Materials and methods
A fresh sample of de novo-localized starch granules (isolated from a transverse section of a potato tuber) was utilized as a positive control for this assessment due to its widespread abundance, ease of preparation [for diascopic brightfield (BF) & PLM techniques], high birefringence and its well-characterized structure. This is contrasted against a freshly-prepared slide of the lower (herein referred to as the distal) epidermis isolated from M. koenigii (observed through episcopic PLM), with a commercially-prepared slide of skeletal myofibrils (possessing very low birefringence which is generally unresolved through Berek and de Sénarmont compensation) being utilized as a negative control for qtPLM in this respect (observations gleaned from this slide are not included in the present study). Anisotropic identification of the specimens was performed using a HC PL Fluotar 50X/0.80 BD objective (Leica P/N: 566504) coupled with a rotary analyzer and a condenser-integrated polarizer (for diascopic polarization) or a corresponding incident light polarizer and a Smith reflector cube (for episcopic polarization). As the analyzed specimens were generally highly birefringent, quantitative analysis of the extent of polarization coupled with determination of object retardation was achieved through the use of a traditional Berek compensator, although an alternative approach was utilized to quantify object retardation (Γobj) in the specimens (as opposed to traditional methods utilizing monochromatic light for determining Γobj). The de Sénarmont and Bräce-Köhler approaches of compensation were (however) not utilized for Γobj or birefringence quantification in this short investigation.
Fig. 1: Starch grains from a potato tuber imaged under diascopic brightfield polarization A without the use of a compensator and B with a corresponding Berek compensator. Exhibition of birefringence in the starch grains is evident, with Γobj = 168.12 nm.
Fig. 2: A: M. koenigii guard cells (with sandwiched stomata) imaged under episcopic brightfield polarization with a corresponding Berek compensator. Notice the orange and yellow pleochroism depicted in the cellulose cell walls of the guard cells which are circled in blue and green respectively, as opposed to the cytosol which depicts a greenish-yellow hue (Γobj = 52.69 nm) (*N.B.: Interestingly, stokes shifted Raman scattering and phosphorescence may be perceived as alternative hypotheses accounting for this phenomenon, as in the latter, cellulose has been characterized to exhibit autofluorescence with emission maxima at λ = 420–430 nm ). B: Stereogram of Berek-compensated PLM-imaged cell walls of the guard cells. A much denser distribution of cellulose microfibrils was noted in the cell wall periphery, as opposed to its mid-section (57 %/40 % = 1.425-fold for the guard cell pair containing the blue circle, for instance).
The results expectably indicate a difference in birefringence exhibited by the potato starch grains and the cellulose cell walls of the guard cells. Both cellulose & amylose occur (in their assayed structures) as birefringent polysaccharide macrofibrils; the spatial arrangement of the individual microfibrils responsible for phase-shifting the E ray relative to the O ray by a designated amount (Γobj), thereby impacting on the curl of the resultant E-vector field (∇ × E) as described by Equations ➀ and ➁ below (assuming a constant sinusoidally-varying B).
Consider the E & O rays emerging from the specimen to have equal amplitude, A, with their E-fields represented as the y- & z-axes respectively. Then, we have [with constant λ and external compensation (if any) designated as Γcomp] for a positively-birefringent specimen:
where x represents the distance traversed by the polarized wave en route through the specimen and Γoc = Γobj + Γcomp.
This would consequentially (utilizing a self-devised context-specific (yet generalized) formula for birefringence determination in R3.5) allow ∇ × E to be derived as follows:
where quantum spatial hopping may be postulated for photons from y1 & z1 to y2 and z2 respectively (where y1 → y0+, z1 → z0+, y2 → y0-, and z2 → z0-, for which Ay or Az = A).
From equation ➁ above, it may be observed that the plot of ∇ × E varies as a curvilinear hyperplane in R3.5, with Γoc introducing a phase-precedent translation to ∇ × E along the w-axis. As would be expected, this is visually manifested as an alternating amplification/extinction of the transmitted analyzed light waves with varying Γoc; consequent w-sections obtained from the plot of ∇ × E would thus permit the identification of Γobj for which dim(∇ × E) is minimal and Γcomp is known. If, however, λ and Γoc are variable (as is often the case for a polychromatically-illuminated specimen having structures of different birefringences when observed under qtPLM, such as the starch grains and the cellulose cell walls of the guard cells depicted in Figures 1 and 2 respectively), one may then surmise that for some defined λa in λ1..n and Γcompa in Γcomp1..n, dim(∇ × E)λa is minimal at Γcompa, accounting for the pleochroism observed in the sample/(s) (a consequence of negative attenuation).
In addition, through the individual determination of Γobj and quantitation of the observed pleochroic distribution in Figure 2, the relative ordering of the primary β(1→4)-glycosidic linkages in the cellulose cell walls of the assessed guard cells may be deduced to be parallel to the major axis of the stomatal pore (as are the cortical microtubules), implicating that the cellulose synthase GT domains  in these cells were oriented in a somewhat orthogonal fashion relative to the minor stomatal axis. This is starkly contrasted against the use of micropolarimetry in determining the radial orientation of the α(1→4)-glycosidic bonds in the starch granules in Figure 1, for which amylose represents a primary constituent.
Some utilization of micropolarimetry in biological analytics was evident in numerous studies conducted from the 1950s – 1970s , . Nonetheless, polarization microscopy (as an integral element facilitating biological research) has faded in relative importance over the decades with greater interest realized in fluorescence microscopy as a bioimaging platform, a notion further fuelled through the development of super-resolution optical microscopy techniques and SIM, amongst others. Nonetheless, unlike the need for specific fluorescent biomarkers (as utilized in confocal laser scanning microscopy & current optical nanoscopy techniques) and/or scanning/light field techniques, biological PLM provides an avenue for realizing potential anomalies in unresolved molecular structures, although significant expertise and work might be required to create a database containing de novo birefringence values for key proteins of interest (e.g. p53 and PrPSc , amongst others). A revival of diascopic and episcopic PLM would thus be congenial in this respect, with micropolarimetry established as an extrapolation of conventional qtPLM approaches.
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