Measurements of complex refractive index change of photoactive yellow protein over a wide wavelength range using hyperspectral quantitative phase imaging

A novel optical holographic technique is presented to simultaneously measure both the real and imaginary components of the complex refractive index (CRI) of a protein solution over a wide visible wavelength range. Quantitative phase imaging was employed to precisely measure the optical field transmitted from a protein solution, from which the CRIs of the protein solution were retrieved using the Fourier light scattering technique. Using this method, we characterized the CRIs of the two dominant structural states of a photoactive yellow protein solution over a broad wavelength range (461–582 nm). The significant CRI deviation between the two structural states was quantified and analysed. The results of both states show the similar overall shape of the expected rRI obtained from the Kramers–Kronig relations.

Light-matter interaction is key to the characterization of a sample and is performed using a diverse range of existing methodologies, including visual inspection, scattering analysis, microscopy, and spectroscopy. The light-matter interaction represents the composite of the electric and magnetic responses of material. It can be quantified by a single parameter called the complex refractive index (CRI) described by the expression

Results
Experimental procedure. In order to obtain the CRI of PYP, we first measured the multi-wavelength light field images of a 100-μm-diameter polymethyl methacrylate (PMMA) microsphere immersed in PYP solution (Fig. 1). The illumination wavelength for the optical field measurement (probe beam) is systemically scanned over a broad visible spectral range. The centre wavelength and bandwidth of the probe beams are defined by the prism, lens, and pinhole used in the illumination 12 . For optical field measurements, a quantitative phase imaging unit (QPIU) 49 was implemented. The QPIU is a common-path full-field interferometer that uses the principle of lateral shearing interferometry (see Supplementary Fig. S1 online for the detailed optical setup). The measurements were performed with and without the pump beam (445 nm peak; M455L3, Thorlabs Inc.) to measure the CRI of both the pG and pB states. To ensure state equilibrium of the PYP solution, we included sufficient idle time (>10 seconds) after switching the LED on or off before taking holograms. For each state, we obtained holograms at eleven different wavelengths ranging from 461 to 582 nm. The wavelengths were carefully selected to fully characterize the CRI of the PYP solution (see Supplementary Table S1 and Fig. S2 for detailed probe beam specifications). The nonlinear effects from the probe beam were negligible, and the intensities of probe beams were normalized in the data analysing process.
From each measured raw holographic image (Fig. 1b), the amplitude and phase images of the immersed microsphere were obtained with a conventional field retrieval algorithm (Fig. 1c,d) 50 . The retrieved light field images were converted into angle-resolved light scattering plots using FTLS, as shown in Fig. 1e. In FTLS, the measured optical field of a sample is numerically propagated to the far-field, which creates a direct 2D Fourier Scientific REPORtS | (2018) 8:3064 | DOI:10.1038/s41598-018-21403-z transformation of the optical field information of the sample. The angle-resolved light scattering plots are achieved by azimuthally averaging the 2D light scattering patterns, which greatly increase the signal-to-noise ratio (SNR) of the measurements. As the spatial analogous to Fourier-transform infrared spectroscopy, FTLS provides an unprecedented SNR in measuring scattered light signals owing to Fellgett's advantage 51 . The azimuthal averaging is possible because the imaging target (a microsphere) is azimuthally symmetric. Then, the CRI was extracted by fitting the obtained FTLS results to the Mie scattering theory, which is the exact solution to Maxwell's equations for light scattering from homogeneous spheres 52,53 . CRI of PYP solutions in pG and pB states. The CRI values of both the pG and pB states of the PYP solution are shown in Fig. 2. The precision or standard deviation of the proposed method at each wavelength is depicted by the error bars. The mean precisions of the rRI and iRI are 8.2 × 10 −5 , and 4.2 × 10 −5 , respectively. The decrease in accuracy with increasing wavelength is caused by the bandwidth widening of the probe beam, which reduces the interference efficiency.
The iRI of the PYP decreases monotonically as the wavelength increases ( Fig. 2a) and converges to zero for wavelengths longer than 500 nm. The iRI values in the presence of the pump beam (pump−on case) are approximately five times smaller than those of the PYP in the absence of the pump beam (pump−off case). The significant decrease in the iRI in the pump-on case indicates a PYP population transition from the pG to the pB state. Since the absorbance of pB is negligible for the current wavelength range (see Supplementary Fig. S3), we deduce that the non-zero iRI in the pump-on case is caused by the presence of a pG population. Therefore, the molecular density or concentration of pG (ρ pG ) can be determined simply by where λ b is the wavelength of the probe beam and ε is the known molecular extinction coefficient of pG. The ρ pG were measured as 3.44 ± 0.1 mM and 0.56 ± 0.1 mM for the pump−off and pump−on cases, respectively (Fig. 2a). Thus, the concentration of the pB state is found to be 2.88 ± 0.1 mM, and the pB population ratio (R pB = amount of pB/amount of pG) is 0.837 ± 0.035. Because a continuous light source was used as the pumping source, R pB = 1 is not achievable in this two-state equilibrium system 54 . A previous report that used LEDs for continuous illumination showed a similar R pB 55 .
The rRI values of the PYP decrease monotonically as the wavelength increases (Fig. 2b), which is a general phenomenon occurring in protein solutions due to the rRI of water. For a wavelength ranging from 461-582 nm, the rRI in the pump−off case decreased from 1.3518 to 1.3452, and the rRI in the pump−on case decreased from 1.3510 to 1.3450. The pump−on case shows lower rRI values than the pump−off case for the entire wavelength range. Unlike the iRI values, the quantitative rRI values have not been reported previously. Therefore, we verified the measured rRI values using a conventional refractometer (R-5000, ATAGO Co., Ltd). The verification was conducted with an identically prepared PYP solution in the pump-off case. The rRI values cannot be measured Similar to the iRI in Eq. (2), the rRI is also a function of molecular density. In order to characterize the PYP independent of the molecular density, we calculated the refractive index increment; that is, the density derivative of the real RI ( rRI/ ρ ∂ ∂ ). The refractive index increment of the PYP in the pG and pB states can be calculated individually using the linear equation: where ρ pG and ρ pB correspond to the molecular densities of the pG and pB states in the pump-on or pump-off cases, as denoted by the subscript; pG α and α pB are the refractive index increments ( rRI/ ρ ∂ ∂ ) of the PYP in the pG and pB states, respectively; − rRI pump off and rRI pump on − are the rRI values of the PYP solution in the pump-off and pump-on cases, respectively; and rRI H O 2 is the known rRI of distilled water 56 . The results are shown in Fig. 2c and tabulated in Supplementary Table S2. The difference between α pG and pB α was maximized (approx. 0.3 M −1 ) at 470 nm, and decreased as the wavelength increased.
The pB population ratio (R pB ) of a state is related to the kinetics of the transition between the pG and pB states of the PYP. The relaxation time τ of the pB state to pG state can be obtained from the measured R pB with the following equation (see Supplementary Information for details) 57,58 .  where h is Planck's constant, N A is Avogadro's constant, λ P is the wavelength of the pump light, φ is the photocycle quantum yield of PYP, and ∂I P /∂λ P is the spectral density of the pump beam. Inserting φ = 0.35 from the literature 59,60 and ∂I P /∂λ P (see Supplementary Fig. S2) and R pB = 0.837 ± 0.035 from the measurement, τ was calculated to be 77 ± 27 ms. We note that the relaxation time determined here is smaller than those reported by typical time-resolved pump-probe experiments (0.15-2 s) 44,61,62 . The discrepancy may be related to the different modes of data collection (continuous illumination vs. pump-probe), but the exact origin is not clear at this stage. The accuracy of this calculated τ is mainly determined from the uncertainty of R pB due to the high R pB sensitivity of τ in Eq. (4).

Discussion
The deviation in the iRI between the two structural states of the PYP is to be expected caused by the well-known different extinction coefficients between two states. The rRI, however, is a quantity that relates to the mass or density of the material 63,64 . Because only conformational change, not mass variation, occurred when the pump beam was turned on or off, the deviation in the rRI can be considered to arise from density variations resulting from the conformational change.
To explain the results of these measurements, we employ the K-K relations. The K-K relations connect the rRI and iRI based on the causality of the response functions. The K-K relations allow for calculation of the rRI from the iRI by Here, ω is the angular frequency of light. Although the K-K relations do not provide an exact quantitative solution due to their inherent infinite-integral form, we are still able to obtain qualitative trends for the rRI from the well-known values of the iRI. The calculated results are shown in Fig. 3. The overall shape of the expected rRI obtained from the K-K relations matches well with the experimental results of both the pump-on and pump-off cases and shows the largest deviation at a wavelength of 470 nm, in agreement with our result. In order to retrieve the reliable rRI results from the K-K relations, we find that the nearest absorption peaks should be considered at least. In this works, we expect iRI measurements over 250 nm -2,000 nm is required for reliable rRI results, regarding the second absorption peak of PYP (280 nm) 65 and water absorption peaks in infrared regime 66 .
To provide a more intuitive explanation for the rRI deviation resulting from the molecular structure change, we introduce the concept of atomic refraction (AR). Historically, AR studies have tabulated the contribution of individual atoms and atomic bonding to the rRI of a molecule, and in this way have been able to closely predict the rRI of unknown chemicals [67][68][69] . By considering the AR as a microscopic version of our preconception about the relationship between the rRI and density/mass, the deviation in the rRI resulting from molecular structure changes can be explained. Further, the AR implies that the rRI results may help to reveal atomic bonding changes that occur during the protein conformational change. The relation between the CRI and electromagnetic susceptibilities [Eq. (1)] 70 should be emphasized once again. Because the susceptibilities of molecules strongly relate to their electric and magnetic dipole moments, measurements of the CRI can provide clues to protein structure. For example, Tamasaku et al. visualized the electron cloud distributions of diamonds with a resolution of 0.54 Å by using extreme-ultraviolet (103 Å-206 Å) light, and the non-linear susceptibility relationship between X-rays and the chosen frequencies 71 . However, in the current study, the CRI measurements of PYP proteins were performed in solution which makes it challenging to directly translate our CRI measurements into structural changes in the PYP. This is because proteins in solution have arbitrary orientations, which result in the smoothing of directional information. When the directional-and/ or polarization-dependent CRIs or the electromagnetic susceptibilities of proteins are measured systematically, they have the potential to provide more useful information on the structure of proteins.

Conclusions
In this work, we presented a method to precisely and quantitatively measure the CRIs of photoactive proteins and their excited states over a wide range of wavelengths. Using a QPI equipped with a wavelength-sweeping source and FTLS, the CRIs of the PYP solution were measured for wavelengths ranging from 461-582 nm.
We found a significant difference in the CRI values of PYP for the absence of a pump beam (pump-off case) and the presence of a pump beam (pump-on case) as a function of wavelength; not only for the iRI, but also for the rRI. We retrieved the refractive index increment values of PYP for both the pG and pB states. The maximum difference between pG α and pB α is approx. 0.3 M −1 at 470 nm. We also explained the reason for the unexpected deviation in rRI by employing the K-K relation and atomic refraction. We expect the extension of measurable wavelength in the UV regime will help the direct examination of the CRI changes in pB states.
The present method measures both the rRI and iRI values of photoactive proteins simultaneously over a wide wavelength range, and it will be useful for real-time measurements as well as adding to the body of comprehensive, precise, and quantitative information on light-matter interactions. We also expect the present method to see widespread application in measuring the CRIs of photoactive proteins in various fields, including structural biology, chemistry, medical science, and pharmacy. Furthermore, precise measurement of the CRI can provide insight into the application of photoactive proteins in the field of material science and optics.

Methods
Sample preparation. The purified PYP solution (see Supplementary Information) was mixed with PMMA microspheres having a diameter of 100 μm (74214 FLUKA, Sigma-Aldrich, Inc.) that had been washed three times with distilled water. Ten microliters of the mixture were sandwiched between the coverslips, and the edges were sealed with epoxy adhesive.

CRI extraction by Mie theory fitting.
To determine the CRI of the PYP solution, we performed nonlinear fitting of the measured FTLS signals using the Mie theory (Fig. 1e). The solution describes the scattering of an electromagnetic plane wave by a homogeneous sphere. We used three fitting parameters: the rRI (n) and the iRI (κ) of the PYP solution, and the diameter of the PMMA microspheres. For a robust and automated fitting analysis, the angle-resolved light scattering signals were prepared as a function of θ n sin instead of θ in order to ensure the same scale for the horizontal axes while adjusting the rRI (n) value because light scattering is dependent on refractive index. During this experiment, we took a photographic image of the microsphere and its surroundings using a CCD camera. We discovered that moving further away from the microsphere's centre caused the thickness of the PYP solution to increase, causing the CRI to change in response.
Due to the highly oscillatory features of the angle-resolved light scattering curves (Fig. 1e), the direct application of conventional fitting algorithms either shows non-convergent results or is dependent on the initial fitting parameters. Instead, we used a process prior to the fitting processes to find appropriate initial fitting parameters by minimizing the total variance of the extreme positions between the measurement and the Mie theory. These initial parameters yield highly reproducible fitting results. Thereafter, we used a nonlinear fitting algorithm (nlinfit, a built-in function of MatLab TM ) with these predetermined initial fitting parameters. We expect the fitting method can be further simplified by employing global optimization algorithms such as genetic algorithms 72 .
PMMA microspheres CRI calibration. Mie scattering is highly dependent on the CRI of both the surrounding medium and the homogeneous sphere. In order to measure the CRI of the PYP solution, the CRI of the PMMA microspheres used should be well-known. However, we found that the CRI of polymers such as PMMA can slightly vary product-by-product, depending on the manufacturing procedure. Therefore, the CRI of the PMMA microspheres should be calibrated beforehand.
For the calibration, we used an identical experimental procedure with PMMA microspheres immersed in distilled water, whose CRI value is known 56 . The fitting parameters were the rRI, the iRI, and the diameter of the PMMA microsphere. The calibrated CRI of the PMMA microsphere was used for the known parameters in the PYP solution CRI measurements.