Angle change of the A-domain in a single SERCA1a molecule detected by defocused orientation imaging

The sarcoendoplasmic reticulum Ca2+-ATPase (SERCA) transports Ca2+ ions across the membrane coupled with ATP hydrolysis. Crystal structures of ligand-stabilized molecules indicate that the movement of actuator (A) domain plays a crucial role in Ca2+ translocation. However, the actual structural movements during the transitions between intermediates remain uncertain, in particular, the structure of E2PCa2 has not been solved. Here, the angle of the A-domain was measured by defocused orientation imaging using isotropic total internal reflection fluorescence microscopy. A single SERCA1a molecule, labeled with fluorophore ReAsH on the A-domain in fixed orientation, was embedded in a nanodisc, and stabilized on Ni–NTA glass. Activation with ATP and Ca2+ caused angle changes of the fluorophore and therefore the A-domain, motions lost by inhibitor, thapsigargin. Our high-speed set-up captured the motion during EP isomerization, and suggests that the A-domain rapidly rotates back and forth from an E1PCa2 position to a position close to the E2P state. This is the first report of the detection in the movement of the A-domain as an angle change. Our method provides a powerful tool to investigate the conformational change of a membrane protein in real-time.

a rabbit SERCA1a that possessed an inserted tetra-cysteine (TC) mutation between residues 196 and 197 in the A-domain 8 . Analysis of crystal structures of this region predicts a large angle change towards the P-domain between the E1PCa 2 state and the E2P state ( Supplementary Fig. S1a). The TC motif was labeled with biarsenical reagent ReAsH (Fig. 1a) 23 . The TC-inserted SERCA1a ) showed slow Ca 2+ -ATPase activity (Fig. 2a), at ~ 20% of wild type. The ReAsH-labeled species formed EP in amounts comparable with that of wild type (Fig. 2b), due to slow EP isomerization, deduced from the reduced ATPase activity. Therefore, the TC insertion and ReAsH labeling, while exhibiting restrained isomerization, still permit the normal conformational changes of SERCA1a. Single ReAsH-attached SERCA1a molecules were embedded in single nanodiscs of POPC (1-palmitoyl-2-oleoyl-glycero phosphatidylcholine) 24 , and stabilized on the surface of Ni-NTA coated glass via two his-tags located in the nanodisc (Figs. 1a, 2c). Importantly, fluorescent images showed that the number of bright spots decreases as the concentration of labeled SERCA1a is reduced and the bright spots completely disappear (protein molecules detach) from the Ni-NTA glass with imidazole treatment (Fig. 2d,e and Supplementary Fig. S2), indicating that we are able to detect a signal from the ReAsH probe attached to the nanodiscembedded SERCA1a.
To observe the defocused image of the ReAsH attached to a single SERCA1a molecule, we constructed an isotropic total internal reflection fluorescence microscope (iTIRF) 21,25 . The evanescent field produced by the www.nature.com/scientificreports/ iTIRF contains all polarization components along x-, y-, and z-directions, thus single fluorophores are efficiently excited even if each fluorophore oriented in a different direction (Fig. 1c). To obtain the defocused image [19][20][21] , the objective lens was typically displaced 640 nm away from the best focal plane (Fig. 1d). Precise and stable displacements of the objective were achieved through the use of the perfect-focus system equipped with a Nikon TE2000E inverted microscope, and the relationship between the adjustment and the amount of displacement was independently calibrated by a combination of a 3D tracking method and a piezo electric stage (see "Methods" section). The change in the pattern of the defocused image describes the change in the zenith angle (θ) of the axis of the fluorophore, and the orientation of the fan-shaped pattern indicates the azimuth angle (φ) of that axis (for a definition of θ and φ see Fig. 1e). The θ and φ angles were derived from the matching algorithm for the defocused image 20 . Accuracy of orientation estimation of this algorithm is ~ 5° according to Ref 20 (see Supplementary Fig. S3).  Total EP at steadystate and E2P fraction. Microsomes expressing wild type or TCi-196/197 mutant were phosphorylated and EP was determined as described under "Methods" section (Left). There was no impairment of EP formation by ReAsH labeling (Right). Microsomes expressing wild type or TCi-196/197 mutant were labeled with ReAsH and EP was determined as described under "Methods" section. The values presented are the mean ± SD (n = 3-4). (c) ReAsH-labeled single SERCA1a molecule was stabilized on Ni-NTA coated glass surface via two his-tags incorporated into the membrane scaffold protein of the nanodisc. The angle of the nanodisc is not controlled because it has only two His-tags, not a three-point attachment. (d) SERCA1a concentration dependence of the number of observed bright pixels. To quantify the bright areas, mainly attributed to ReAsH, we calculated the area of bright pixels above the threshold value (inset). The number of bright pixels clearly depends on the concentration of the ReAsH-labeled SERCA1a embedded nanodisc (cf. Supplementary Fig. S2). (e) Fluorescent images of the glass surface (in the absence of protein), of the ReAsH-treated non-TC-tagged SERCA1a (WT), of the ReAsH-labeled SERCA1a ) and of the imidazole-treated ReAsH-labeled SERCA1a   Fig. 3b) and plotted the changes in angles in Fig. 3c. Note that we selected molecules which show the same intensity as that of a molecule bleached in a single step (cf. Fig. 4b) and which keep the same xy-position after infusion. This ensures that each fluorescence is derived from the same molecule and that it is a single molecule 26 . In the E2P analogous state (E2·BeF 3 − ), the θ angle shows a single population of 65.4° ± 14.3° (n = 49; Red Bars in Fig. 3d), suggesting that these SERCA1a-nanodiscs are likely bound to the glass at a particular angle (see "Discussion" section).

Angle change in the
We selected only the molecules that could transform from the E2P analogous state (E2·BeF 3 − ) to the E1Ca 2 state and ignored both the molecules unable to undergo the transition and the bright spots derived intrinsically from the glass surface (cf. bright spots in the image of Ni-NTA coated glass in Fig. 2e). To determine the angle distribution of a stabilized control molecule, thapsigargin (TG), a highly specific and subnanomolar affinity SERCA inhibitor, was perfused over the ReAsH labeled SERCA1a, which brings it to an extremely stable E2 (TG), a form analogous to the E2 state 27,28 . The fluorophores then showed random changes in angles before and after the solution exchange (n = 180; Fig. 3f,g). The width of this distribution, σ = 11.8°, was estimated by fitting to the derived histogram (Fig. 3g). Even though the molecules undergo perfusion, this narrow distribution indicates the stability of the angle of the fluorophore, and therefore the stability in the angle of the SERCA1a molecules and nanodiscs, ensuring that detection of a larger angle change must reflect angular changes in the A-domain associated with the SERCA1a conformational changes.
To select molecules that transform from the E2P analogous state (E2·BeF 3 − ) to the E1Ca 2 state, we defined that those molecules showing a significant change in angle, i.e., > 3σ of 35.4°, have undergone the conformational change. We found that 19.4% of molecules showed a significant change in angle between the E2P analogous state (E2·BeF 3 − ) and the E1Ca 2 state when compared to the TG bound molecules (n = 253; Fig. 3b,c) and used these for analysis. Here, we defined the ψ angle as the angle change derived from an inner product of two angles (see Fig. 3a). The ψ E2E1 angle between E2P (E2·BeF 3 − ) and E1Ca 2 was determined as 67.3° ± 12.3° (n = 49 molecules; red in Fig. 3c,e). This angle change is in excellent agreement with the expected value of 50°-60°, estimated from the crystal structures (PDB ID code: 3B9B for E2·BeF 3 − and 1SU4, 2C9M, 3J7T, and 5XA7 for E1Ca 2 ; see "Methods" section).  (e) Histogram of ψ S1S2 angle between the two dwell states [e.g., ψ S1S2 angle is calculated as 58.8° using cyan and orange points in (d)]. The calculated ψ S1S2 angle is 59.5° ± 24.3° (n = 11 molecules). Inset schematically shows the definition of the ψ S1S2 angle. www.nature.com/scientificreports/ subsaturating ATP concentration (0.01 µM, cf. the reported K d for MgATP binding to SERCA1a is 6.1 µM 30 ) and in the molecules stabilized by TG, confirming that this angle change is driven by Ca 2+ and ATP ( Fig. 4g-i, Supplementary Fig. S4). We selected the molecules that show obvious changes in angles with time and obtained the trajectory of θ and φ angles by analyzing the defocused images. Note again that these selected molecules show a single bleaching step or have the same intensity of a molecule that exhibits a single bleaching step (Fig. 4b), ensuring that each fluorescence is derived from a single molecule 26 . The trajectory of these molecules showed ~ 2 dwell states, separated along the φ axis (light cyan and light orange in Fig. 4c). The time course of the φ angle change and its histogram clearly indicate two distinct dwell states (Fig. 4d). Considering the major accumulation of ADP-sensitive EP at steady-state (E1P; compare the amount of total EP and E2P in Fig. 2b), we can assume that most molecules dwell in the E1PCa 2 state. Therefore, the longest dwell, termed State 1 (light cyan in Fig. 4c,d), comprises molecules mainly at E1PCa 2 (cf. Fig. 4j). Because of the reaction cycle rate constants of SERCA1a and the expected change in angle derived from crystal structures ( Supplementary Fig. S1b) 29 , the other dwell, termed State 2 (light orange in Fig. 4c,d; cf. Fig. 4j), should occur during E2P processing, i.e., E2PCa 2 → E2P → E2 (cf. Fig. 1b). The dwell time of State 2 was fitted by an exponential function of which the rate constant was ~ 0.2 s −1 (n = 6 molecules; Fig. 4f). This rate is comparable to the postulated rate of the E2P processing (see "Discussion" section). To compare the angle with the crystal structure, we calculated the change in angles between State 1 and State 2 as an inner product of ψ S1S2 using the θ and φ angles. The molecules forming the E1Ca 2 state show a stable angle with a narrow distribution of σ ~ 5° (Fig. 4h,i; (σ θ , σ φ ) = (2.9°, 5.5°)). Therefore, we defined the stable point as the frame in which the fluorophore remains at the same angle within ~ 3σ of 15° in each frame. The ψ S1S2 angle of 59.5° ± 24.3° (n = 11 molecules; Fig. 4e) was calculated using the points that show stable angles in θ and φ over 3 s (cyan and orange points in Fig. 4d). This is comparable to the estimated angle of ~ 60°, derived from the crystal structures of E1Ca 2 ·AlF 4 − ·ADP (an E1PCa 2 model, transition state analog of E1PCa 2 formation by ATP) and E2·BeF 3 − (PDB ID code: 1T5T and 2ZBE; see "Methods" section). To dissect the motion during the transition between State 1 to State 2, we increased the time resolution to 128 ms and 45 ms (Fig. 5a,f and Supplementary Video 2, 3). Note that this high-speed observation was performed within ~ 9 s (128 ms resolution) and ~ 4.5 s (45 ms resolution) because of the photo-bleaching limitation. Under these conditions, the whole ATP-turnover cycle is probably not completed and thus not observable, therefore, we chose only the molecules that showed obvious changes in the angle during observation, and this angle change should occur during the transition between these two states. In the presence of 100 µM ATP and 100 µM Ca 2+ , the trajectory also showed two distinct states (Fig. 5b,g), while the motion was completely absent at low ATP concentration (much lower than K d for ATP; 0.001 µM ATP and 1 µM Ca 2+ ) or in the presence of TG (Supplementary Fig. S4 and S5). The time course of the φ angle revealed back and forth motion between two states (Fig. 5c,h). When the angle changes from State 1 to State 2, the fluorophore stays at State 2 for ~ hundreds of msec, then goes back to State 1. The dwell times of States1 and 2 were fitted by an exponential function of which the rate constants are ~ 1.2 s −1 and ~ 4.8 s −1 in measurements at 128 ms resolution, respectively (n = 13 and 7 dwells from 3 molecules; Fig. 5e). The observed population of State 1 and State 2 are ~ 70% and ~ 30%, respectively, in measurements both at 128 and 45 ms resolution (Fig. 5d,i). These observations suggest the existence of a rapid transition in angle and so a rapid equilibrium of E1P and E2P-like states ( Fig. 5j; see "Discussion" section).

Discussion
In the last two decades, outstanding studies have revealed the 3D crystal structures of SERCA1a at atomic resolution. The structures of Ca 2+ -transport intermediates provide detailed insights into the conformation of the pump at specific points of the transport cycle. However, the dynamic processes underlying these conformational changes between intermediates are still unknown. Approaches estimating the precise angle of a single fluorescent dipole have enabled us to reveal the true behavior of working proteins in real-time at the single-molecule level 26,[31][32][33] . Even though the spatial resolution has limitation, the time course of the angular changes of a fluorophore, attached to a domain in a single molecule, can reveal an intermediate state 34 . Recently, there have been reports of single-molecule measurements of a P-type ATPase by Förster resonance energy transfer (FRET) 18 and time-resolved X-ray solution scattering (TR-XSS) combined with Molecular Dynamics simulation 35 . However, our work is the first report of the direct measurement of the angle change of the A-domain in SERCA1a in 3-D space, a critical conformational change in the transport cycle. It provides unprecedented and unequivocal evidence of a swinging A-domain during the structural transition. Further, the spatial and temporal resolution obtained during the dynamics of a working SERCA1a allow measurements of the rates of change and point to an equilibrium of substates in back and forth motion.
In measurements using ligand-stabilized molecules, the observed angle change between the E2P analogous state (E2·BeF 3 − ) and the E1Ca 2 state is in excellent agreement with the expected value estimated from crystal structures (Fig. 3e). Therefore, our experimental system is likely capable of measuring the structural changes in other reaction steps of a single SERCA1a molecule. Interestingly, the θ angle shows a single population in the E2P analogous state (E2·BeF 3 − ) (Fig. 3d). In this state, the cytoplasmic domains of the molecule are tightly gathered into a closed structure 17 . Therefore, if the angle between the fluorophore and the plane parallel to the nanodisc is almost constant, this θ angle represents the angle between the glass surface and the nanodisc. Even though the angle of the nanodisc is not controlled because it has only two His-tags, not three points, this single population with a small deviation of σ θ E2BeF = 14.3° (Red Bars in Fig. 3d) suggests that the nanodiscs bind to the glass at a particular angle.
In the experiment performed in the presence of Ca 2+ and ATP, we monitored only the ~ 1% of molecules that showed an obvious change in the angle of the fluorophore. The low percentage can be partially attributed to the selection criteria and to the slow EP isomerization compared to the observation time. However, it is possible this proportion may be improved upon. We did test the linker length for the Ni-NTA glass and chose the www.nature.com/scientificreports/ best linker, C 3 (~ 1.5 nm; cf. Fig. 2c inset), but there may be other approaches. In our conditions, namely in the presence of ATP and an oxygen scavenger, the fluorophore often showed changes in intensity over long periods of time. This may be caused by the motion of the fluorophore, because the intensity of the electromagnetic field generated by the iTIRF depends on the θ angle of the fluorophore and on the distance between the glass surface and the fluorophore. A frame of insufficient fluorescence intensity failed to yield estimates or estimated wrong angles from the matching algorithm, and was eliminated (Gray frames of reconstructed images in Figs. 4a,g, 5a,f, Supplementary Fig. S4a,d, S5a,d). This may warrant investigation to improve the proportion of active molecules. www.nature.com/scientificreports/ Indeed, the accuracy of the orientation estimation of defocus imaging is reported to be high 19,20 (accuracy of theta angle estimation was evaluated to ~ 6.3°; see Supplementary Fig. S3), and our measurement system too is highly precise. In the live imaging of the E1Ca 2 state molecule and that stabilized by TG there is a small distribution of σ θ = 5.7°-9.2° and σ φ = 6.7°-12.7° even under high-speed imaging ( Supplementary Fig. S4 and S5). These results indicate ~ 10° precision in our system, as well as showing the molecule is stabilized on the glass. This high accuracy and precision of our system allows for reliable detection of the angle change of the A-domain in EP isomerization under live imaging.
In the observation at 1 s resolution (Fig. 4), the trajectory of the angle change in the A-domain is consistent with the direction and track expected from the crystal structures ( Supplementary Fig. S1b Lower). The observed dwell states, State 1 and State 2, should occur mainly when E1PCa 2 is formed and during E2P processing, respectively ( Fig. 4j; see "Results" section). The calculated rate constant of the dwell-time in State 2 is ~ 0.2 s −1 (Fig. 4f). Considering that State 2 includes transformation of E2PCa 2 to E2P and E2P hydrolysis, this rate constant is consistent with the rate constant of E2P hydrolysis of 0.6-0.3 s −1 in the wild type 29,36,37 . The observed angular change between the two dwell states of ψ S1S2 = 59.5° ± 24.3° (Fig. 4e; see "Results" section), is close to the value expected from the crystal structures, which strongly supports the existence of the dwell states and the pertinence of the rate measurements for the catalytic cycle. Therefore, we can conclude that we succeeded in capturing the motion of a functioning SERCA1a molecule during the reaction cycle, which validates our observation system for measuring domain movements in live imaging.
Under turnover conditions in the observations at both 128 and 45 ms resolution (Fig. 5a-e, f-i, respectively), a frequent change in angle between State 1 and State 2 was detected with an obvious high rate constant of ~ 4.8 s −1 from State 2 ( Fig. 5e; compare the dwell 2 of E2P processing in Fig. 4f and dwell 2 of high-speed observation in Fig. 5e lower). This rapid change in angle was never detected at 0.001 µM ATP and 1 µM Ca 2+ , a condition under which the molecules can transform between E1Ca 2 and E2 states but cannot form the E1PCa 2 state, and where almost all molecules are in the E1Ca 2 state (Supplementary Fig. S5). Therefore, this rapid change in the angle of the A-domain likely occurs during EP isomerization. Since in our experimental condition E2P, the ADP-insensitive EP, is not accumulated in the steady-state (Fig. 2b left), we assigned this State 2 as E1P'Ca 2 . This State 2 molecule has an A-domain that is already largely rotated from State 1 (E1PCa 2 ) to a position close to the E2P state, yet seems able to move back rapidly to the State 1 (E1PCa 2 ) position, indicative of a rapid equilibrium, thus behaving as if it is a transient intermediate E2PCa 2 state (Fig. 5j). This interpretation is consistent with our previous finding in which a proteolytic analysis of E2PCa 2 , (captured and stable in the 4Gi-46/47 insertion mutant) indicated that in this state the A-domain is rotated and associated with the P-domain at the tryptic T2 site region (Arg198 on the Val200 loop) 7,8 . Furthermore, the populations of State 1 and State 2, in which the back and forth motion between two states is occurring, suggest E1P'Ca 2 has slightly higher free energy than E1PCa 2 . In the observations at both 128 and 45 ms resolution (Fig. 5a-e,f-i, respectively), the ratio of populations of State 2 to State 1 is ~ 0.4 (Fig. 5d,i), which, assuming equilibrium, yields an equilibrium constant of K eq ~ 0.4. This value is consistent with the rate constants of dwell-times 1 and 2 (compare the ~ 1.2 s −1 in Fig. 5e upper and the ~ 4.8 s −1 in Fig. 5e lower). According to the Gibbs relationship of ΔG 0′ = −RT·ln K eq , the Gibbs free energy of this reaction is only ΔG 0′ ~ 2 kJ/mol. This observation of frequent back and forth motion and the small difference in free energies suggest that the structural change of the A-domain occurs first, then there is the reduction of Ca 2+ affinity and gate-opening through the A-domain's three linkers with large free energy changes, and finally by releasing Ca 2+ the protein is transformed to the E2P ground state, abbreviated as E2PCa 2 [occluded] → E2PCa 2 (lumenally open highaffinity) → E2PCa 2 (lumenally open low-affinity) → E2P + 2Ca 2+ , see Ref. 38 . Thus, our live imaging of a single membrane molecule at work is important not only for further understanding of the Ca 2+ transport mechanism of SERCA but also for exploring the properties of other P-type ATPases.

Methods
Mutagenesis and expression. The pMT2 expression vector 39 carrying rabbit SERCA1a cDNA 40 with TC motif (Cys-Cys-Pro-Gly-Cys-Cys) inserted between Asp196 and Pro197 in the A-domain (TCi-196/197 mutant) was constructed as described previously 8 . Transfection of pMT2 DNA into COS-1 cells and preparation of microsomes from the cells were performed as described 41 . The monkey cell line derived from kidney, COS-1 (RCB0143), was provided by the RIKEN BRC through the National Bio-Resource Project of the MEXT/AMED, Japan. The labeled SERCA1a protein was inserted into nanodiscs using the following procedure. The ReAsH-labeled microsomes (0.224 mg mL −1 ) were incubated with 0.5 mg mL −1 native sarcoplasmic reticulum vesicles, 20 μM www.nature.com/scientificreports/ MSP1D1 (membrane scaffold protein 1D1) in 1.1 mM 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine (POPC), 10 mM CaCl 2 , 20 mM Tris/HCl (pH 7.5), and 10 mg mL −1 octaethylene glycol monododecyl ether (C 12 E 8 ) on ice for 30 min. Then the nanodiscs containing a ReAsH labeled or a native SERCA1a were reconstituted, and this mixture was purified by size exclusion column chromatography as described previously 24 . Both the ReAsH-labeled SERCA1a and the native SERCA1a embedded nanodiscs were infused into the flow chamber for single-molecule imaging, and fluorescence from only the ReAsH-labeled molecules was observed.

ReAsH-labeling of SERCA1a microsomes and preparation of nanodisc containing a single labeled Ca
Ca 2+ -ATPase activity. Ca 2+ -ATPase activity of expressed SERCA1a was obtained essentially as described previously 43 . The rate of ATP hydrolysis was determined at 25 °C in a mixture containing 1 μg of microsomal protein, 0.1 mM [γ-32 P]ATP, 0.1 M KCl, 7 mM MgCl 2 , 10 μM CaCl 2 , 1 μM A23187, and 50 mM MOPS/Tris (pH 7.0). The Ca 2+ -ATPase activity of expressed SERCA1a was obtained by subtracting the ATPase activity determined in the presence of 1 μM TG, a highly specific and subnanomolar affinity SERCA inhibitor with conditions otherwise as above.
Total amount of EP (total EP, sum of E1P and E2P) at steady state and E2P fraction. Phosphorylation of SERCA1a in microsomes with [γ-32 P]ATP was performed essentially as described previously 37  Precipitated proteins were separated by 5% SDS-PAGE at pH 6.0 according to Weber and Osborn 44 . The radioactivity associated with the separated Ca 2+ -ATPase was quantified by digital autoradiography as described 45 . The amount of EP in expressed SERCA1a was obtained by subtracting the background radioactivity determined in the presence of 1 μM TG, with conditions otherwise as above. We confirmed that 1 μM TG reduces the EP value in the wild type and all mutants to a background radioactivity level (i.e., 1% of the maximum EP level, which is the same as that obtained in the absence of Ca 2+ without TG).
Observation of the ligand-stabilized SERCA1a. E2·BeF 3 − formation was induced by a method based on that previously described 49  buffer containing the oxygen scavenger system. E1Ca 2 formation was induced by the following method: The SERCA1a embedded nanodiscs were incubated for 5 min at 23 °C in observation-buffer containing 20 mM CaCl 2 and the oxygen scavenger system. E2(TG) formation was induced by the following method: The SERCA1a embedded nanodiscs were incubated for 5 min at 23 °C (for Fig. 3f,g) or at 5 °C (for Supplementary Fig. S4) in observation-buffer containing 1 µM CaCl 2 , 100 µM ATP, 3 µM thapsigargin, and the oxygen scavenger system. Microscopy and defocused imaging. A ReAsH-attached SERCA1a was visualized under an inverted microscope (TE2000E; Nikon Instruments) equipped with a 100 × objective lens (Apo TIRF 1.49 N.A.; Nikon Instruments), a 532-nm laser (JUNO532-800; Showa Optronics) with custom-made dichroic mirror to keep the laser polarization after reflection (Chroma), two emission filters (NF03-532E & NF01-532U; Semrock), an EMCCD camera (iXon + DU897; Andor), a highly stable customized stage (Chukosha), and an optical table (Newport). The detailed optical setup was described previously 21,25 . All systems were set into a compartment (Nihon Freezer) under which the temperature was stabilized at 23.0 ± 0.1 °C or 5.0 ± 0.1 °C by PID regulation with a heater and a cooler.
The isotropic TIRFM was constructed using a diffractive diffuser (D0740A, Thorlabs) based on the method reported previously 21,25 . To observe defocused images, the distance between the objective lens and the specimen www.nature.com/scientificreports/ was kept constant by the perfect focus system (Nikon Instruments) 21 . The custom-made piezoelectric stage (P-620.ZCL; Physik Instrumente GmbH & Co) was used for calibration of the exact distance 50 .
In the observation performed at 5 °C, to decrease the aberration, the objective lens was typically placed ~ 640 nm closer beyond the best focal plane, and the correction collar was set at the position fully turned counterclockwise.
Analyzing the defocused image. Each defocused image was analyzed using 3D steerable filters, based on the method reported previously 20,21 . To solve the equation for estimation of dipole orientation (see Appendix A1 in Ref 20 ), we used the custom software reported in the Ref 21 . Parameters for determining the optical path difference 51 used the following values: Refractive index of glass is 1.526; Refractive index of immersion is 1.518; Thickness of glass is measured by the micrometer (MDC-25MX, Mitutoyo; typically 0.144 mm); Design value for thickness of glass is the value of the objective correction collar (typically 0.150 mm). Note that we take into consideration the effect of the glass equipped in the iXon EMCCD camera.
Evaluation of the theta angle estimation was performed by the following method. The diffraction patterns were rendered by the method reported previously 20 . Poisson noise was generated using RandomJ plugin for Image J and added to the simulated patterns. RMSE (root mean square error) was calculated by Igor Pro (WaveMetrics). Orientation estimation of θ angle was performed under the condition in which the φ value was fixed at 0°.

Data availability
The data that support the findings of this study are available from the corresponding authors upon reasonable request.