Ratios, proportions, and mixtures of chlorophylls: corrections to spectrophotometric methods and an
approach to diagnosis.
I. M. Henderson
Ecology Group
Institute of Agriculture and Environment
Massey University
Palmerston North
New Zealand.
Supporting Information Appendix 1.
Further detail and explanation of mathematical derivations were requested by a reviewer.
For
consistency, equations are numbered the same as in the main text and additional equations are given
letter subscripts where needed.
A.
The relationship between an absorbance ratio and the proportions of a mixture.
Let ai,j be the specific absorbance of one unit of chlorophyll i at the characteristic peak (λj) of
chlorophyll j, Ai is the measured absorbance at wavelength λi, and Ci is the concentration of
chlorophyll i in the sample . A mixture of two chlorophylls has proportion P of chlorophyll 1 and (1P) of chlorophyll 2, and the observed absorbances will be:
𝐴1 = 𝐶1 𝑎1,1 + 𝐶2 𝑎2,1
𝐴2 = 𝐶1 𝑎1,2 + 𝐶2 𝑎2,2
Since 𝑃 = 𝐶1 ⁄(𝐶1 + 𝐶2 )
𝐴1 = 𝐶1 𝑎1,1 + 𝐶1 ((1 − 𝑃)⁄𝑃)𝑎2,1
𝐴2 = 𝐶1 𝑎1,1 + 𝐶1 ((1 − 𝑃)⁄𝑃)𝑎2,2
In a ratio the concentration (C1) is eliminated and:
𝑃𝑎
𝐴
+(1−𝑃)𝑎
𝑟 = 𝐴1 = 𝑃𝑎1,1 +(1−𝑃)𝑎2,1
2
1,2
2,2
(1)
This can also be expressed in terms of the ratio (x) of the two chlorophylls: x = P/(1-P) so P = x/(x+1)
and (1-P) = 1/(x+1). Substituting these into (1):
𝑥𝑎
+𝑎
𝑟 = 𝑥𝑎1,1 +𝑎2,1
1,2
2,2
(1a)
The ai,j can also be expressed in terms of dimensionless ratios. The minimum value of r will be when
only chlorophyll 2 is present so P = 0 and from (1), rmin = a2,1/a2,2. The maximum ratio is when only
chlorophyll 1 is present, P = 1 and rmax = a1,1/a1,2. Let S be the ratio of specific absorbances at the
respective peaks, a1,1/a2,2. Substituting these ratios into (1a) and simplifying gives:
𝑆𝑥+𝑟𝑚𝑖𝑛
𝑚𝑎𝑥 )+1
𝑟 = (𝑆𝑥⁄𝑟
(1b)
The inverse relationships are (from 1a):
𝑎
𝑥 = 𝑟𝑎2,1
−𝑟𝑎2,2
1,2 −𝑎1,1
And from (1b)
1
𝑟
𝑥 = 𝑆 ((𝑟⁄𝑟𝑚𝑖𝑛
−𝑟
𝑚𝑎𝑥 )−1
)
(1c)
The ratio x will be undefined when one of the chlorophylls is absent (r = rmax), so it will be more
useful to express this relationship in terms of the proportion of one of the chlorophylls:
𝑃 =
𝑃=
𝑥
𝑥+1
=
1
(1+1⁄𝑥)
1
(𝑟⁄𝑟𝑚𝑎𝑥 )−1
)
1+𝑆(
𝑟𝑚𝑖𝑛 −𝑟
(2)
This expression is undefined when r = rmin but in that case P is clearly zero which is an easier error to
trap. Multiplying (2) through by (rmin – r) gives the alternative form
𝑃=𝑟
𝑟𝑚𝑖𝑛 −𝑟
𝑚𝑖𝑛 −𝑟+𝑆𝑟 ⁄𝑟𝑚𝑎𝑥 −𝑆
From which it can more easily be seen that this relationship will be linear in r only if S = rmax.
B.
What ranges of parameters are realistic?
It would be possible to insert a wide range of parameter values into (2) but these may not be realistic
for a combination of two chlorophyll spectra. The shape of a spectral peak can be approximated by a
Gaussian curve (Aguirre-Gomez et al. 2001) and for the red peaks of chlorophylls in acetone the
spread usually corresponds to a standard deviation of 8-10 nm. If the peaks are identical in height (S
= 1) but differ by one standard deviation rmax will be 1.645 and rmin = 1/rmax = 0.608. This set of
parameters is similar to those defining the upper curve in Fig. 2 of the main text. Given that these
methods are likely to be considered when the peaks are distinct enough to provide resolution but still
overlap considerably (e.g. the 10 nm difference between chlorophyll a and bacteriochlorophyll d) the
general features of that curve are to be expected. However, if the peaks differed by 20 nm the range
of ratios will be much greater (rmax = 7.4 and rmin = 0.14. If the specific absorbance coefficients (ai,i)
of the two pigments differ (S ≠1) or the peak widths differ, other forms are possible. For example
with a 9 nm difference between peaks, equal standard deviations but a1,1 50% higher than a2,2, rmax =
1.5, rmin = 0.67 and S = 1.5 so the relationship between P and r will be linear as in the middle curve of
Fig. 2. The parameters of the lower curve in Fig. 2 can be obtained from the case of a 13 nm
difference in peak location, 20% greater spread and 4-fold greater specific absorbance in the peak for
one of the chlorophylls; an admittedly extreme case.
C.
Elaboration of the method of Moss (1967a & b)
Moss (1967a) empirically established the relationship between the ratio of absorbances A 430/A410 and
the proportion of phaeophytin in a mixture with chlorophyll a. In Moss (1967b) he shows how this
proportion, and absorbance at 665nm, can be used to estimate concentrations of both pigments. Both
pigments have red absorbance peaks close to 665nm but the specific absorbance of chlorophyll a is
greater than that of phaeophytin a (90.8 and 55.2 L g-l cm-l are the values Moss used).
Following Moss (1967b) but in the notation used here
𝐴665 = 𝐶1 𝑎1 + 𝐶2 𝑎2
Where C1 is the concentration of chlorophyll a, C2 the concentration of phaeophytin a and ai their
respective specific absorbances.
Let x = C1/C2 so:
𝐴665 = 𝐶1 𝑎1 +
𝐶1
𝑎
𝑥 2
𝑥𝐴665
1 +𝑎2
𝐶1 = 𝑥𝑎
And similarly:
𝐴
𝐶2 = 𝑥𝑎 665
+𝑎
1
2
Substituting x from (1c)
𝐶1 =
𝐴665
𝑟⁄𝑟𝑚𝑎𝑥 −1
))
𝑟𝑚𝑖𝑛 −𝑟
(𝑎1 +𝑎2 𝑆(
𝐶2 =
𝐴665
𝑟
−𝑟
𝑎
(𝑎2 + 1 ( ⁄ 𝑚𝑖𝑛
))
𝑆 𝑟 𝑟𝑚𝑎𝑥 −1
The ratio of a1/a2 is sometimes called the acid ratio as it would be the ratio of absorbances at 665nm
before and after addition of acid to convert chlorophyll to phaeophytin. This ratio is usually taken as
1.7 for pure chlorophyll a (Lorenzen 1967) but Moss (1967b) used the equivalent of 1.64. It is also
convenient to use the inverse of the specific absorbance coefficients scaled to mg in equations; e.g.
1000/a1 = 11.01 mg cm L-1 using the figures of Moss, but 11.5 is more reasonable given recently
derived specific absorbances of approximately 87 L g-1 cm-1 (Ritchie 2006). The absorbance peak of
chlorophyll in 90% acetone is also now generally reported as 663 nm. With these modifications
equations in a more familiar form can be produced:
𝑆
(𝑟 ⁄𝑟𝑚𝑎𝑥 )−1
𝐶ℎ𝑙𝑎 (mg L-1 ) = 𝐹 × 𝐴663 × 11.5⁄(1 + 1.7 (
1
1
𝑟𝑚𝑖𝑛 −𝑟
𝑟
))
−𝑟
))
𝑚𝑎𝑥 )−1
𝑃ℎ𝑎 (mg L-1 ) = 𝐹 × 𝐴663 × 11.5⁄(1.7 + 𝑆 ((𝑟⁄𝑟𝑚𝑖𝑛
(3)
(4)
Where F = volume of acetone (ml) / (volume of lake water filtered (L) x cuvette path length (cm))
D.
Corrections to the method of Parkin & Brock (1981).
Parkin and Brock (1981) based the form of their equations on the proportion of absorbance at a
particular wavelength due to each of the chlorophylls in a mixture. As before, let ai,j be the specific
absorbance of chlorophyll i at the peak of chlorophyll j, Ci be the concentration of chlorophyll i in the
sample, Di be the proportion of absorbance at the peak of chlorophyll i that is due to chlorophyll i, and
P is the proportion of chlorophyll 1 (P = C1/(C1+C2)).
𝐶1 𝑎1,1
𝐷1 = 𝐶
1 𝑎1,1 +𝐶2 𝑎2,1
1
Since 𝐶2 = 𝐶1 (𝑃 − 1) the concentrations can be eliminated and
𝐷1 =
𝑎1,1
𝑎1,1 +(𝑎2,1 (1⁄𝑃 −1))
Substituting Eq. 2 for P
𝐷1 =
𝑎1,1
𝑟⁄
𝑟𝑚𝑎𝑥 −1
)))
𝑟𝑚𝑖𝑛 −𝑟
𝑎1,1 +(𝑎2,1 (𝑆(
and noting that a2,1/a1,1 = rmin/S, this simplifies to:
𝐷1 =
𝑎1,1
𝑟⁄
𝑟𝑚𝑎𝑥 −1
))
𝑟𝑚𝑖𝑛 −𝑟
𝑎1,1 +(𝑟𝑚𝑖𝑛 𝑎1,1 (
Now the specific absorbance (a1,1) can be eliminated and simplified to
𝐷1 =
𝑟𝑚𝑖𝑛 −𝑟
𝑟
𝑟( 𝑚𝑖𝑛 −1)
𝑟𝑚𝑎𝑥
And rearranged to
𝐷1 =
𝑟𝑚𝑎𝑥 ((𝑟𝑚𝑖𝑛 ⁄𝑟 )−1)
𝑟𝑚𝑖𝑛 −𝑟𝑚𝑎𝑥
(7)
Following similar steps
𝐷2 = 𝑟
𝑟−𝑟𝑚𝑎𝑥
𝑚𝑖𝑛 −𝑟𝑚𝑎𝑥
(8)
(7-8) can be expressed in a similar form to that used by Parkin & Brock (1981, see Eq. 5a and 6a in
main text) as
𝐷1 = [(1⁄𝑟 × 𝑐) − 𝑔]
(7a)
𝐷2 = [(𝑟 × 𝑒) − 𝑔]
(8a)
Where c = (rmax x rmin)/(rmin - rmax), g = rmax /(rmin - rmax), e = 1/(rmin - rmax).
Their full method could thus be written
𝐶ℎ𝑙𝑎 = [(𝑑⁄𝑎 × 𝑐) − 𝑔] × 𝑎 × 11.9 × 𝐹
(7b)
𝐵𝐶ℎ𝑙𝑑 = [(𝑎⁄𝑑 × 𝑒) − 𝑔] × 𝑑 × 10.2 × 𝐹
(8b)
Where a = absorbance at 663nm, d = absorbance at 654nm, F = volume of acetone (ml) / (volume of
lake water filtered (L) x cuvette path length (cm)). 11.9 and 10.2 are derived from the inverse specific
absorbance coefficients (1/ai,i) with units g cm L-1. Based on Parkin & Brock’s (1981) equations,
likely values for the remaining parameters are c = −0.87, e = −0.99, g = −1.55.
E.
Corrections to the method of Caraco & Puccoon (1986)
Equations 7b and 8b can be rearranged to closely match the form of Caraco & Puccoon’s (1986)
equations. Their “Ext” parameters are the inverse of the type used by Parkin & Brock (1981) with
units L mg-1 cm-1. Let r = a/d, b = g/f, d =g/e
𝐶ℎ𝑙1 = 𝐹 × 𝐴1 (1⁄𝑟 − 𝑏) × (1000⁄𝐸𝑥𝑡1 × 𝑐)
(11)
𝐶ℎ𝑙2 = 𝐹 × 𝐴2 (𝑟 − 𝑑) × (1000⁄𝐸𝑥𝑡2 × 𝑒)
(12)
where Ai = Measured absorbance at the peak of chlorophyll i. The factor of 1000 is to give
chlorophyll concentration in mg L-1.
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