Spectral reflectance, or indexes that characterize spectral reflectance at concrete wavelength, is commonly used as an indicator of plant stress, or its photosynthetic apparatus status. In this paper, new leaf optical model is presented. Within this paper, experimental determination of surface and internal reflectance of Spring barley leaves and mathematical-physical modelling of internal reflectance were performed. It was proven that a new proposed theoretical model and the experimental spectra of internal reflectance are strongly correlated. It can be concluded that the total reflectance is not a function of epidermis condition, but it testifies about overall functional condition of Spring barley leaves.
1. Introduction
Leaf optical properties are associated with physiological stress, that affects these plants. Stress factors can be different: dehydration, freezing, ozone, diseases, herbicides, intraspecies and interspecies competition, insect, lack of mineral nutrition, high salinity, extreme temperatures, and so forth [1–3]. Reflectance spectrum changes are similar for different plant species [1]. Because of this fact, spectral reflectance, or indexes that characterize spectral reflectance at concrete wavelengths, is commonly used such as reliable indicators of plant condition and/or their photosynthetic apparatus status. The study of spectral characteristics has not only scientific, agricultural, and environmental reasons but it is also important from the economical point of view. The spectral characteristics are predominantly used for detection and prevention of potential stress factors [3, 4].
When the electromagnetic radiation impacts air—leaf interface, three competition processes may occur (absorption, reflection, and transmission). Mathematically, it can be described using (1) as a(1)EA(λ,αD)=E(λ)-(ER(λ,αD,gHG)+ET(λ,αD)),
where E(λ) is electromagnetic flux that impacts the sample, λ is wavelength, ER(λ,αD,gHG) is reflected radiation to the upper halfspace, αD is angle of incidence, and gHG is leaf surface anisotropy function. EA(λ,αD) is absorbed radiation and ET(λ,αD) is radiation, which passed through the leaf to the lower halfspace. Let us assume that the radiation is not leaking by leaf sides and the leaf fluorescence is not counted into the energy balance described in (1). If the following variables are assumed: reflectance R(λ,αD,gHG), absorbance A(λ,αD), and transmittance T(λ,αD), (1) can be rewritten in the form of
(2)A(λ,αD)=1-(R(λ,αD,gHG)+T(λ,αD)).
Total light intensity reflected from the leaf surface R(λ,αD,gHG) is formed by two components, surface reflectance Rs(λ,αD,gHG) and internal reflectance Ri(λ,αD). Reflectance (surface and internal) is a function of a wavelength λ and angle of impact αD[0,π/2] if it is measured considering normal [5]. Surface reflectance can be divided into the two components: mirror reflectance Rsspec(λ,αD) and diffuse reflectance Rsdif(λ,αD,gHG). Mirror reflectance is typical for homogeneous objects with perfectly smooth surface. For ideal mirror-like surfaces, it holds that the angle of incidence is equal to the angle of reflection. For surface diffuse reflectance, it is typical that the light is reflected randomly due to rough surface structures [6]. Leaf internal reflectance is always diffuse.
In fact, there are no surfaces that can be characterized as strictly mirror-like or diffuse. Surface reflectance has always two components, mirror-like and diffuse [7]. If the simplification is used, that leaf surface reflectance is mirror-like, Fresnel equations can be used for surface reflectance components calculation. Fresnel equations represent relations among vector amplitudes of incident, reflected, and refracted waves. Quantity of refracted transverse electrical wave rTE is defined as
(3)rTE=-sin(αD-αL)sin(αD+αL),
where αD and αL are angles of incidence and reflection. Transverse reflected magnetic wave quantity can be defined as
(4)rTM=tg(αD-αL)tg(αD+αL).
Due to the fact that the sunlight is nonpolarized electromagnetic radiation, neither transverse reflected electric nor magnetic waves are equal zero. Surface reflectance can be expressed through the squares of relevant amplitude coefficients
(5)Rsspec(λ,αD)=12[|rTE|2+|rTM|2]=12[|-sin(αD-αL)sin(αD+αL)|2+|tg(αD-αL)tg(αD+αL)|2].
The light, that is not reflected to the upper halfspace, enters to the leaf, according to the Snell law:
(6)n1·sinαD=n2sinαL,
where n1, respectively, and n2 are leaf refractive indexes. With respect to the energy conservation law, light intensity for radiation that passes through the air-leaf interface can be defined as
(7)T(λ,αD)=1-Rs(λ,αD,gHG).
It is necessary to define leaf surface anisotropy factor gHG for calculation of diffuse surface reflectance. The anisotropy factor is the input parameter for calculation of Henyey-Greenstein probability function pHG [8]. Let us assume that the light beam is reflected to the upper halfspace, then the following can be used:
(8)pHG=1-gHG2gHG[1+gHG(1+gHG2)-1],0≤gHG≤1.
Surface reflectance diffuse component can be expressed through radiation flux using (1) and (8):
(9)ERsdif(λ,αD,gHG)=pHG·E(λ)-ERsspec(λ,αD).
2. Materials and Methods
Spectrophotometric measurements were realized by double beam scanning spectrophotometer UV-VIS 550 UNICAM within 400–800 nm wavelength intervals. Diffuse reflectance measurements were performed using integral spheroid RSA-UC-40 Labsphere, which was incorporated into the spectrophotometer. Microscopic measurements were realized by light microscope Amplival. All reflectance measurements were performed on the adaxial leaf side. Primary leaf blades of Spring barley (Hordeum vulgare, cv. Akcent) were used for all measurements. The plants were growing in the growth chamber with the following parameters: temperature 21°C, light regime: 16 hours of light, 8 hours of darkness, and the light source was fluorescence tubes with light intensity 90 μmol·m^{−2}·s^{−1}. According to Feekes phenomenological scale, barley leaves can be characterized by growing phase 1,3. Firstly, the reference group that consisted of nonstressed plants (no stress) was measure; then the plants stressed by drought were measured, too (3rd, 6th, and 9th day). Diffuse reflectance spectra were achieved by the following process. At first, diffuse reflectance was measured separately for whole leaf and epidermis. For epidermis reflectance measurements were used strongly absorbing background to minimize light reflected at epidermis-background interface. Internal reflectance spectrum was calculated as a difference between total diffuse reflectance and the whole epidermis reflectance. Pigments content was calculated according to equations proposed by Lichtenthaler [9] as an average of six samples. To achieve more trustworthy model, rough measurements of cells size and shape were realized by light microscope. Also the size, shape, and number of chloroplasts in the cells were studied. In presented model, cube shape of cells and spherical shape of chloroplasts are assumed. For microscopic measurements, the control groups of nonstressed and barley stressed plants for nine days were used.
3. Results and Discussion
The presented model is based on few presumptions. The cell walls have transmittance τ and reflectance ρ both of interval 〈0;1〉; light during its propagation passes through k chloroplasts; in chloroplasts the Lambert-Beer law holds (it is presumed that chlorophyll content in chloroplasts is constant). Chloroplasts are the only one absorbing components in the intracellular medium. Absorption of nonchloroplasts structures is neglected.
In the simplest case, when the light penetrates into the cell, reflects from the wall, and passes k chloroplasts and after the next reflection it leaves the cell; internal reflectance can be defined by
(10)IRi(λ)=τ2ρ2qk(λ)I0(λ),(11)q(λ)=10-ε(λ)cd=10-D*(λ),
where q(λ) represents absorption properties of one or more chloroplasts, ε(λ) is the molar extinction coefficient, c is the chlorophyll concentration in chloroplasts, and D*(λ) is the optical density of one chloroplast. A more sophisticated case is represented by multiple scattering in one cell with one chloroplast. Those numbers may be theoretically infinite. For the light that passes through the cell wall and after reflection it passes through the chloroplast; the following relation can be used:
(12)I1(λ)=τρq(λ)I0(λ).
Thereafter, the same light beam reflects from the wall and leaves the cell; in this case intensity of internally reflected light is:
(13)I1A(λ)=I1(λ)τρ=τ2ρ2q(λ)I0(λ).
Using the same procedure, other wall reflections and chloroplast passes are:
(14)I1B(λ)=I1(λ)ρ,(15)I2(λ)=I1(λ)[ρq(λ)]2,(16)I2A*(λ)=τ2ρ2q(λ)[ρq(λ)]2I0(λ),(17)I3A*(λ)=τ2ρ2q(λ)[ρq(λ)]4I0(λ).
For ∞ cell reflections, the sum of geometrical series is realized:
(18)IRi(λ)=τ∑j=1∞IjA(λ)=τ3ρ2q(λ)I0(λ)[1-ρ2q2(λ)].
There are some extreme situations that may occur. If the τ=0 (nontransparent) walls, than it is clear that IRi(λ)=0. The second extreme situation occurs when τ=1, but ρ→0 (nonreflecting walls); in this case internal reflectance is also equal zero. The third extreme situation occurs when τ→1;ρ≻0;q(λ)=1; in this case plant tissue does not absorb:
(19)IRi(λ)=τ3ρ21-ρ2I0(λ).
If the last term is changed in comparison with the previous case, q(λ)<1, especially if q(λ) converges to zero (strong absorption), then also IRi(λ)→0.
Generally, if the light beam passes through n1 cell walls and reflects on n2 interfaces, passes through n3 cells and in every of those cells there is k chloroplasts, relation for IRi(λ) calculation is modified to
(20)IRi(λ)=τn1ρn2{qk(λ)1-[ρqk(λ)]2}n3I0(λ).
After approximation, when [ρqk(λ)]2 is set to zero, the relation for IRi(λ) calculation is simplified to
(21)IRi(λ)=τ2ρ2qk(λ)I0(λ).
It is nothing else than an equation for a simple case (two passes, two reflections, and k chloroplasts).
Within the discussed model, it is considered that light beam passes through the cell (b); light beam geometrical length is x. In the cell with the volume Vb, there are Nb chloroplasts. Another presumptions are chloroplasts spherical shape and cells cubic shape. Chloroplasts radius is R and the diameter is D. Chloroplast geometrical cross-section is σc (main cut):
(22)σc=π(D2)2=πD24=πR2.
Chloroplasts concentration in cell is nb; said in different way, number of chloroplasts Nb in cell volume unit Vb can be expressed as
(23)nb=NbVb.
Let’s assume light beam surrounding, if the chloroplast center is inside the cylindrical volume:
(24)Vp=πR2x.
So if the distance between chloroplast center and light beam is smaller than R, the light beam passes through the chloroplast. In the other case, when the distance between the light beam and chloroplast center is bigger than R, the light beam does not pass through the chloroplast. Number of the chloroplasts with the centers inside the Vp can be expressed as
(25)np=nbVp=NbπR2xVb=σcNbxVb.
On the light beam length x, there are np chloroplasts, while the mean distance among chloroplasts is b; if we assume their regular arrangement, b can be defined by
(26)b=xnp=xVbσcNbx=VbσcNb
or equivalently according to
(27)b=x(np-1)x=x(σc(Nb/Vb)x-1)x=1σc(Nb/Vb)-1
based on calculation manner. Equation (26) holds in case when the light beam hits the chloroplast after it has passed length equal to the average chloroplasts distance. On the contrary, (27) holds in the case when the light beam hits the chloroplast without any preceding pass (x=0). For simplicity, the first case is mentioned in the following text. If the spherical shape of the chloroplast (with radius R and diameter D) is presumed, refractive indexes in the chloroplasts and its surrounding are the same and the light beams have the same optical density (intensity of incident collimated radiation is homogeneous); then average optical length, that is passed by the light beam in the spherical chloroplast, can be determined. Let us divide chloroplasts into the cylindrical sections (with radius r), where r∈〈0;R〉. Optical length in cylindrical section is defined by
(28)2x(r)=2R2-r2.
The mean optical length is
(29)x-=1π·R2∫0R2R2-r22πrdr,
or alternatively, after substitution,
(30)y=R2-r2,dy=121R2-r2(-2r)dr,ydy=-rdr.
The mean optical length can be expressed as
(31)x-=4R2∫0Ry2dy=4R2[y33]0R=4R33R2=4R3=2D3.
It is clear that average optical length in spherical chloroplast is 2D/3. According to Poisson probability distribution, on the light trajectory x, there are k chloroplasts which can be defined as
(32)Pk(x)=e(-x/b)(x/b)kk!.
General formula for internal reflectance calculation is expressed by
(33)Ri(λ)=τnρmqk·(2/3),
where indexes n and m define the number of the light that passes through and number of reflections realized on the cell walls. With respect to (11), (33) can be redefined to the following form:
(34)Ri(λ)=τnρm∑k=0∞e-(x/b)(x/b)kk!(q2/3)k,
after modification of
(35)Ri(λ)=τnρme-x/b∑k=0∞(x/b)k(q2/3)kk!,Ri(λ)=τnρme-x/be(x/b)·q2/3.
The formula that fundamentally includes all parameters that are necessary for internal reflectance theoretical spectra modelling is achieved:
(36)Ri(λ)=τnρme-xσcNb/Vbe(xσcNb/Vb)10-ε(λ)c(2/3)D.
Some of these parameters can be measured independently, the others may be achieved using fitting procedures. The relation expressed by (36) can be simplified to
(37)Ri(λ)=τnρmexp(-π·D24NbVbx)×exp[π·D24NbVbx10-ε(λ)c(2/3)D].
There are two unknown variables remaining (chlorophylls concentration in chloroplasts and light beam geometrical length); then if one of these parameters is determined properly, (see following text) the last unknown parameter can be calculated by modification of (37). Let us make a substitution according to
(38)[u=π·D24NbVbx].
After which modification of (37) can be expressed via
(39)lnRi(λ)τnρm=u(10-ε(λ)c(2/3)D-1),log[-(ln(Ri(λ)/τnρm)u+1)]=ε(λ)c23D.
Formula for chlorophylls concentration calculation (for known light beam geometrical length) is defined as
(40)c=321ε(λ)Dlog[-(ln(Ri(λ)/τnρm)(π·D2/4)(Nb/Vb)x+1)].
On the contrary, if the chlorophylls concentration is known, the light beam geometrical length can be calculated after modification:
(41)lnRi(λ)τnρm=π·D24NbVbx(10-ε(λ)c(2/3)D-1).
Then x can be expressed as
(42)x=4·Vbln(Ri(λ)/τnρm)πD2Nb(10-ε(λ)c(2/3)D-1).
Another procedure, alternative to the method mentioned above, is a presumption that light beam hits k chloroplasts, when it passes through the cells which can be defined as
(43)P=P0+P1+P2+⋯Pn,
where P means the sum of all partial probabilities, while it holds that P=1. P0 is the probability that the light beam does not hit any chloroplast during its pass through the leaf tissue, P1 is the probability that the light beam hits exactly one chloroplast during its pass through the leaf tissue, and so on. If all necessary parameters, such as cells and chloroplasts size and shape, cell walls transmittance and reflectance, chlorophylls concentration in chloroplasts, and the distance, which have to be passed by light beam in the cell, are known, the formula for internal reflectance calculation can be expressed by
(44)Ri(λ)=τnρm∑k=0∞Pk(10-ε(λ)(2/3)D)k.
With respect to the fact that the main contributors to the internal reflectance are especially light beams that pass only through a few chloroplasts, (44) can be rewritten in the form of progression:
(45)Ri(λ)=τnρmP0+τnρmP1(10-ε(λ)c(2/3)D)+τnρmP2(10-ε(λ)c(4/3)D)+⋯τnρmP10(10-ε(λ)c(20/3)D).
The other progression members are not mentioned anymore due to the fact that their contribution can be neglected. For determination of average distance among chloroplasts, the following procedure can be used, assuming that chloroplasts distribution in cells is homogeneous. The mean chloroplasts distance was calculated using Monte Carlo method (MATLAB 7.1). For every chloroplast, three pseudorandom numbers ξi were generated, for interval 〈0;1〉. Meanwhile, chloroplasts geometrical centers in Cartesian coordinates system are defined according to
(46)x=ξ1a,y=ξ2a,z=ξ3a,
where ξ1,ξ2, and ξ3 are pseudorandom numbers and a[μm] is size. Due to the fact that within the presented model there are only chloroplasts mentioned with the volume completely located in the cell, the following criterion was used. Chloroplasts must accomplish boundary conditions expressed by
(47)lx=ξ1a-rc≥0,ly=ξ2a-rc≥0,lz=ξ3a-rc≥0,
where lx,ly, and lz are the distances from the origin of coordinates system, and rc[μm] is the chloroplast radius. Another limiting criterion is prohibited overlap of the chloroplasts:
(48)λi≥2rc,
where λi[μm] is the distance between two chloroplasts in the cell. The same procedure was repeated until all Nb chloroplasts center positions were successfully simulated. In experiment, it was investigated that Nb=15. In general, if two chloroplasts are mentioned, namely, chloroplasts A[xA,yA,zA] and B[xB,yB,zB], where x, y, z indexes Cartesian coordinates, then their mutual position can be calculated from
(49)λi=(xB-xA)2+(yB-yA)2+(zB-zA)2-2rc=|AB|-2rc.
Chloroplasts distance is not defined by the distance of their geometrical centers but by the distance of two nearest mutual points. The mean distance among all chloroplasts present in the cell is λ-[μm]; this is expressed by
(50)λ-=1n∑i=1nλi.
When all necessary calculations were performed, the following results were achieved. The mean distance among chloroplasts in Spring barley cells, which was not stressed by dryness, was b=14,6μm. The barley stressed by drought for 9 days is b=12,4μm. For the other calculations according to
(51)q(λ)=10-ε(λ)cd2/3,
spectrum of pigments molar extinction coefficients (Figure 1) was used.
Spectrum of pigments molar extinction coefficients.
Using optical microscope, it was determined that the number of chloroplasts was the same for both groups (15). There was also difference in chloroplasts sizes. For the control group, the average chloroplasts diameter (D) was 3,6 μm; meanwhile, in case of stress plants it was only 2,1 μm. There were also differences in cells sizes. The barley stressed by drying had approximately 20% smaller cells diameter (ab=26,5μm for stressed plants and ab=33μm for control group).
If the chlorophylls content per surface area is known, it is possible to calculate rough chlorophylls concentration in chloroplasts. If we assume that during the stress influencing period cells and chloroplasts size are changing linearly, but chloroplasts number remains constant, then the chlorophylls concentration can be calculated for each day of stress conditions:
(52)c*=3mnπR02abn230πRn2ab02=mnR02abn210Rn2ab02,
where c* (mol·L^{−1}) is chlorophylls concentration in chloroplasts for concrete day, R0 (μm) is chloroplasts radius that was not stressed by drought, Rn (μm) is the radius of the chloroplasts that was stressed by dryness, for n days, ab0 (μm) is the barley cell that was not stressed by drought and abn (μm) is the size of the cell that was stressed by dryness for n days. If the chlorophylls concetration in chloroplasts is known, it is possible to determine the probability of the light beam pass p(k) through the k chloroplasts for different wavelengths:
(53)p(k)=qk.
Based on the knowledge of molar extinction coefficients for pigments and spectrophotometry measurements of pigments content probability of light beam passes through the chloroplasts for wavelengths 550 nm and 680 nm was calculated; see Figures 2(a) and 2(b).
Probability of the light beam passes through k chloroplasts for wavelengths 550 nm (a) and 680 nm (b).
It is necessary to do some simplifications in case of the internal reflectance theoretical spectra modelling. The first of them is precondition, in which cell walls transmittance and reflectance are not a function of wavelength (at least for concrete day of measurement). Chlorophylls concetration in chloroplasts, pigments molar extinction coefficients, and cell walls transmittance and reflectance are input parameters for (44). In case of cells and chloroplasts size, it is presumed that these parameters are changing linearly during the dryness. In the next step, the mean optical length, number of cell walls passes, and reflections are optimized; see Tables 1 and 2.
Parameters used for modelling of internal reflectance theoretical spectrum of Spring barley (control group).
Wavelength interval (nm)
n
m
τ
ρ
x (μm)
400–500
2
3
0,9
0,1
5–50
500–640
8–12
2
0,9
0,1
200–250
640–700
2
3
0,9
0,1
80
n: number of cell wall passes, m: number of cell walls reflections, τ: cell walls transmittance; ρ: cell walls reflectance, x: light beam geometrical length.
Parameters used for modelling of internal reflectance theoretical spectrum of Spring barley (after 9 days of dryness).
Wavelength interval (nm)
n
m
τ
ρ
x (μm)
400–500
2–4
3
0,88
0,12
20–45
500–640
6–12
2
0,88
0,12
200–270
640–700
2
3
0,88
0,12
20
n: number of cell wall passes, m: number of cell walls reflections, τ: cell walls transmitance; ρ: cell walls reflectance, x: light beam geometrical length.
Good agreement between the experimental internal reflectance spectrum and proposed theoretical model was achieved; see Figure 3.
Comparison of the experimental and theoretical internal reflectance spectrum of Spring barley stressed by dryness. No stress—control group, Day 9—duration of drought stress, *experimental spectrum, **theoretical spectrum.
Experimentally measured spectra of diffuse reflectance may be quantitatively very different compared to theoretical spectra. It can happen due to the total reflection of some light beams on the cell walls. The cells do not have to be close to each other, so in fact, total reflection may occur when the light passes from the optically more dense environment (palisade cells) to the optically less dense environment (air in the intercell space).
If the experimentally measured spectra of the internal reflectance are compared with the theoretical spectra calculated according to proposed model, good agreement is achieved. This good agreement is achieved both for control group and plants stressed for 9 days by dryness. Theoretical spectra exhibit interesting informations. Firstly, it is useful to divide the reflectance spectrum into three sections according to the wavelength interval (400–500 nm; 500–640 nm and 640–800 nm). Wavelength interval 400–500 nm is described by relative strong chlorophylls and carotenoids absorption; the mean optical length of the light beam is significantly shorter than for the light with wavelength from interval 500–640 nm; this interval may be characterized by a relative weak pigments absorption. The shortest light mean optical length was calculated for the wavelength 680 nm due to the strong Chl a absorption. The rapid increase of the internal reflectance for wavelengths longer than 700 nm is mainly caused by water content decrease during the drought. The changes in the internal reflectance spectrum are not influenced only by cells and chloroplasts number, size, shape, and pigments content. Very important is also total internal reflection contribution. Weight of the total reflection increases with decreasing water content in the intracellular space.
4. Conclusions
Experimentally determined spectra of the internal reflectance of Spring barley leafs were compared with proposed theoretical model. Good agreement between theoretical model and experimental data was observed. It was revealed that most influencing factors for nonstressed leafs internal reflectance spectra modelling are cells and chloroplasts number, size and shape, pigments content, cell walls transmittance, and reflectance. However, for the Spring barley stressed by drought, water content was critical. Although water absorption can be neglected in visible light spectrum, intercellular spaces fulfilled by air are responsible for the total reflection, especially for the light with wavelength longer than 700 nm. However, internal reflectance is two orders lower than surface reflectance; it has significant information value about plant physiological state.
Acknowledgments
This paper has been elaborated in the framework of the project opportunity for young researchers, reg. no. CZ.1.07/2.3.00/30.0016, supported by Operational Programme Education for Competitiveness and cofinanced by the European Social Fund and the state budget of the Czech Republic. The authors are grateful to the Ministry of Education of the Czech Republic for the financial support of the project of OP VaVPI “Regional Materials Science and Technology Centre (RMSTC)” no. CZ.1.05/2.1.00/01.0040.
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