2.1.

Example of ${\mathit{TiO}}_2$ Crystal Polymorphs Discrimination

Here, as a model system, type discrimination of ${\mathrm{TiO}}_2$ crystal polymorphs, anatase and rutile, is performed using the MCR-ALS method. Anatase and rutile ${\mathrm{TiO}}_2$ powder were mixed and placed on quartz cover slips. Space-resolved Raman spectra were obtained by using a 785-nm excitation Raman microspectroscopic system. Raman spectra of the powder mixture were collected over a $20\times 20\mu \text{m}$ region with a 0.25-μm scanning interval, hence, 6400 spectra were obtained. Each Raman spectrum consisted of 1340 wavenumber points corresponding to the 1340 elements of a charge-coupled device detector. Figure 1 shows the representative Raman spectra. These spectra have a number of overlapped Raman bands, interpreted as the superposition of the two intrinsic Raman spectra of anatase and rutile ${\mathrm{TiO}}_2$ shown in Fig. 2.

Fig. 1

Space-resolved Raman spectra of ${\mathrm{TiO}}_2$ powder containing anatase and rutile forms.

Fig. 2

Raman spectra of pure anatase (solid line) and rutile (dotted line) ${\mathrm{TiO}}_2$.

In the present MCR-ALS analysis, the Raman spectra sets were combined to make a $1340\times 6400$ matrix. The number of pure spectral components was set to $k=2$, and the initial guess of $1340\times 2$ matrix $\mathbf{W}$ was set by random numbers. In the ALS optimization, the constraints were set as follows: (1) $\mathbf{W}\ge 0$ and $\mathbf{H}\ge 0$, (2) L2-norm penalty term in Eq. (2) was set to be $\beta =0.002$, (3) L1-norm penalty term in Eq. (6) was set to be $\alpha =0.002$. These equations were iteratively solved with non-negative matrix factorization algorithm, and in every iteration step, column vectors of the matrix $\mathbf{W}$ were all normalized.⁶⁰ After the 500 iteration cycle, ensuring that ${||\mathbf{A}-\mathbf{WH}||}^2$ converged to a sufficiently small value constant value, the pure spectral components were obtained as shown in Fig. 3.

Fig. 3

(a) Multivariate curve resolution-alternating least squares (MCR-ALS) retrieved Raman spectra, (b) optical microscope image of ${\mathrm{TiO}}_2$ powder, and (c) pseudocolor distribution images of components 1 (red) and 2 (green).

The MCR-ALS based factorization successfully decomposes the raw spectral data sets into two pure component spectra, i.e., the components 1 and 2 spectra are identical to the Raman spectra of anatase and rutile, respectively [Figs. 2 and 3(a)]. Based on this discrimination, each MCDI is constructed by rearranging the row vectors of $\mathbf{H}$. As shown in Fig. 3(c), a clear distribution image has been obtained, providing the qualitative and quantitative information about the ${\mathrm{TiO}}_2$ mixture sample. As shown here, the MCR-ALS technique automatically resolves, without a priori spectral information, the observed set of space-resolved Raman spectra into physically interpretable spectra of the two polymorphs.

2.2.

Analysis of Living Cells

Taking advantage of the physically interpretable factorization, the MCR-ALS technique can be effectively applied to molecular-level analysis of living cells. Living cells are highly complicated molecular systems and contain a large number of spectral components with no a priori information. This is particularly the case with in vivo analysis. Furthermore, they contain many compounds that have similar molecular structures. Consequently, Raman spectra tend to show many overlapped bands. This situation makes it difficult to analyze living cell Raman spectra at a detailed molecular level. The MCR-ALS analysis has great advantages to overcome these difficulties: (1) It does not need a priori information on the spectral and concentration profiles. (2) Decomposed spectra are ready for physical and/or chemical interpretation. (3) Sparseness constraints can be effectively used to achieve high contrast MCDI.

The MCR-ALS analysis is capable of extracting dynamic information from living cells. Figure 4 shows the time-lapse MCDI of a single dividing Schizosaccharomyces pombe, fission yeast cell.⁶¹ Raman mapping measurements were performed at 600 to 800 points (depending on the image size) at an interval of 0.5 μm and at nine different times (1, 2, 4, 6, 6.5, 10, 14, 18, and 22 h after inoculation of yeast cells into medium) in the cell cycle. The resultant 6885 Raman spectra were assembled to construct one $\mathbf{A}$ matrix; two spatial and one temporal dimensions were combined to a single dimension. The ALS optimization was conducted with an SVD-based initialization (six SVD spectral components were used as the initial guess of $\mathbf{W}$ matrix) and L1-norm regularization, yielding sparse solutions. The resulting six components are denoted 1 to 6 as given in Figs. 4(a) and 4(b). It should be noted that the MCR-ALS optimization started with random initialization was in vain for decomposition in this complicated cell system. In order to avoid falling into a false local minimum rather than the global minimum, the initialization of $\mathbf{W}$ or $\mathbf{H}$ was a key step.

Fig. 4

Time-lapse Raman imaging of single dividing fission yeast cell. (a) Six spectral components (1 to 6) derived from MCR-ALS and (b) distribution images of components 1 to 6 together with the optical microscope images. $\text{Scale bar}=2\mu \text{m}$.

By the MCR-ALS method, the separation of background signals is easily carried out; the component 1 is interpreted as due to the background because it shows a featureless spectral profile [Fig. 4(a) 1] and high intensity distribution patterns outside the cells [Fig. 4(b) 1]. The distributions of polysaccharides are clearly observed from the component 2, whose spectrum [Fig. 4(a) 2] very much resembles the spectra of glucan and mannan.⁷ It localizes at the cell wall and septum at all measurement times [Fig. 4(b) 2]. The component 3 [Fig. 4(a) 3] shows typical lipid spectrum with unsaturated lipids⁷ and ergosterol.²⁰ From the time dependent changes of the corresponding spatial distribution [Fig. 4(a) 3], we know that the concentration of lipids shows a drastic decrease just after the cell division and gradually comes back as time goes on. The spectral components 4 and 5 [Fig. 4(a) 4 and 5] contain well-known protein bands. The concentrations and distributions of proteins [Fig. 4(b) 4 and 5] show time dependence that is totally different from that of lipids. The fact that we have two spectral components for proteins means that we have two groups of protein molecules with different structures (and hence different spectra) that show distinct time- and space-dependence during the cell cycle. We need additional information on those protein groupings in order to resolve this set of Raman spectra into more physically meaningful protein spectra. The origin of the component 6 is still unclear, although the MCR-ALS analysis ends up with much less clear results without this component. In this way, the intrinsic spectra and MCDI obtained from MCR-ALS has elucidated unknown and unexpected molecular-level dynamics taking place during the process of cell division.

Another study with the MCR-ALS shows the existence of organelle-specific water structures in a living budding yeast cell.⁶² Water molecules inside a cell are believed to play critical roles in physiological processes, creating distinct structural and chemical properties as compared to bulk water.⁶³ A detailed intracellular water structural information in living yeast cells (diploid Saccharomyces cerevisiae) has been obtained using Raman microspectroscopy, in which the OH stretch Raman band of water is sensitive to the changes in the hydrogen-bonding networks. In the following MCR-ALS analysis, 247 mapping measured Raman spectra ranging from 3100 to $3800{\mathrm{cm}}^{-1}$ (572 wavenumber points) were used as an input. From an SVD analysis, the number of components was determined to be five. For the initial guess of $\mathbf{W}$, the bulk water spectrum was used as the only “fixed” component and the other spectral profiles were randomly set. The ALS iteration was performed with L2-norm regularization, which is known to be effective for MCR-ALS analysis of data sets including like component spectra.

The resultant five components are shown in Fig. 5. The five pseudocolor MCDIs are almost mutually exclusive and the five spectral components show varying relative intensities of the OH stretch bands. The component 1 corresponds to bulk water whose spectrum is fixed. Its spatial distribution shows high value outside the cell. The component 5 has a significantly lower intensity in the $3200{\mathrm{cm}}^{-1}$ OH stretch region as compared to the intensity in the $3400{\mathrm{cm}}^{-1}$ region, indicating a lower proportion of stable hydrogen-bonding network in component 5 than in bulk water. From the analysis of the average spectra in the fingerprint region (data not shown here), the components 2 to 5 have been indicated to originate from the cell wall, cytoplasm, nuclear, and lipid bodies, respectively. Thus, organelle-specific water structures in living yeast cells are successfully retrieved and elucidated by using the MCR-ALS method. It is well recognized that water is the most difficult molecule to study in living cells. Only the combination of Raman microspectroscopy and MCR-ALS can provide this unique information of organelle-specific water structures in a cell in vivo.

Fig. 5

MCR-ALS analysis of the OH stretch Raman band of intracellular water in a living budding yeast cell. Raman spectral components 1 to 5 and their distribution images. The top image is the optical microscope image.