 µ m g. Horizontalprofile of light intensity corresponding to the image f .The lateral shift between the two interference profile allows measuring theaxial position z . h. Dependency of the distance between the interference fringes on the position of the focus for an alga fixedon the surface. i. A 3D trajectory obtained thanks to the
SDI modified objective. Acquisition frequency : 10 Hz. Each pointcorresponds to one frame. Colors represent z . j. Distribution of the Z position of the algae over 3314 positions taken from 117individual trajectories. k. Distribution of the vertical angles while an alga crosses the middle of the flow cell ( z between 60 and90 µ m). 332 crossing events. 117 trajectories. technological development that could be transposed to magnetic tweezers in the future is the use of ex-8remely stiff handles since they should allow a betterspatial separation from the surface without havingany notable impact on the stiffness of the construct.Regarding its applications to the tracking ofmicro-organisms, the method presents the advantageto be linear and thus to avoid the constitution oflookup tables. However, it must be acknowledgedthat a large amount of light going through the sam-ple (around 90%) is absorbed by the black filter in-side the SDI objective. It means that an intense lightshould be sent to the sample in order to get a brightenough signal. This can be problematic for the ob-servation of light-sensitive species. In the case of Chlamydomonas , the algae studied in this paper, thisproblem was overcome by using a wavelength (730nm) that is thought to be large enough to not disturbthe behavior of the organisms.The main point that needs to be carefully thoughtabout in the process of developing new applicationsis the design of the slits, and the corresponding an-gle of the illumination. There are three parametersthat have to be adjusted to the scatterer to be ob-served : their width, their length and their separation.For a given averaged angle, a larger separation be-tween the slits allows for a better sensitivity and thusa better precision in the z-direction but decreases thedepth of field. Larger widths and lengths (howeverfully contained inside the pupil of the objective) willincrease the quantity of light that goes through theobjective but will also reduce the width of the enve-lope, as well as reduce the depth-of-field of the sig-nal. Once these three parameters are taken into ac-count, the method is easy to learn and use, especiallybecause it can be, as we showed, directly imple-mented to any existing microscope by engineeringthe objective. Furthermore, the signal analysis pro-cedure, described in the Supplementary S.V is sim-ple and computationally inexpensive. For this rea-son, we claim that Stereo Darkfield Interferometryhas the ability to extensively increase the adoptionof 3D tracking methods in the biological communityespecially with the increase in quality and speed oc-curring in the camera field.
Superluminescent LEDs (Exalos, EXS210030-03,650 nm, 10 mW) are placed in the object focalplane of a diode collimation package (Thorlabs,LTN330A) and the outcoming parallel beams aredirected to the pupil of the oil objectives (respec-tively UPLSAPO100X/UPLSAPO40X) with oppo-site incident angles. Their intensity is regulatedthrough a feedback loop based on the signal of themonitor photodiode included in the chip. The op-tical axis is then switched from vertical to horizon-tal using a mirror (BBEA1-E0Z, Thorlabs). The 4f-setup , needed to have access to the Fourier planesas it is not possible with these objectives, is madeof two lenses of focal 100mm (AC254, Thorlabs).A sheet of black paper (Canson, 160g/m2) is thencut with a laser machine in order to get the pairof slits needed for the SDI technique. Slits con-sist of four identical rectangles (200 µ mx700 µ m)whose centers are placed at the following positionswith respect to the optical axis : ([-1.1mm,0],[-0.4mm,0],[0.4mm,0],[1.1mm,0]). The slits arebrought in contact with a thin lens of focal 100mm(Thorlabs, LBF254-100-A). Then, a pair of opticallyclear glass slides (4mmx8mm) of thickness 1mm isplaced in contact with this lens. Both are tilted in they-direction with opposite angles of ±
17 degrees inorder to shift the beams stemming from both pair ofslits in opposite y-directions. Finally, the resultingpattern is imaged on a monochromatic CMOS cam-era (UI-3030CP-M, IDS Ueye), placed at 100mmfrom the titled glass slides. The camera is linkedthrough USB3.1 to a computer that processes the im-ages in real-time. Conjugation relations are shownin Figure S1 and the protocol of the alignment is de-scribed in the Supplementary Materials S.XII.
An achromatic objective 20X from Olympus wasdismantled. We have replaced a cylindrical spacer14 mm wide and 10 mm long just after the frontlens component of the objective by a holder support-ing a black paper with slits followed by prisms anda spacer so that this system also spans 10 mm. Around glass slide of radius 13 mm was inserted inthe holder to hold the prisms. We have machined a98 mm x 8 mm) rectangle slit in the holder allow-ing us to introduce two optically clear glass slides(4 mm x 8 mm) of thickness 1 mm in contact withthe round one. One of the rectangular glass slideswas slightly titled by introducing a thin sheet of pa-per of thickness ≈ µ m (cigarette paper ) betweenit and the round slide. The other rectangular glassslide was tilted in the opposite direction by intro-ducing an identical sheet of paper on the oppositeside. The space left between the round slide and thetiled rectangular slides was filled by capillarity witha drop of Olympus immersion oil, in order to createtwo prisms of opposite angles. A sheet of black pa-per in which the SDI slits were cut were then addedon top of the prisms. Slits consist in four rectan-gles which centers are placed at the following posi-tions with respect to the optical axis : ([-2.2mm,0],[-0.9mm,0],[0.9mm,0],[2.2mm,0]). Outer slits (width: 0.85 mm, height : 1.25 mm) are wider than in-ner slits (width : 0.55 mm, height : 1.25 mm) inorder to compensate for the smaller light intensity atlarge scattering angles. The height of the cylindri-cal spacer was adjusted so that, once the objectiveis reassembled, the slits are located at the back focalplane of the objective.
In order to measure accurately the optical noise ofthe setup and compare it to theory, it is necessaryto uncouple it from the Brownian motion and fromslow thermal or mechanical drifts that can affect themeasurement. In order to do so, we melt MyOne T1beads (ThermoFisher) on a glass surface. The stocksolution was diluted 10000 times and then spread ona glass slide. The slide was then heated at 110 ◦ Cfor 5 minutes following the evaporation of the solu-tion. A standard flow cell was then assembled withthis slide and water was injected. The position of 15beads was then tracked for 5 seconds at 160 Hz atdifferent light intensity. The average of the trajec-tory of the 15 beads was then subtracted from eachindividual trajectory in order to get rid of mechanicalor thermal drifts.
In order to minimize optical drifts that are due totemperature changes, the objective is inserted intoa box made of duralumin. The box temperature iscontrolled through 4 Peltier moduli placed in par- allel (ET-063-08-15, Adaptive). The PID feedbackloop then allows to reach a temperature stabiliza-tion with the precision of 0.0001K. The temperatureis read in the following way : thermistors (TDK,B57703M0103) are connected to Wheastone bridgesconverting the resistance difference into a voltage.The voltage is then read with a 32-bit Sigma-Deltaanalog to digital converter (ADC).
For the linearity test of the setup with 40x or100x magnification, the objective is moved with apiezo nanofocusing device (P-725.2CL, Physik In-strumente). For the other measurements on thissetup, the piezo nanofocusing device is replaced bya deformable stage to avoid any noise inherent to thepiezoelectric device. The position of the screw con-trolling the stage is measured with a rotational mag-netic sensor (AMS,AS5048A). For the linearity testof the 20x setup used for algae tracking, the objec-tive is moved by the stepper motor driving the mi-croscope focus (IX81, Olympus).
All single-molecule experiments are performed at25 ◦ C. All the DNA substrate used in the presentedassays are synthesized single stranded oligonu-cleotides. Their sequences are indicated in Ta-ble S2. Their 5’end is complementary to a 57 bases3’ DBCO modified long oligonucleotide (Oli1) thatis attached to azide-functionalized surfaces (PolyAn2D Azide) through a 2-hour-long incubation (100nMOli1, 500 mM Nacl). Their 3’end is complementaryto a 58-base oligonucleotide (Oli2). Oli2 containstwo biotin modifications at its 5’end. The ssDNAsubstrate is first hybridized with Oli2 by mixing botholigos at 100nM in 100 mM Nacl, 30 mM, Tris-HclpH 7.6. 5uL of streptatividin coated Dynabeads My-One T1 (Thermofisher) are washed three times in200 uL of passivation buffer (140 mM NaCl, 3 mMKCl, 10 mM Na2HPO4, 1.76 mM KH2PO4, BSA2%, Pluronic F-127 2%, 5mM EDTA, 10 mM NaN3,pH 7.4). The result of the hybridization between thesubstrate and Oli2 is diluted down to 2 pM buffer andthen incubated 10 minutes with the beads in a totalvolume of 20uL of passivation buffer. The beads arethen rinsed three times with passivation buffer in or-der to remove unbound DNA. 1uL of the bead so-lution is then introduced in the cell coated with Oli1and filled with passivation buffer. They are incubated10 minutes. Excess unbound beads are washed out byflowing passivation buffer.
The substrate used in these assays (HP10STACK) isa 153-base ssDNA strand. Once hybridized to Oli1and Oli2, the size of the remaining single-strandedDNA reduces to 38 bp. 24 of these bases fold intoa 10 bp hairpin with a 4-base apex loop. This hair-pin is used to test that only one DNA molecule isbound to the beads by testing the fluctuation be-tween the open and closed state at 10 pN. Oncethis is checked, the hairpin is blocked by a 16bpoligonucleotide (OliBlock-half-stack) for the half-stack configuration or by a 22bp oligonucleotide(Oliblock-full-stack) for the fully stacked configura-tion. The remaining bases of single-stranded DNAcontain the 8-base sequence that is complementaryto the oligonucleotide whose hybridization kineticsis measured. The assay is realized in HybridizationBuffer (100mM Nacl, 40mM Tris-Hcl pH 7.6).
A 193-base ssDNA is used in this assay (HP30).Once hybridized to Oli1 and Oli2, the size of thesingle-stranded DNA reduces to 78 bases. Amongthem, a 30-bp is included with a loop of 4 bases. Itis flanked by two handles of 7 successive thymines.Once the beads are attached to the surface of the cell,the buffer is changed to Upf1 buffer (100mM KCl,3mM MgCl2, 40 mM Tris-Hcl pH 7.6). The closingand opening of the hairpin are tested using a forcescan going from 20 to 8 pN. The force is then fixedto 8 pN on average and the helicase domain of yeast-Upf1 purified as described in [31] is injected withATP in Upf1 buffer at a concentration of 10 nM un-less otherwise mentioned.
The substrate used in this assay (HP10FLUC) isa 160-base long ssDNA strand. Once hybridizedto Oli1 and Oli2, the size of the remaining single-stranded DNA reduces to 45 bases. 24 of these basesfold into a 10 bp hairpin with a 4-base apex loop.The kinetic of the folding and unfolding transitionsof the hairpin at different forces are acquired in pas-sivation buffer.
Cultures of
Chlamydmonas reinhardtii strain CC125were grown axenically in a Tris-Acetate-Phosphate(TAP) medium at 20 ◦ C under periodic fluorescentillumination ( ≈ µ E . m . s − , cycle 16h light/8hdark). Cells were harvested in the exponentiallygrowing phase, then centrifuged at 800 rpm. for 10min and the supernatants replaced with fresh TAP. Adilute suspension ( ≈ − %) was then loaded intoa simple microfluidic chip ( L ≈ W ≈ H = µ m; made with standard soft photo-lithography techniques) where the algae dynamicwas acquired at 10 fps (camera uEye UI-3000SE)on an inverted IX81 Olympus microscope with theSDI objective described in the main text and usinga 730nm LED for illumination in order to preventphototactic responses from the cells. Images are transmitted in real-time to a computerand analyzed on the fly at the frequencies indicatedin the main text. For high-magnification and high-precision applications (force spectroscopy), imagesare analyzed through Fourier decomposition usingthe algorithm described in the Supplementary S.V.In the case of algae tracking, the position of thefringes is simply measured through the computationof the barycenter of their intensity in the x and y di-rection and the resulting signal is processed througha Finite Impulse Response (FIR) filter of width 0.8 s(8 frames). M.R. designed, performed and analyzed the biologi-cal experiments, performed the signal treatment andthe theoretical analysis, contributed to the setup andwrote the manuscript. T.V. built, calibrated and char-acterized the optical setup and contributed signifi-cantly to its design. G.R built the electronics. T.V.and G.R. built the mechanics. N.R. expressed andpurified Upf1. B.D. contributed to the hairpin con-structions. R.J. performed the culture of
Chlamy-domonas reinhardtii and supervised the analysis oftheir trajectories. V.C. supervised the research, de-signed the optical setup, its mechanics and its elec-tronics and contributed to its experimental imple-mentation. J-F.A. contributed to the design of theexperiments and helped supervise the project. All11uthors discussed the results and commented on themanuscript.
This work has originated in collaboration withThomas Lepic. We wish to acknowledge stimulat-ing discussions with H. Le Hir, J. Ouellet, DavidBensimon, Nicolas Desprat, Ding Fangyuan, MariaManosas and their help with hairpins and ds-DNAconstructs.
This study was supported by the ANR CLEANMDgrant (ANR-14-CE10-0014), ANR G4-CRASH(G4-crash - 19-CE11-0021-01) from the French Agence Nationale de la Recherche to V.C., , bythe European Research Council grant Magreps [267862] to V.C. and by continuous financial supportfrom the Centre National de Recherche Scientifique,the Ecole Normale Supérieure France and the LabexIPGG.
The authors declare the following competing inter-ests : J-F.A. and V.C. own shares of the companyDepixus R (cid:13) that makes a commercial use of StereoDarkfield Interferometry . T.V. and G.R. are now em-ployed by the same company. V.C. and T.V. and J-F.A have filed the patent US9933609B2 for the tech-nique described in this paper. Application has beengranted on 2018-04-03.
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S.I Conjugation relations in the optical setup of high magnification (100X and 40X)
Usually, the object focal plane of an objective is not accessible as it is located inside the structure of thelatter. In order to access the focal plane, a common way is to use a 4f-setup that allows reimaging it furtheraway. Figure S1 describes the whole setup.
Figure S1:
Scheme of the whole
SDI setup with all the optical elements being materialized. The pair of lenses after the objectiveis an afocal system that allows reimaging the Fourier plane of the objective.Red and blue dummy rays allow materializingconjugation relations. .II SDI : depth of field and linearity Figure S2:
Dependence of the distance between interference fringes on the focal position of the emitter obtained by moving a20X objective with a stepper motor.
Figure S3:
Dependence of the distance between interference fringes on the focal position of the emitter obtained by moving a40X objective with a piezo stage. igure S4: Residues from the linear fit of the dependence between the bead-to-objective axial (z) distance and the distancebetween the interference fringes for 16 different tracked beads. Objective 40X.
Figure S5:
Residues from the linear fit of the dependence between the bead-to-objective axial (z) distance and the distancebetween the interference fringes. Objective 20X. .III Theoretical precision The precision of the localization of the particles depends on the minimal noise that can be obtained whenmeasuring the translation of the interference patterns. Because the spread of the SDI Point Spread Function(PSF) is large compared to the pixel size a , we neglect in the following the error due to pixelization a [44].The Cramér-Rao bound gives a theoretical lower bound for the standard deviation of the position measure-ment due to photon noise. The photon horizontal density profile depends on the translation parameter ofthe fringe ε in a simple way : f ( x | ε ) = f ( x − ε ) (2)The Cramér-Rao bound stipulates that the variance of the estimator of the hidden parameter ε from thesampling of the profile f with N f photons verifies : σ ε > N f − E (cid:16) ∂ ln f ( x | ε ) ∂ ε (cid:17)(cid:12)(cid:12)(cid:12) ε (3), where the expectation value E() is taken with respect to f ( x | ε ) and ε is the true value of the parameterto be estimated. The expectation value in the denominator is the Fisher information about the parameter ε contained in one photon. Using equation 2, it is straightforward to check that in the case of the estimationof a translation parameter, σ ε does not depend on the true value of the parameter ε and that equation 3becomes σ x , pattern > N f (cid:82) f (cid:48) ( x ) f ( x ) dx (4)Equivalently, the precision of the localization of one fringe pattern in the y direction is given by : σ y , pattern = N f (cid:82) f (cid:48) ( y ) f ( y ) dy = w y N f (5), where the last equality is due to the fact that the y-profile is Gaussian of waist w y .We computed above the Cramér-Rao bound in the image space. The corresponding bounds in the objectspace can be deduced from the transverse magnification g and the axial sensitivity g z that is related to g through equation 17.In order to get the transverse position of the bead, the positions of both interference pattern need to beaveraged, and then divided by the magnification g : σ x , bead = g N f (cid:82) f (cid:48) ( x ) f ( x ) dx = g N tot (cid:82) f (cid:48) ( x ) f ( x ) dx (6) σ y , bead = w y g N f = w y g N tot (7) N tot being the total number of photons scattered (in both fringes pattern).The axial position of the bead is given by the difference between the position between the two fringes,and thus the minimal localization error in this direction reads : σ z , bead = g z N f (cid:82) f (cid:48) ( x ) f ( x ) dx = g z N tot (cid:82) f (cid:48) ( x ) f ( x ) dx = σ x , bead tan ( α + α ) (8)These theoretical errors are compared with experimental ones on the figure 1 in the main text.19maging profile Fisher Maximal TheoreticalInformation photon number localization(px -2 .photon -1 ) per fringe and per frame precision (px)SDI pattern 0 ,
18 7 . × . × − One fringe 0 ,
23 1 . × . × − Envelope 0 ,
006 11 × . × − Table S1:
Comparison of information related properties of the different profiles presented in Figure 1.
S.IV Computation of axial magnification
Figure S6: Top :
Figure defining the notations used to calculate the transverse magnification of the setup. The object to beimaged is placed in O. The interferences are observed in I, the plane of the camera, i.e. the back focal plane of the setup. I isconjugated to the point B, placed in the front focal plane of the setup. The α i are the angles of the rays going through the slitsat the position X i . Bottom :
Figure defining the construction point J i , belonging to the same wavefront as B with respect to thewave of angle α i . In this part we compute the phase shift between two rays stemming from the object O ( x o , z o ) when theyreach a point of the screen I ( x , z ) by assuming that the objective is perfectly aplanetic but without assumingthe astigmatism along the axial coordinate. For this purpose we use a construction point belonging to theobject focal plane whose image would be I. Given the hypothesis of perfect aplanetism, the coordinates ofthis point are B ( x / g , ) , where g is the transverse magnification of the setup and all the rays stemming fromthis point and converging to I have no relative phase shift. Thus, denoting α et α the angles correspondingto the slits, we have : ( BI ) α = ( BI ) α (9), so, ( BX ) + ( X I ) = ( BX ) + ( X I ) (10)20ncoming parallel rays being conjugated to the plane of the slits, the following relation also holds : ( J X ) = ( BX ) (11), and, ( J X ) = ( BX ) (12), J being defined figure S6 and J being the corresponding construction point for α . They verify (seefigure S6): ( OJ i ) = cos ( α i ) z o − sin ( α i )( x o − xg ) (13)Bringing these relations together, we get δ ( OI ) = ( OI ) α − ( OI ) α = ( OJ ) + ( J X ) + ( X I ) − ( OJ ) − ( J X ) − ( X I )= ( OJ ) − ( OJ )= ( cos ( α ) − cos ( α )) z o − ( sin ( α ) − sin ( α ))( x o − xg ) (14)Thus, δ φ = π n δ ( OI ) λ = π n λ ( cos ( α ) − cos ( α )) z o − ( sin ( α ) − sin ( α ))( x o − xg ) (15)We deduce in particular the lateral displacement of the first fringe when the bead is displaced of δ z b in theaxial direction : dx = g cos ( α ) − cos ( α ) sin ( α ) − sin ( α ) δ z b (16)Thus, the axial magnification is written (recalling that g is the transverse magnification) : g z = g cos ( α ) − cos ( α ) sin ( α ) − sin ( α ) = g tan ( α + α ) (17)We also deduce the interfringe : i = g λ n ( sin ( α ) − sin ( α )) (18) S.V Localization algorithm
As seen in the previous section, the localization of the bead in the three directions is based on the precisesub-pixel localization of the interference fringes in the X and Y. Here we detail the algorithms of localiza-tion. i X localization
The translation of a pattern in X is measured through the following steps :1. A Blackmann-Nuttal window of width 128 pixels is applied on the pattern.2. The windowed signal f ( x ) is decomposed on the Fourier basis using discrete Fourier transform(DFT). Figure S7.b shows the amplitude | ˆ f k | of the Fourier modes. The envelope correspondsto the slow modes while the oscillating part of the signal is contained in the second peak of thespectrum. 21. As the signal translates of a length δ x , the phase of each mode k is shifted by a quantity 2 π k δ x . Thisproperty is used in order to infer δ x . Using a fixed reference profile f re fx , we compute the phase shiftof each mode between f re f and f : δ φ k = φ re fk − φ k .4. δ φ k is eventually unwrapped and its values at the modes of interest are fitted by a linear function (seeFigure S7.c). The value of the x-displacement of the pattern δ x is inferred from the slope a of thefitted linear function: δ x = a π . Figure S7: a.
Instantaneous intensity X-profile of an interference pattern. b. Amplitude of the Fourier modes of the patternshowed in a. The sum is normalized to 1. The red square shows the modes that correspond to the oscillating part of the pattern. c .Unwrapped phase shift between the reference profile and the instantaneous profile showed in a. The phases of the modes insidethe red square are fitted by a linear function. The displacement of the pattern in the direction x is inferred from the slope. Errorson the phase are taken as the inverse of the Fourier modulus shown in b. ii Y localization Figure S8 shows the intensity profile in the Y direction of
SDI pattern. In order to avoid the need forbackground correction, the center of the profile is obtained by computing the barycenter ∑ ni = i f ( x i ) , where i is stands for the pixel and f for the normalized light intensity.22 igure S8: Intensity profile of the
SDI pattern in the Y direction. Relative errors are equal to √ N , where N is the number ofphotons received in each pixel. iii Comparison to theoretical precision The previously described algorithms are evaluated with regard to the theoretical precision (Cramér-Raobound) expected for the
SDI pattern in Figure S9. The X-localization almost performs as well as expectedby the theory. While the Y-localization also allows reaching subnanometric localization error, there is asignificant discrepancy with the theoretical prediction. This might be due to non-linear effects related to theoscillations of the Airy function that cannot be easily filtered out by Fourier decomposition, in oppositionwith the case of the X-direction where the fast modes can be isolated from slowly varying optical artefacts.
Figure S9:
Experimental localization error in the X ( a ) and the Y ( b ) directions are compared to the Cramér-Rao lower bound.Protocols for the measurements are the same as presented in the legend of Figure 1 and in the section Material and Methods . .VI Oligonucleotide hybridization assay Figure S10: k o f f and k on of the oligonucleotides binding as a function of the concentration of oligonucleotide in the half-stackedconfiguration. The linearity of k on as a function of the concentration confirms the nature of the steps observed in the data. Figure S11: k on of the oligonucleotides binding as a function of applied force in both configurations, half-stacked and fullystacked. No significant dependence on the force can be observed, confirming previous results, like in [29]. igure S12: Dependence of the stacking free energy with the force. The free energy is computed in the following way : ∆ G stacking = kT log k of f , FS k on , HS k of f , HS k on , FS , where FS and HS stand respectively for fully-stacked and half-stacked configuration. T = 298.15K S.VII Upf1 helicase
Figure S13:
Distribution of measured step size over 190 stepping events during Upf1 unwiding at 500 nM ATP. igure S14: Trace showing the unwinding and the reclosing of a 30-bp Hairpin by the helicase domain of yeast-Upf1. The blacklines show identified blockage. A blockage is identified with the following empirical criterium : The measured position must liewithin a range of 2bp during at least 0.5s.
Figure S15:
Distribution of the stalling positions of 80 stalling events of yeast-Upf1 identified as explained in the previous figure.Negative positions correspond to blockages happening during the closing of the hairpin while positive positions correspond toblockages happening during the opening of the hairpin. .VIII Step detection In order to detect steps in noisy data, we use the algorithm
Moving Step Fit (MSF) described in [45], fittingthe signal by horizontal lines instead of fitting it by any linear function. A window w is defined. For eachpoint in the data x i , the mean of the signal m li between i − w / i is computed along with the residues ofthe fit, called left residues RSS l : RSS li = i ∑ j = i − w / ( x j − m li ) (19)Equivalently, the mean of the signal m ri between i and i + w / RSS r : RSS ri = i + w / ∑ j = i ( x j − m ri ) (20)Finally, the mean of the signal m i over the whole window w is computed, along with the whole residues RSS : RSS i = i + w / ∑ j = i − w / ( x j − m i ) (21)The MSF score is defined as : MSF i = | m ri − m li | ( RSS i − RSS ri − RSS li ) (22)A step is detected at the position x i if the following conditions are true : • The step is larger than a given threshold t s : | m ri − m li | > t s (23) • The score
MF S i is larger than a score threshold t MFS .We set the score threshold t MFS by computing the distribution of scores on a signal that precedes the in-jection of the helicase, i.e. where no step is expected. The threshold is defined as the score value wherethe cumulative probability of scores on such a step-free trace equals 99.99%. For such a threshold, a falsepositive detection every 10000 points (62.5 seconds at 160 Hz) is expected.The parameters used to detect steps in the helicase data described in this paper are the following : • The window w equals 100 points (0.8 seconds at an acquisition frequency of 160 Hz). • The step threshold t s equals 0.3 nm. • The score threshold t MFS equals 1 × − µ m Figure S16 shows the number of false positive per unit of time obtained by analyzing, with the algorithmdescribed above, 628s seconds of signals before the helicase injection, compared to the number of steppingevents per unit of time during helicase unwinding at 500 nM ATP. Figure S13 shows the distribution ofmeasured step sizes of 190 stepping events of the helicase Upf1.27 igure S16:
Left : Rate of false positive detection (steps detected without helicase). Nsteps=18, T = 628s. Right : Rate of stepdetection (steps detected during helicase unwiding). Nsteps = 190s, T = 689s
S.IX Viscosity and axial noise in the vicinity of the surface
Figure S17:
Evolution of the correction factor to Stoke’s law λ z ( L ) for the drag of a sphere of radius 0.5 µ m when the spherelies close to the surface as a function of the distance L from the surface, according to the analytical formula of Brenner [42]. Inour experiment, the drag is 15 times larger than the one that would experience the bead infinitely far from the surface. In this section we briefly discuss the impact of the increased hydrodynamic drag close to the surface onthe measurement noise. The noise density below the z-cutoff frequencies (in our case several kHz) for an28verdamped Brownian sphere attached to a molecule of stiffness k is written : < P ( ) > ∝ k B T γ k (24)The drag γ thus plays an important role in the measurement noise. However, this drag deviates fromStoke’s law in the vicinity of a surface. γ ( z ) = πη R λ z ( L ) , where R is the radius of the bead and λ hasbeen computed analytically by Brenner ([42]). By looking at the figure S17, this drag can be significantlyreduced by working further away from the surface. This could be done by using larger dsDNA handles.However, the stiffness of such handles decreases with the number of monomers and thus with their lengths: k ∝ N ∝ L . Thus, the dependence of the noise density as a function of the length of the handles verifies : < P ( ) > ∝ k B T λ z ( L ) k ( L ) ∝ L λ z ( L ) (25)We draw this function as a function of L in figure S18. As can be seen, it is a growing function of L , thusshowing that the loss of noise due to the decreased stiffness of the handles is worse than the gain in noisedue to the decrease of the drag. For this reason, there is no hope that increasing the length of our dsDNAhandles could help fighting against this surface effect. That is the reason why we propose in our discussionto use handles that are much stiffer than the DNA substrate whose change of extension we are measuring,in order to bring the bead further away from the surface without increasing significantly the stiffness of theconstruct. Figure S18:
Evolution of the product λ z ( L ) × L as a function of L. .X Hairpin dynamics Figure S19:
Kinetics of folding/unfolding of a 10 bp hairpin measured with
SDI . a : Trace representing the closing and theopening of a 10bp-hairpin at a acquisition frequency of 1500Hz. The high optical resolution of SDI allows a frame-by-framedetermination of the state of the hairpin and thus for a millisecond time resolution. b : Kinetics of opening/closing of the HP as afunction of the applied force. There is an ≈
10% uncertaintity on the force because of the variability of the bead magnetization.This uncertainty is not represented in the graph as it is systematic. c Distribution of folding/unfolding times at three differentforces. Exponential fitting of the distribution allows for the determination of the kinetic parameters represented in b. .XI Algae dynamics Figure S20:
Instantaneous velocity as a function of the vertical position of the algae of 97 trajectories are displayed. For thesake of clarity, among the 332 original single trajectories, we only displayed the ones that had no point in the excluded regionrepresented by the red dashed rectangle. Represented trajectories are displayed in their entirety.
Figure S21:
Mean velocity of the algae
Chlamydomonas Reinhardtii over 332 single trajectories. One count corresponds to onetrajectory.
S.XII Alignment procedure
We develop the following alignment method with the goal of positioning the amplitude mask 1) in theFourier plane with a sub-millimetric accuracy and 2) centered on the optical axis of the system with theaccuracy of ten microns. As the Fourier plane is located inside the microscope objective, a 4f line is31nstalled to provide a mechanical access on its image plane (pair of 1” achromatic doublet lens of 100mm,the focal length has been chosen to avoid 1) vignetting effect and 2) off-axis aberrations of more powerfuldoublet lens).
Figure S22: A . A folded 4f line (elliptic mirror + 2 100mm Thorlabs) is positioned below the objective to provide mechanicalaccess to its Fourier plane. B . The SDI module comprises 1) a compact assembly of amplitude mask-tube lens-tilted glasswindows, 2) a removable Bertrand lens and 3) a camera. Phase 1 : Assembly and Installation of the 4f line
1. The distance between the two lenses of the 4f line is locked (SM1 tubes from Thorlabs) once obtainedsharp images of objects at the infinity. To do so, a temporary cage system assembles the 4f line and amodule composed by a 50mm achromatic lens and a camera at its focal plane.2. The 4f line is installed in the cage system, attention is paid to align it along the optical axis :(a) First, a temporary module composed by SM1 tubes and a camera are assembled such that oncebrought in contact with the 4f-module, the camera is placed at the focal plane of the second lensof the 4f-module.(b) The whole module (4f line + a temporary module) is then moved along the optical axis until onegets a sharp image of the pupil (see Figure S23).32 igure S23:
Image of the entrance pupil (telecentric objective) observed on a camera temporarily placed at the focal plane ofthe second lens of the 4f line, after proper positioning of the 4f line in the system: the large black squares pieces of cardboardpositioned on the ceiling of the room.
The assembly and integration of the SDI module itself employ a Bertrand lens, used to simultaneouslyobtain a sharp image of the mask containing the slits and of the back focal plane of the microscope objective(typically an object attached on the ceiling of the room).
Phase 2: assembly of the SDI module.
1. 3 optical components (mask containing the slits - tube lens - pair of tilted glass slides, see Figure S22)are assembled in a cage system such that the tube lens is sandwiched tightly between the mask andthe tilted glass slides.2. Once this module is assembled, the camera is added at the focal point of the tube lens (the tilted glassslides must be in place during this tuning).3. A Bertrand lens (25mm of focal length, to perform 1:1 imaging between the place containing theslits and the camera plane) is added in the assembly, by the means of a magnetic mount (CPF90,Thorlabs), the position of which along the optical axis being tuned along the cage system axis untilone gets a sharp image of the slits. 33 igure S24:
Image of the Fourier plane of the objective, here observed through the Bertrand lens of the SDI module. Slits havebeen removed, and the glass slits have been centered such that the spot of a HeNe alignment laser is shown split and largely cutby the line between the glass windows.
Figure S25:
Same type of image, but with a mask added: slits of the mask and objects placed at an infinite distance of theobjective are simultaneously sharp.
Phase 3: coarse implementation of the SDI module on the system
1. With Bertrand lens on position, the SDI module is inserted on the main cage system of the instrumentand its position along the optical axis is manually tuned until one gets a sharp image of the back focalplane through the mask (Figure S27).2. The lateral position of the mask containing the slits and the glass slides are then adjusted to becentered onto the central of the pupil image (an alignment laser on top of the system can help).34. At that point, the illumination angles of the two light sources are optimized by ensuring that theyconverge in the center of each pair of slits.This protocol typically results in good quality SDI images. However : • Residual error in the transverse positioning of the mask in the Fourier plane may result in the selectionof slightly different angles for the left and right pairs of slits : this leads to a common differentialsensitivity (variation of phase vs. defocus) between the two sets of fringes on the field of view. Forexample, all top fringes share the same offset of sensitivity compared to their corresponding bottomfringes. • Residual error in the positioning of the mask along the optical axis (i.e., not truly in the Fourier plane)involves that the selected averaged angles may according to the position in the field of view.Consequently, a final alignment step is performed directly on the signals generated by fixed objects inthe field of view of the microscope.
Phase 4: fine adjustment of the SDI module.
1. 3 images obtained at focus are recorded, as well as images taken at large and symmetrical values ofdefocus (above the depth of field allowed by the setup, i.e. until the two optical rays for each fringeare separated).2. 3 objects are selected, located on the center and on both sides (along the mask axis) of the field ofview.3. The transverse position of the mask is corrected (using micrometric screws), until on-axis imageshows no differential sensitivity between both sets of fringes.4. Longitudinal position of the complete SDI module (sliding along the optical axis on the cage sys-tem) is corrected until one sees no different sensitivities between on-axis image and off-axis images(symmetric).5. Steps 1-3-4 are iterated until convergence.A typical set of images corresponding to one iteration of this process is shown on Figure S26 andFigure S27. They have been recorded during the alignment of an SDI instrument featuring a 40X oil-immersion system. 35 igure S26:
Images recorded for -23,0,+23 µ m of defocus at the final position of the mask. Three objects are selected (seeassociated text). The large white spots are caused by the edges of a thermalization box present in the light path and are notrelated to the SDI imaging itself. igure S27: Images obtained for the 3 objects selected on the field of view and for the 3 focus positions.
DNA name Sequence(cf. Material and methods)Oli1 5’ ATTCGAAGAGCACCAGAAAGACCAAAAGACACGGTGAAGGATTAGACAGAAGAAGAC 3’-DBCOOli2 5’ Double-biotin TGGGAGTAGCGGATCATGATGGATGTTGCCAGCTGGTATGGAAGCTAATAGCGCCGGT 3’HP10STACK 5’ GTCTTCTTCTGTCTAATCCTTCACCGTGTCTTTTGGTCTTTCTGGTGCTCTTCGAATACTGCCAGAGTTTTCTCTGGC(stacking assay) AGTGCGTGCTCGCAGTGACCGGCGCTATTAGCTTCCATACCAGCTGGCAACATCCATCATGATCCGCTACTCCCA 3’Oliblock-half-stack 5’ ACTGCCAGAGAAAACT 3’Oliblock-full-stack 5’ GCACGCACTGCCAGAGAAAACT 3’HP10FLUC 5’ GTCTTCTTCTGTCTAATCCTTCACCGTGTCTTTTGGTCTTTCTGGTGCTCTTCGAATTTTTTTTACTGCCAGAGTTT(fluctuation assay) TCTCTGGCAGTGCGTGCTCGCAGTGACCGGCGCTATTAGCTTCCATACCAGCTGGCAACATCCATCATGATCCGCTACTCCCA 3’HP30 (UPF1) 5’ GTCTTCTTCTGTCTAATCCTTCACCGTGTCTTTTGGTCTTTCTGGTGCTCTTCGAATTTTTTTTAGTGCAGATGCTTCCTATAGACTGCCAGAGTTTTCTCTGGCAGTCTATAGGAAGCATCTGCACTTTTTTTTACCGGCGCTATTAGCTTCCATACCAGCTGGCAACATCCATCATGATCCGCTACTCCCA 3’
Table S2:
DNA sequences used in this paper. Names are defined in the section Material and Methods. .XIII Rotational postprocessing Figure S28:
Schematic explanation of the notations for the following section. We claim that a slight misalignment between themagnetic axis, which is relevant for the thermodynamic understanding of the Brownian noise, and the optical axis, explains partof the noise in magnetic tweezers experiments and can be easily corrected by a post-processing rotation. On this picture, theangle is exaggerated on purpose, but even an angle as small as 1 ◦ can have a very important effect on the observed noise observedin the optical direction z o . A slight misalignment of the optical axis ( z o ) with the axis of the magnetic force ( z ) can cause a largeincrease of the measured noise in the z o direction.Let’s call θ the small angle between z o and z and u the coordinate of the bead position along the axis ofrotation (that belongs to the plane (x,y)) perpendicular to z . For small θ , z o verifies : z o = z + θ u (26)and the fluctuations verify ( x , y and z being uncorrelated at the first order approximation): σ z o = (cid:113) σ z + θ σ u (27) σ z o (cid:39) σ z ( + θ σ u σ z ) (28)We recall the formulation of the power spectrum density of an overdamped trapped bead : < P ( ω ) > ∝ k B T γ ω + ω c (29)38 where γ is the viscous coefficient of the bead, k the stiffness of the trap, and ω c = k γ For the frequencies below the cut-off frequencies, the noise density is roughly : < P ( ω ) > = < P ( ) > ∝ k B T γ k (30)For a polymer of length L whose thermodynamics properties are encoded in its force-extension curve F ( L ) , and that is subject to a force F in the z direction; k (cid:107) = k z = ∂ F ∂ L (31) k ⊥ = k x = k y = FL (32)For a 120 bp double-stranded DNA subject to a force of 10 pN , we have roughly : k z = ∂ F ∂ L (cid:39) pN / nm (33) k x = k y (cid:39) . pN / nm (34)Thus, the noise density below the cut-off frequency is larger in the lateral direction by a factor k (cid:107) k ⊥ = σ ⊥ )compared to the axial direction ( σ (cid:107) ) by a factor √ = ◦ between the optical axis and the magnetic axis is largeenough to double the measured noise in the optical axial position z o compared to the noise in the magneticdirection z .To overcome this issue, we post-rotate the experimental data in order to recover the magnetic direction z from the measurement in the orthogonal referential ( x o , y o , z o ). We look for the 3D rotation angles ( α , β )that minimize the noise in the rotated z direction. Formally, we minimize σ z with respect to α and β where: xyz = R ( α , β ) x o y o z o This minimization is obtained through standard multidimensional gradient-descent algorithms. In fig-ure S29, we show σ z o as a function of α and β and materialize the angles α ∗ and β ∗ that minimizesthe noise. Figure S30 shows the effect of the post-processing rotation on the spectrum of a tethered bead.Figure S31 shows how the post-processing allows recovering signal of a much higher quality for the hy-bridization assay presented in the main text. 39 igure S29: Measured noise in the z-direction obtained through rotation of the measured data ( x o ( t ) , y o ( t ) , z o ( t ) ) with angles( α , β ) respectively around the x and the y axis. The curve as a function of α is drawn for β = β ∗ . The curve as a function of β is drawn for α = α ∗ . Figure S30:
Power spectrum comparison. In turquoise, the power spectrum of x o ( t ) . In purple, the power spectrum of z o ( t ) . Inorange, the power spectrum of z ( t ) with z being taken as the axis corresponding to the rotation that minimizes the noise in the z direction. The cutoff frequency of the X-signal is visible in the z o ( t ) trace, demonstrating that this axis is coupled with the x direction. The orange curve shows that rotating the signal allows decoupling the signals. igure S31: Traces obtained during the oligonucleotide hybridization assay described in the main text. In purple, raw z o ( t ) . Inorange, optimally rotated z ( t ) . The rotation allows to better distinguish the 1 nm steps in the data. The curves are vertically offsetfor clarity.. The rotation allows to better distinguish the 1 nm steps in the data. The curves are vertically offsetfor clarity.