DFT-based Raman spectral study of astaxanthin geometrical isomers

Highlights • 10 different astaxanthin (AST) geometrical isomers were analyzed by Raman spectroscopy.• Raman spectroscopy can distinguish all the geometrical isomers of AST molecules.• Structures and vibrational modes of all the AST isomers are obtained due to DFT calculations.


Introduction
Astaxanthin (AST) is a carotenoid widely distributed in nature, found in salmon, shrimp, crab and alga. AST is known as a super antioxidant, which can effectively remove free radicals from the body, and promote a variety of physiological functions, such as suppressing oxidation, lowering lipid and enhancing immunity activities (Liu and Huang, 2016;Visioli and Artaria, 2017). Therefore, it has recently attracted a lot of attention in both scientific research and applied areas (Ambati et al., 2014;de Bruijn et al., 2016;Honda et al., 2021a;Honda et al., 2021b;Visioli and Artaria, 2017;Yang et al., 2021;Yu and Liu, 2020). Currently, AST products have been commercially used in safe food additives, nutritional products, cosmetics, aquaculture (Ambati et al., 2014;Yang et al., 2021). The main source of human commercial food AST is extracted from algae and aquatic crustaceans.
Since AST has been widely studied and applied, it is important to study whether there are specific functional differences between AST isomers . It is now recognized that the geometrical configuration of AST isomers is closely related to the functional performance and biological activity of AST (Gong et al., 2020;Su et al., 2020;Visioli and Artaria, 2017;Yu et al., 2021;Yu and Liu, 2020). For example, during in vitro digestion, the 13-cis shows higher bioavailability than 9-cis and all-trans ASTs, and 9-cis shows higher transport efficiency than all-trans and 13-cis (Visioli and Artaria, 2017;Yang et al., 2017). Based on AST fed to male rats, it was supported that the cis-AST isomers, the 13-cis have greater bioavailability and tissue accumulation efficiency than the all-trans-isomer (Honda et al., 2021b). The cis isomers of AST show much higher antioxidant efficacy than all-trans isomer in vitro (Liu and Osawa, 2007). The cis-AST, especially 9-cis AST exhibits greater anti-inflammatory effect than all-trans isomer (Yang et al., 2019). The 9-cis and 13-cis ASTs show a higher protection against oxidative stress than all-trans AST against oxidative stress (Yang et al., 2017). The 9-cis has higher storage stability than the other cis-isomers in oils and fats, because during storage, the 13-cis and 15-cis isomers will be isomerized into all-trans isomer during the storage but 9-cis AST is barely isomerized (Honda et al., 2021a). It is known that isomers of other carotenoids also exhibit different biological properties. For example, it is reported that the mono-cis configuration (15-cis) of neurosporene is bound to the reaction center, while the all-trans-neurosporene is bound to the light-harvesting complex (Koyama et al., 1988;Koyama et al., 1982).
In this regard, therefore, more in-depth and detailed research is required to verify the existence of isomerization and distinguish the structural differences of the isomers. Previous studies used analytical methods such as Nuclear Magnetic Resonance (NMR) and High Performance Liquid Chromatography (HPLC) to detect and study geometry isomers of AST (Buchwald and Jencks, 1968;Euglert and Vecchi, 1980;Holtin et al., 2009;Osterlie et al., 1999;Zajac et al., 2018). For example, High-performance liquid chromatography-mass spectrometry (HPLC-MS) and NMR spectroscopy were applied to separate the components of the complex AST and distinguish the all-trans, 9-cis and 13-cis isomers (Holtin et al., 2009;Osterlie et al., 1999). X-ray diffraction was used to resolve crystal structures of unbound, chloroform solvate, pyridine solvate of all-trans AST (Bartalucci et al., 2007). However, these techniques usually require considerable time, sample volume and cost for the inspection and analysis. More importantly, these tools can hardly be applied to the in vivo identification of AST isomers in living cells. For example, to clarify the bioavailability and tissue accumulation efficiency of AST isomers, current researchers could only evaluate the AST isomer contents by HPLC in vitro after the tested rats were killed (Honda et al., 2021b).
If different astaxanthin isomers can be traced in a living body, it is certain that the properties and biological significance of the different isomers can be studied more deeply, so as to expand the application of AST in food science, nutrition, medical, aquatic feed, etc. In recent years, Raman spectroscopy as in vitro and in vivo detection technology, has also been introduced to detect and study AST, mainly all-trans isomer (Huang et al., 2010;Liu and Huang, 2016;Shao et al., 2019;Sharma et al., 2015;Subramanian et al., 2014). However, there are still many problems to be solved in the structure and Raman analysis of AST, which restrict the better application of Raman spectroscopy on AST. Since AST is also a type of carotenoids, there are many similarities in the vibrational spectra of AST and other carotenoids (such as typical beta-carotene). The Raman spectrum of β-carotene exhibits the main vibrational bands at about 1512, 1156, 1006 and 955 cm − 1 which represent the ν 1 -C--C-, ν 2 -C-C-, ν 3 -CH-rocking vibrations of -CH 3 groups attached to the polyene chain coupled with -C -C-bonds and ν 4 -C-H out-of-plane wagging vibrations, respectively (Macernis et al., 2021;Saito and Tasumi, 1983;Streckaite et al., 2020). When the structure of the functional group changes, the corresponding characteristic bands will also change. For instance, the different Raman ν 1 band shifts were found in the β-carotene conformers with the different β-ring dihedral angles and conjugated chain lengths (Macernis et al., 2021;Macernis et al., 2014;Streckaite et al., 2020). It is clear that the vibrational spectral signals are very sensitive to the molecular structure. The AST molecule and β-carotene have the same conjugated polyene chain but different substitution pattern in the ionone ring, since AST has an hydroxyl group (OH) at the C3 site and a carbonyl group (C--O) at the C4 site. Moreover, the AST has extended conjugation into the carbonyl group of ending ring. Therefore, AST and β-carotene should have many different features on the vibrational modes and spectral signatures (Requena et al., 2008;Saito and Tasumi, 1983;Subramanian et al., 2014). However, to the best of our knowledge, there was no systematic quantum chemistry theoretical work to study Raman spectra and vibrational modes of the cis-trans isomers of AST. In the previous studies, when people analyzed the experimental Raman spectrum of AST, they often referred to the vibrational mode assignments of β-carotene, which could lead to some inappropriate structural and spectral analysis (Requena et al., 2008;Subramanian et al., 2014). It was reported that only six astaxanthin isomers could be distinguished from Raman spectroscopy, but ten geometric isomers have been reported (Subramanian et al., 2014). As the intensities of Raman bands of astaxanthin are extremely strong, whether this is caused by resonance Raman is also controversial (Bergamonti et al., 2011;Meinhardt-Wollweber et al., 2018;Subramanian et al., 2014). For the same Raman peak, different isomers sometimes have significantly different linewidths, which has not been reasonably explained. The reason why different isomers have different proportions in organisms has not been explained, either. For instance, all-trans configuration is roughly approximated as 70%, 9-cis and 13-cis are about 10%, the other isomers are less than 2% . The experimental crystal structure of all-trans isomer of AST was obtained through X-ray diffraction (Bartalucci et al., 2007). However, the geometric structure parameters of the cis-isomer of astaxanthin have not been reported. It is noteworthy that Raman spectroscopy technique has now been used for the detection of all-trans astaxanthin in living cells . If the spectral problems could be solved, people may be able to achieve the identification and imaging of all geometric isomers of AST in living organisms, such as cells, living tissues, algae. Therefore, to address the above questions, this present work aimed to identify all-known geometrical isomers of AST and solve the problems in structure and Raman spectroscopy. For this purpose, we employed quantum chemistry methods and Raman spectra to systematically study the structural and spectral properties of AST isomers. The structures and Raman spectra of all geometric isomers of astaxanthin were therefore systematically explored and interpreted.

Materials and methods
The all-trans isomer of AST was purchased from Aladdin Bio-Chem Technology Co., LTD, Shanghai, China. The 9-cis and 13-cis isomers of AST were purchased from ZZBIO Co., LTD, Shanghai, China.
For the Raman spectroscopy measurement, the samples were placed on a quartz plate to collect the Raman spectra, and the Stokes Raman lines were recorded in the region of 200-3700 cm − 1 using XploRA Raman spectrometer (HORIBA JOBIN YVON, Paris, France) with a 785 nm laser. The laser power on the sample was approximately 1.2 mW, and exposure time was 5 s. The spectra were measured at least three times.
For the computation of the Raman spectra of AST isomers, density functional theory (DFT) was employed using Gaussian 09 packages (Frisch et al., 2009). All calculations were performed by applying the B3LYP functionals, with the hybrid of Becke's nonlocal three parameter exchange functional and the Lee-Yang-Parr correlation functional. The triple-zeta 6-311 + G(d,p) split valence-shell basis set was used, which was augmented by d polarization functions on heavy atoms and p polarization functions on hydrogen atoms as well as diffuse functions for heavy atoms (Perdew et al., 1996). The geometries were optimized without any constraint and the optimized geometries do not have imaginary frequencies. Then the Raman spectra were simulated with a resolution of 8 cm − 1 at the same theories. The vibrational frequency scaling factor for B3LYP/6-311 + G (d,p) is 0.9808 (Laury et al., 2012). The calculated Raman activities were converted into Raman intensities by using the following relation derived from the basic theory of Raman where ν 0 is the exciting frequency (in cm − 1 ), ν i is the vibrational frequency (in cm − 1 ) of the ith normal mode, h, c, and k are fundamental constants, and f is a common normalization factor for all band intensities. The assignments of vibrational models and their potential energy distributions (PEDs) were obtained using the Gaussview 5.0.8 program and VEDA 4xx program (Dennington et al., 2009;Jamróz, 2013).
The initial geometry structure of all-trans AST was derived from the crystal structure data of unbound all-trans AST which is the mixture of 3S,3 ′ S: 3R,3 ′ S: 3R,3 ′ R in a 1:2:1 ratio (Bartalucci et al., 2007). By statistical probability when two chiral hydroxy group are added to the same molecule the result is 1:2:1 for the SS, RS and RR isomers. This AXT sample consists of the R and S form ionone ring in equal amounts. Therefore, the chirality of crystal structure data is represented by 3R,3 ′ S ( Bartalucci et al., 2007). In the structural optimization calculation of this work, the calculated structures are all with 3R,3 ′ S chirality, and the initial input structure of all-trans AST is from the crystal data (Bartalucci et al., 2007). The experimental crystal structure data of other geometry isomers of AST was not found. Therefore, the initial input structures of Fig. 1. The optimized geometry structure and atom labeling of all-trans, three mono-cis isomers and six di-cis isomers of astaxanthin molecules. the cis ASTs in geometry optimizations are constructed based on the crystal data of all-trans AST.
Astaxanthin (3,3 ′ -dihydroxy-β,β ′ -carotene-4,4 ′ -dione, C 40 H 52 O 4 ) consists of a nonpolar central conjugated polyene chain and two polar cyclized end groups (hydroxyl and keto groups). The existence of multiple carbon-carbon double bonds makes AST have multiple cis-trans isomers. As shown in Fig. 1, all-trans-AST can be isomerized in to monocis forms, such as 9-cis, 13-cis, 15-cis, and 6 kinds of di-cis isomers, including 9,9 ′ -di-cis, 9,13-di-cis, 9, 13 ′ -di-cis, 9,15-di-cis, 13,13 ′ -di-cis, 13, 15-di-cis which were found in plants and animals (Euglert and Vecchi, 1980;Yu and Liu, 2020). The optimized molecular structures of 10 known AST isomers are illustrated in Fig. 1. The calculated structure parameters of all-trans AST are compared with the experiment ( Bartalucci et al., 2007), involving the bond, lengths and dihedral angles in Table S1. From Table S1, it is clear that the experimental and theoretical calculated structural parameters are very close, and the RMSD value is extremely small. The RMSD value of the calculated bond lengths with respect to experiment is only 0.016 Å. The RMSD values of the calculated angles and dihedral angles with respect to experiment are only 1.1 • and 4.4 • , respectively. This also means that this level of theoretical calculation is suitable for the structural simulation of AST, and the calculated structural parameters of other nine isomers may also be very valuable.

Interpretation of Raman spectrum of all-trans isomer
Since the proportions of all-trans, 9-cis and 13-cis AST isomers are usually more than 5% , the functional performance and biological activity of these isomers are more broadly studied (Honda et al., 2021b;Visioli and Artaria, 2017;Yang et al., 2017). The standard all-trans, 9-cis and 13-cis AST are commercially available. The experimental Raman spectra were measured using 785 nm laser, which are shown in Fig. 2. Similar to the Raman spectrum of β-carotene, the main Raman bands of AST at about 1520 cm − 1 , 1160 cm − 1 and 1006 cm − 1 are noted as the ν 1 , ν 2 and ν 3 bands, respectively (Macernis et al., 2021;Saito and Tasumi, 1983;Streckaite et al., 2020). The Raman bands of different isomers in the ν 2 region are very different from each other, for there are big differences in peak wavenumber and intensity. For the ν 1 and ν 3 bands, the difference is mainly at the change of full width at half maxima intensity (FWHM). The different spectra of these three AST isomers were analyzed by combining experiments with theoretically calculated spectra.
The AST molecule contains 96 atoms, therefore, it has 282 vibrational modes. The calculated Raman activity intensities of vibrations of ten isomers are list in the Tables 1 and 2, and they are also shown in Figs. 2 and 3 in blue lines. The calculated Raman activity intensities can be converted into the Raman intensities by the theoretical formula of Raman scattering using the wavelength of laser in the experiment (Novikov et al., 2022). The calculated Raman activity intensities of 16 vibrational modes of all-trans isomer exceeds 10000, and the Raman activity intensities of 237 vibrational modes is less than 1000. As shown in Fig. S1 in Supporting Information, the Raman activity intensities of vibrational modes below 800 cm − 1 and above 1800 cm − 1 are very weak. These weak vibrational modes are relatively difficult to be observed on both experimental and calculated Raman spectra. Therefore, only Raman bands in the 800-1800 cm − 1 range can generally be observed in the experiment. In this work, the vibrational modes of strong Raman bands are assigned corresponding to the experimental spectra. And as shown in Tables 1 and 2, most of the vibrational modes and PEDs of these ten isomers are different. But most of previous work used the vibrational modes of β-carotene to analyze the Raman spectra of different isomers of AST, which seriously affect the correct analysis of astaxanthin Raman spectroscopy, especially in-situ Raman analysis in vivo (Requena et al., 2008;Subramanian et al., 2014).
The Raman spectrum of all-trans AST, which is the most common isomer in nature, is firstly discussed. The experimental and calculated Raman spectra are shown in Fig. 2, while the wavenumbers, intensities and assignments of the recognizable Raman bands are shown in Table 1. The structure of all-trans AST molecule has Ci symmetry, therefore, the distributions of the vibrational modes are also centrosymmetric. In the 1500-1700 cm − 1 regions, the bands are mainly from the stretching vibrational modes of carbon-carbon double bonds. In the Fig. 2 and Table 1, the most intense ν 1 band at 1519 cm − 1 is from the symmetry stretching of carbon-carbon double bonds of the whole polyene chain, the PED of C13 = C14, C13 ′ =C14 ′ and C15 = C15 ′ stretching vibrations are above 10 percent, and the PED of C11 = C12, C11 ′ =C12 ′ , C9 = C10 and C9 = C10 ′ are less than 10 percent. It should be noted that there may be a very weak shoulder peak covered in this strong band, whose simulated wavenumber is 1535 cm − 1 , comes from the asymmetric stretching of C15 = C15 ′ , C13 = C14 and C13 ′ =C14 ′ . In the 800 to 1500 cm − 1 regions in Fig. 2, the bands are mainly from the stretching vibration of C -C single bonds, and bending of C -H bonds. In the ν 2 region, the moderate intensity band at 1282 cm − 1 is mainly from the rocking vibration of C11-H and C11 ′ -H bonds. There is one shoulder band from the bending of C3-O20-H and C3 ′ -O20 ′ -H angles. The weak band at 1218 cm − 1 is from the stretching of carbon-carbon single bonds of the polyene chain, mainly from C10-C11, C10 ′ -C11 ′ , C12-C13, C12 ′ -C13 ′ bonds. The strong band at 1197 cm − 1 mainly is from the stretching of C8-C9, C8 ′ -C9 ′ bonds. The very strong band at 1162 cm − 1 is mainly from symmetry stretching of C14-C15, There is a weak shoulder band at about 1137 cm − 1 , which is from asymmetric stretching of C14-C15, C14 ′ -C15 ′ , C10-C11, C10 ′ -C11 ′ bonds. The strong band at 1010 cm − 1 is from rocking vibration of methyl groups on the chain, which are C20H 3 , C20 ′ H 3 , C19H 3 , C19 ′ H 3 .

Interpretation of Raman spectrum of 9-cis isomer
As shown in Fig. 1, for the all-trans isomer, the substituent C7 = C8 group and C11 = C12 group are on different side of the C9 = C10 double bond plane. While for the 9-cis isomer, the substituent C7 = C8 group and C11 = C12 group are on one side of the C9 = C10 double bond plane. This major structural difference would change the vibrational modes and Raman spectrum of the 9-cis isomer. As shown in Table 1, the assignments and PEDs of 9-cis isomer are not symmetric, since it is not a symmetric molecule. In the 1500-1700 cm − 1 regions, the bands are also from the stretching of carbon-carbon double bonds. The most intense ν 1 band at 1524 cm − 1 is mainly from the stretching of C13 ′ =C14 ′ and C13 = C14. There is also a shoulder band, which is mainly from str C15 = C15 ′ and C11 ′ =C12 ′ , its simulated wavenumber is still at 1536 cm − 1 . But the intensity of this shoulder is a little stronger than that of the alltrans, which explains the fact that the ν 1 FWHM observed in the experiment is a little wider than the all-trans isomer, and the experimental ν 1 wavenumber of 9-cis (1524 cm − 1 ) is higher than all-trans isomer (1519 cm − 1 ). The very strong band in the ν 2 region at 1159 cm − 1 is from the stretching of C14-C15 and C10 ′ -C11 ′ . As shown in the Fig. 2 and Table 1, the most significant difference from all-trans isomers is that 9-cis has a moderate intensity band at 1135 cm − 1 . The Raman spectral difference between cis-trans isomers can also be found in many carotenoids in previous reports, like β-carotene (Koyama et al., 1982), neurosporene (Koyama et al., 1988), β-Apo-8 ′ -carotenal (Hashimoto et al., 1993), spheroidene (Jiang et al., 1996). Based on this 1135 cm − 1 Raman band in Fig. 2, these two isomers can be easily distinguished. This band is composed of two overlapping peaks. The vibrational mode of first one (simulated at 1139 cm − 1 ) is mainly from stretching of C14-C15, with part of C6-C7 stretching vibration. The vibrational mode of the second one (simulated at 1131 cm − 1 ) is mainly from stretching of C6-C7, with part of C14-C15 stretching vibration.

Interpretation of Raman spectrum of 13-cis isomer
The 13-cis isomer is not a symmetric molecule, therefore the assignments and PEDs are also not symmetric. As shown in Fig. 2, the differences between 13-cis, 9-cis and all-trans isomers are mainly in the 1500-1700 cm − 1 region and the 1100-1300 cm − 1 region. The most intense ν 1 band at 1532 cm − 1 is also mainly from the stretching of C13 ′ =C14 ′ and C13 = C14 (simulated at 1525 cm − 1 ), but there are also several shoulder peaks. The discernible shoulder band at 1562 cm − 1 is from the stretching of C5 ′ =C6 ′ and C5 = C6. The overlapping shoulder band (simulated at 1551 cm − 1 ) is mainly from stretching of C5 ′ =C6 ′ , C5 = C6 and C7 = C8 bonds. The other overlapping strong shoulder band (simulated at 1539 cm − 1 ) is mainly from stretching of C15 = C15 ′ and C11 ′ =C12 ′ bonds. One of the possible reasons for this band broadening is the superimposition of several shoulder peaks. Therefore, this band shows a much larger FWHM in the experiment than the all-trans isomer (Subramanian et al., 2014). In the 1100-1300 cm − 1 ν 2 region (Fig. 2), the strong band at 1141 cm − 1 is from the stretching of C14-C15, C13-C20 and C14 ′ -C15 ′ . Its Raman activity intensity is 133809, while the intensity of 9-cis is 63504, and the intensity of all-trans is 37477. Therefore, it can be clearly observed in the experimental spectra that the band of 13-cis is the strongest, and the intensity of 9-cis and all-trans gradually decreases. The strong bands at 1162 and 1197 cm − 1 are actually formed by the superposition of a series of peaks, so the FWHMs are much wider than the all-trans isomer. The simulated Raman band shows some difference at the ν 2 region, maybe due to that the simulated 1167 cm − 1 band is not strong enough.

Interpretation of Raman spectrum of 15-cis isomer
The structures and Raman spectra of other geometric isomers (15-cis and di-cis isomers) of AST are theoretically studied, whose proportions are usually less than 5% . The theoretical and experimental spectra of the above three isomers (all-trans, 9-cis, 13-cis) are very consistent, so the theoretical simulation spectra of other isomers are also of great reference value. Their optimized structures are shown in Fig. 1, and the geometry parameters are provided in the Tables S2-S4 of supporting information. The 15-cis AST molecule has C2 symmetry, therefore, the PEDs of the vibrational modes are symmetric. The calculated Raman spectrum of 15-cis is shown in Fig. 3, while the Table 1 Comparison between the experimental and theoretical calculated Raman spectra of the all-trans, 9-cis and 13-cis isomers of astaxanthin, with the assignments and PED of vibrational modes at B3LYP/6-311 + G(d,p) Level.  Abbreviations: PED, potential energy distribution (PED above 10 percent are listed); Exp, the wavenumber of experimental Raman shift (in cm − 1 ); Cal, the wavenumber of calculated Raman shift (in cm − 1 ); Act Int, the calculated Raman activity intensity is in A 4 /AMU; Δv, the calculated Raman shift minus experimental Raman shift of the vibrational mode (in cm − 1 ); vs, very strong; s, strong; m, medium; w, weak; vw, very weak; sh, shoulder peak; β, in plane bending; ρ, rocking; δ, scissoring; ν, stretching; ω, wagging; as, asymmetry; s, symmetry.
Therefore, when the experimental Raman spectrum of an unknown pure AST isomer is measured, the calculated spectra of ten isomers can help to the identification of the structure of this AST isomer. In principle, these characteristic bands could also be useful for Raman recognition of AST in a biological system. For example, Raman spectroscopy may be used for in situ recognition and conversion mechanism of AST isomers in different living organism, and it may also be used for the study of the different stability of AST isomer in soybean and sunflower oils. Of course, much more work needs to be done for practical applications in the complex biological system.

Discussion of Raman spectra and structure of AST cis-trans isomers.
Many early works considered that the strong Raman intensity of the polyenic or carotenoid pigment was due to resonance Raman effect (Bergamonti et al., 2011;Meinhardt-Wollweber et al., 2018). However, our data in this work seem not support this assumption. There are several evidence to prove our viewpoint in this work. Firstly, the 785 nm laser used in this work is far away from the AST 460 nm absorption band. But still, the Raman signals are very strong with 785 nm laser, and the spectrum is similar to the spectrum measured by 532 nm laser (Shao et al., 2019;Sharma et al., 2015). Secondly, for the simulation of Raman activity intensity in this present quantum chemistry calculations, the laser excitation was not taken into account. As can be seen from Tables 1 and 2 that the Raman activity intensities of many vibrational modes are extremely strong, even reaching the order of millions, while the Raman activity intensity of most common molecules is only in the range of one hundred. Therefore, this strong Raman intensity of the experiment mainly come from its own strong Raman activity, not the resonance effect. Thirdly, as shown in Fig. 2, the simulated normal Raman spectrum (without resonance effect) is quite consistent with the experimental result, also confirming that it is not a resonance Raman effect.
As shown in Fig. 1, all-trans-AST can be isomerized in to mono-cis forms, like 9-cis, 13-cis, 15-cis, and 6 kinds of di-cis isomers, like 9, 9 ′ -dicis, 9,13-di-cis, 9, 13 ′ -di-cis, 9,15-di-cis, 13,13 ′ -di-cis, 13, 15-di-cis. In the statistics of previous review work on organisms, different aquatic organisms have different proportion of AST isomers, and the all-trans isomer accounts for the main proportion . For instance, the flesh of rainbow trout, salmon, arctic charr and halibut contains about 75% to 100% all-trans AST; the flesh of cod, sprat, herring contain about 50% to 65% all-trans AST; the flesh of crustacean organisms, such as krill, shrimp, white prawn, contains about 60% to 90% all-trans AST . While 9-cis and 13-cis are the other isomers with the largest proportion, sometimes reaching more than 10%. 15-cis is usually less than 3%, and the content of di-cis is even lower. Therefore, in this work, we mainly focused on all-trans, 9-cis and 13-cis these three kinds of geometric isomers from experiments and calculations, and other geometric isomers are analyzed by theoretical calculations.
The single-point energies of these ten isomers are all calculated. The energy of all-trans isomer is the lowest, and the energy differences of the other isomers relative to the all-trans isomer are listed in Table 4. Based on the Maxwell-Boltzmann distribution, the ratios of different isomers are estimated through their relative energies. The ratio of all-trans configuration is approximately 70%, 9-cis and 13-cis are about 10%, the other isomers are less than 2%. The total content of di-isomers does not exceed 10%. The 15-cis is the least among mono-cis isomers, while 9,15di-cis and 13, 15-di-cis are the least among di-cis isomers. The calculated ratios are well consistent with the previously reported ratios of isomers in natural organisms or chemical conditions (Euglert and Vecchi, 1980;Holtin et al., 2009;Yu and Liu, 2020). Therefore, based on Boltzmann distribution calculated from the energies of isomers, it can be reasonably explained why the isomers have these different proportions.
The experimental crystal structure of all-trans isomer of AST was obtained through X-ray diffraction (Bartalucci et al., 2007). The calculated structure parameters of this work are compared with the experiment, involving the bond lengths, angles and dihedral angles in Table S1. Since the all-trans AST molecule has Ci symmetry, the structural parameters are also symmetric. For instance, the bond lengths of C1-C2 and C1 ′ -C2 ′ bonds are the same. In the all-trans molecule, C6 to C15 with C6 ′ to C15 ′ constitute a linear conjugated chain of 40 carbon atoms. The dihedral angle in the chain, like C7-C8-C9-C10 and C14-C15-C15 ′ -C14 ′ , are all near to 180 • , and this conjugated chain is nearly flat. The C5-C6, C7-C8, C9-C10, C11-C12, C13-C14 and C15-C15 ′ bonds are carbon-carbon double bonds, their bond lengths are shorter than 1.4 Å. The C8-C9, C10-C11, C12-C13 and C14-C15 bonds are single bonds in Table 3 Raman bands used to distinguish AST cis-trans isomers.  , k is Boltzmann constant, T is set as 298.15 K, Ei is the energy of ith isomer, Ni is the molecule number of ith isomer, N is the total molecule number of the isomers (Liu et al., 2011). conjugated system, their bond lengths are about 1.45 Å. They are shorter than the traditional carbon-carbon single bond, for example, the bond length of C1-C16 exceeds 1.5 Å.
The molecular geometry structures of these ten isomers are all optimized. Their structural parameters like bond lengths, angles and dihedral angles are listed in the Tables S2-S4 of supporting information, respectively. It can be seen from Tables S2 and S3 that the bond lengths and angles of different isomers are very close. The differences in structural parameters between ten isomers mainly lie in the dihedral angles where cis-trans isomerism occurs, as shown in Table S4. For instance, the C8-C9-C10-C11 dihedral angles of 9-cis, 9,9 ′ -di-cis, 9,13-di-cis, 9,13 ′ -dicis, 9,15-di-cis isomers are near to 0 • , while the C8-C9-C10-C11 of other isomers are near to 180 • . The C12-C13-C14-C15 dihedral angles of 13cis, 9,13-di-cis, 13,13 ′ -di-cis, 13,15-di-cis isomers are near to 0 • , while the C12-C13-C14-C15 of other isomer are near to 180 • . These structural parameters may greatly promote in-depth study of the physiological properties of different AST isomers, such as the analysis of their interactions with proteins.

Conclusion
In this work, the structures and Raman spectra of all geometric isomers of astaxanthin have been systematically explored. The energies, structure, and Raman spectra of these 10 isomers are obtained based on DFT calculations. From the relative energy of isomers, it can be reasonably explained why the isomers have these different proportions in the nature or chemical synthesis. The vibrational modes and PEDs of these isomers are analyzed. Structural parameters such as bond length, bond angle and dihedral angle of all isomers are also obtained. The Raman spectra simulated by DFT are in good agreement with the available experimental data, explaining well the experimental spectra in terms of Raman shift, intensity change and peak broadening. Moreover, we demonstrate that the different AST isomers have characteristic Raman bands in the 1000 to 1350 cm − 1 . Therefore, by combining DFT and Raman spectroscopy, all the known geometric isomers of astaxanthin can be well distinguished. The research of molecular vibrational modes and molecular structures have important significance for the indepth understanding of the chemical and biological properties of AST, and may promote the development of the research on food, nutrition, medicine and other fields in food molecular science.