1. Introduction

When an absorbing-and-scattering specimen is investigated by using a standard absorption spectrophotometer, a measurement of the attenuance lg(P₀/P) does not provide the true absorbance of the sample, and the measured attenuance depends on the acceptance angle of the instrument (or the sample-to-detector ratio); here P₀ denotes the output of the detector when it views the collimated incident beam leaving the reference cell and P is the output when the cell is filled by the sample under investigation. Reliable methods for finding the true absorbance, either by separating the contribution made by scattering [1] or by circumventing it altogether by placing the sample inside an integrating cavity [2], have been available for some time. A 2004 paper [3], written by the present author and his associates, suggested a practicable procedure for a quantitative analysis of the scattering spectrum by dividing it into two parts, one caused by selective scattering and the other by non-selective scattering; the approach, which relied on the assumption that absorption and selective scattering from a single pool of pigments satisfy the Kramers-Kronig relations and on a milder assumption concerning the wavelength dependence of non-selective scattering, was shown to account for the scattering and absorption spectra of a wide variety of suspensions, including those of red blood cells. An independent approach, utilizing similar ingredients, was presented some years later by Nonoyama and coauthors [4], who also addressed an issue not treated in the earlier work, namely the lower absorbance (or hypochromism) of a suspension of pigmented cells as compared to the absorbance recorded by disrupting the cells and allowing the pigments to disperse throughout the sample holder. They commented that “this perceived hypochromism can be accounted for by considering two important issues: the acceptance angle of the instrument and the combined scattering and absorption effect of light on the particles”. The purpose of the present paper is to point out that hypochromism, properly so called, should be used when one compares two spectra neither of which suffers from scattering losses, and to show that Duysens’ theory of screening hypochromism can be applied, in its original form [5] or in a trivially amended fashion [6], to understand the hypochromism of red blood cells.

2. Theoretical background

The problem before us, when stripped of all the distracting details specific to a particular system, may be stated as follows: Find the relation between A′ and A, where A′ is the true absorbance of a suspension of randomly distributed, identical clusters of subunits, and A is the absorbance recorded after disrupting the clusters, the sole purpose of disruption being that of turning the suspension of clusters at hand into a suspension of randomly distributed subunits. The reader should observe that a “subunit” (which will henceforth be called a particle) may be a single molecule or a macromolecule carrying one or more chromophores. Closely entwined with clustering, yet demanding a separate investigation, is the phenomenon of scattering. We will assume that each cluster contains k subunits, and that scattering is either negligible or that an appropriate correction has been made to procure a scattering-free absorption spectrum of the suspension of clusters. It will be more convenient to work with the napierian absorbances ℰ = ln(10) × A and ℰ′ = ln(10) × A′, and no confusion is likely to arise if the shorter term absorance is applied also to ℰ and ℰ′.

We consider two homogeneous samples of absorbing entities, each with a pathlength l; in one sample, the absorbing entities are randomly dispersed particles at a concentration N, whereas the other sample has clusters, also randomly dispersed, whose concentration is N′ = N/k. Using unprimed symbols for the former sample and primed symbols for the latter, we apply the BLB law to each sample by writing

Amesz, Duysens and Brandt [9] proposed the following expression for σ′, the absorption cross section of a cluster characterized by a physical cross section s (defined as the area of the projection of a cluster on a plane perpendicular to the monitoring beam):

We now divide Eq. (1b) by Eq. (1a), and use Eq. (2) to get the relations

In order to interpret Y, it is necessary to propose a concrete model for the shape of a cluster, which in turn makes it necessary to distinguish, by using an appropriate subscript, one model from another. If we assume that each cluster is a right circular cylinder of cross section s and length d, and assume further that each cluster is aligned so that its plane faces are perpendicular to the monitoring beam, a cluster can be treated as a minuscule absorption cell (containing k particles in a volume v_c = sd) and we can apply the BLB law to it, which gives the following expressions for the transmittance (T_c) and absorbance (ℰ_c):

If, on the other hand, we assume that each cluster is a sphere of diameter d, cross section s = πd²/4 and volume $v_s=\frac{2}{3}sd$, we can calculate its transmittance by using the relation

For clusters of other shapes, it will be more convenient to use a numerical simulation, such as that developed by Halling [10] and found, in the case of spherical clusters, to be in agreement with the analytical approach discussed above.

3. Flattening in the spectrum of red blood cells

Once A becomes available, one can see whether a particular model succeeds in reproducing the absorption spectrum of a suspension of clusters. Though ℰ_c and ℰ_d are not known, each is proportional to A, and the proportionality constant (= m, say) can be treated as an adjustable parameter in a non-linear least squares fitting procedure involving a comparison of the experimental spectrum A′ and a particular model (A′_c = ln(10) × ℰ′_c or A′_s = ln(10) × ℰ′_s). Figure 1 displays the absorption spectra of red blood cells suspended in 0.6% NaCl solution (intact cells) and in distilled water (lysed cells); a flattened spectrum A′_c calculated by using the relation [cf. Eq. (7)]

 

Fig. 1 Plots of A(λ), A′(λ) (representing absorption spectra of suspensions of lysed and intact human red blood cells, respectively), and of A′_c(λ), a model spectrum calculated by applying Eq. (11). Corrections for scattering were applied by recording two spectra at different distances from the input port of an integrating sphere and analyzing the data in accordance with the method developed by Latimer and Eubanks [11]. For obtaining red blood cells, approximately 150 μl of freshly collected human peripheral blood was added to 5 ml of 0.6% NaCl solution containing 7 mM trisodium citrate as an anticoagulant; erythrocytes were sedimented at 1000 g, washed twice and resuspended either in 0.6% NaCl (intact cells) or in distilled water (lysed cells). For more experimental details, see [3].

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A different expression for the ration ℰ′/ℰ was proposed by Vekshin [13, 14]; though his model can account for the observed hypochromism of red blood cells, it is physically unreasonable because the best fit between the model and the experimental data leads to the conclusion that the number of heme subunits per cluster is smaller than ten [14].

A serious discrepancy between the experimental spectrum A′(λ) and the predicted spectrum A′_m(λ) (where m = c, s, ···), would have implied that either the procedure used for finding the scattering-corrected absorption spectrum is at fault or that at least one of the basic assumptions underlying Duysens’ analysis is unrealistic. The absorbance of a cluster must always be lower than that of a solution containing the same number of subunits, and it will be noted in the region close to 500 nm, A′ is slightly larger than A, but the difference is negligible for practical purposes. Figure 1 also reveals a noticeable difference between A′ and A′_c at wavelengths shorter than 300 nm, but this is probably due to the contribution (in the latter spectrum) of absorption by the proteins. The central assumption in the Duysens approach is that the spectral properties of subunits do not change when they are inside a cluster, and this is most likely to hold good when the pigments are buried inside proteins, as in hemoglobin or in light-harvesting complexes [6], since pigment-pigment distances remain large even when neighboring proteins are tightly packed.

4. Concluding remarks

Since Duysens’ analysis has been criticized in the past [13, 15], a few clarifying remarks appear to be in order. Bustamante and Maestre [15] chose to treat a homogeneous suspension of clusters as an inhomogeneous assembly of subunits, where inhomogeneity is defined as the non-uniformity of the concentration of the absorbing entities across a plane perpendicular to the monitoring beam. When one replaces a homogeneous sample with an inhomogeneous sample, one can calculate an average transmittance, but one cannot speak of an absorbance, since the number of absorbing entities encountered by each ray in the monitoring beam is no longer a constant. As has been pointed out above, each of the two samples whose absorbances are compared here may be considered to be homogeneous, since the absorbing entities (subunits or clusters) are distributed uniformly throughout the cuvette in either case, and the BLB law is applicable to the subunits in one case and to the clusters in the other. Vekshin’s criticism of the papers of Duysens [5], Papageorgiou [16] and Fukshansky [17] has been marred by his unsympathetic reading, if not exactly by a wilful misreading, of these works, and he has failed to spot what a sympathetic and patient reader would sooner or later recognize as obvious typing errors or notational ambiguities. His concluding paragraph, which ends with the remark “Thus the “sieve effect” is erroneous”, merely shows that he has not grasped the fact that what he calls “screening hypochromism” is another name for the phenonmenon named flattening by Duysens, and subsequently called “sieve effect” [18] or “package effect” [19]. The former term is infelicitous and misleading, for it carries the connotation that sieving is a consequence of clustering or packing. Even a sample in which chromophores (or subunits) are randomly dispersed may be said to act as a sieve if its transmission is not too low; all that clustering does is to make the sample a better (more transmissive) or leakier sieve.