New Phytol (1975) 75» 11-20. A THEORETICAL ANALYSIS OF THE CONTRIBUTION OF ALGAL CELLS TO THE ATTENUATION OF LIGHT WITHIN NATURAL WATERS I. GENERAL TREATMENT OF SUSPENSIONS OF PIGMENTED CELLS BY J. T. O. KIRK CSIRO, Division of Plant Industry, Post Office Box 1600, Canberra City, A.C.T. 2601, Australia {Received 20 November 1974) SUMMARY A theoretical treatment of light attenuation within suspensions of phytoplankton, which takes account of the heterogeneous distribution of pigment in the system, is described. As a measure of the effect of canopy structure on light attenuation, a new parameter, the penetration coefficient (PA) is introduced, this being defined as the ratio of the downward monochromatic radiation flux at a given depth in the suspension to that within an equivalent solution of the algal pigments. A set of rules is established, applicable to cells of any shape or orientation, which characterizes the relationship between the penetration coefficient and the parameters of canopy structure (cell number, size, pigment composition etc.). A standardized procedure for calculating the spectral distribution of, and the vertical attenuation coefficients for, photosynthetically active radiation within model phytoplankton suspensions, is outlined. INTRODUCTION In water bodies of moderate productivity the phytoplankton make only a small contribution to the attenuation of light with depth (Tailing, i960). In highly productive waters, by contrast, suspended algal cells can intercept a substantial proportion of the incident light. With increasing eutrophication of lakes, rivers and coastal w aters in all parts of the world, situations in which light absorption by the phytoplankton is quantitatively important 1 are becoming increasingly common. It has been pointed out by Tallmg (1970) that as yet we have little understanding of the way in which the structure of the phytoplankton 'canopy' modifies its light absorption properties. As a step towards such an understanding an analysis, using model systems, of the contribution of suspended algal cells of varying size, shape, pigment content and pigment composition, to the attenuation of light within natural waters is being carried out: the theoretical basis of this analysis is presented here. A brief account of some of the conclusions of this w ork has been given recently (Kirk, 1974).
12 J. T. O. KIRK THEORETICAL CONSIDERATIONS (i) Aims and assumptions The primary aim of this work is to elucidate the relationship between the structure of the phytoplankton canopy and light attenuation within it. This aim would be achieved by establishing a set of rules with the help of which it would be possible to understand the changes in light attenuation that would result from specific kinds of change in canopy structure. A subsidiary aim is to develop a standardized procedure for calculating the spectral distribution of photosynthetically active radiation (PAR), and the vertical attenuation coefficient for PAR M'ithin a phytoplankton suspension, and to apply it to hypothetical model suspensions of various types of algae. With the help of such a procedure, the changes in light attenuation brought about by specific modifications in the canopy could be expressed in quantitative terms. The approach used is similar to that adopted by Duysens (1956) and Rabinowitch (1956) in their treatment of absorption spectra of suspensions of pigmented particles. The particles are assumed to be randomly distributed within the medium. It is assumed that they are non-scattering and that transmission of a light beam through a particle obeys the laws of geometrical optics. Duysens showed, on theoretical grounds that a dilute suspension of cells of this type should obey Beer's law, i.e. the optical density is proportional to cell concentration. A more general proof has recently been developed by Stokes (1975). Using an equation derived on the basis of these assumptions Duysens was able to account for the observed flattening of the absorption spectrum of a Chlorella cell suspension compared to that of an equivalent pigment solution. The treatment used in the present work differs from that of Duysens in a number of respects. Duysens was interested in absorption spectra of cell suspensions as seen in the spectrophotometer and concerned himself mainly with ^^^Joi^^^, the ratio of the optical density of the suspension to that of an equivalent solution. In the present work it is the absolute amount of radiation available for photosynthesis which is of interest, and the most relevant parameters are I^^^ and /sus/^sob which respectively are the downward intensity of monochromatic radiation on a horizontal plane at some given depth in the suspension, and the ratio of that intensity in the suspension to the intensity in an equivalent solution. By an equivalent solution is meant a system in which the cellular material is so finely subdivided that it behaves, in effect, like a solution, but with the proviso that the extinction coefficients of the pigments are the same as in the living cell. The value of this imaginary system is that it corresponds to one extreme of canopy structure: that in which the suspension contains a very large number of very small cells. It can therefore be used as a constant reference system when we wish to examine the effects of changes in canopy properties (cell size, shape, number, pigment content etc.) on light attenuation. The ratio of downward intensity at a given depth in the suspension to that in the solution at a given wavelength is referred to in this paper as the penetration coefficient, P^. p = () Duysens established relationships between a^yjo^^^i ^^^d the absorption properties of individual cells for oriented cubical, and for spherical, cells. His generalized equation for randomly oriented cells of arbitrary shape is not of a form which permits conclusions to be reached concerning the relationship between the characteristics of the cell and
Attenuation of light by algae 13 light attenuation by the suspension. An aim of the present work is to treat the system in such a way as to arrive at conclusions applicable to suspensions of cells of any shape or orientation. To simplify the theoretical treatment attention in this paper has been confined to direct parallel sunlight, which is assumed to be refracted, but not substantially refiected, at the water surface. (2) Attenuation of light in a suspension of pigmented cells Consider a dilute suspension containing n randomly oriented pigmented cells, of arbitrary size and shape, per cubic metre illuminated at normal incidence by a parallel beam of monochromatic light, of i m^ cross-section. Initially we shall ignore light absorption by the medium. The7th cell in the illuminated column would, in the absence of the other cells absorb a proportion Ajaj of the light, where Aj is the projected area (in m^) of the cell on a plane at right angles to the beam, and aj is the fraction of the light incident upon the cell which it absorbs. For a i-m layer of suspension the optical density (In ij?;,) due to the;th cell is In 1/(1 -Ajaj), which, since Ajaj is small, is approximately equal to A a : Applying Beer's law (Duysens, 1956) the optical density Z),^,, due to all the cells is given by n /),, = I Ajaj = naa where Aa is the mean value for all the cells in the suspension of the product of the projected area and the fractional light absorption. It follows that We shall now consider the case in which the suspension is illuminated from above with a parallel monochromatic beam which, within the water, is at an angle, j?, to the horizontal. The downward flux of radiation per unit area on a horizontal plane just below the water surface is / ; the downward flux on a horizontal plane at depth z metres is /jug. At the wavelength in question the water has an absorption coefficient k{m~^), given by k = w+g where w is the absorption coefficient of pure water, and g is the absorption coefficient due to gelbstoff (dissolved yellow plant breakdown products, present in all natural waters). We may now write T _ T ^ sus z cosec P being the pathlength of the beam between the surface and depth z metres. (3) Comparison of a suspension with the equivalent solution If the intracellular pigment concentration is Cmgm"-', and the average volume per cell is v nr', then if the pigments were uniformly distributed throughout the system they would be present at a concentration of Cnv mgm~^. If the natural logarithm specific absorption coefficient of the pigment mixture {In iji for i mg m"^, i m pathlength) at the appropriate wavelength is 7 m^ mg~s then for a pathlength of z cosec ^ metres the optical density, D^^i* due to the dispersed pigment is given by JDJOI = Cnvy z cosec j8
14 J. T. O. KIRK The downward flux of radiation per unit area on a horizontal plane at depth z metres in the completely dispersed system, /^^h is given by J J p~ {k +Cnvy) z Using equations (i), (3) and (4) the penetration coefficient can now be expressed in terms of the various parameters of the system :^ - Aa)nz cosec fi /_\ Because of the complex nature of the factor Aa in any suspension of randomly oriented cells of arbitrary shape, not all the implications of equation (5) for light attenuation in phytoplankton canopies are immediately obvious. To arrive at some general conclusions we shall first consider the properties of suspensions of an imaginary class of cells that we shall refer to as constant pathlength cells, and then extend the conclusions to cells of any type. (4) Constant pathlength cells Constant pathlength cells are defined to be shaped, and oriented with respect to the incident light, in such a way that a thin light beam passes through the same pathlength of cell material no matter at what point it traverses the cell. Consider a suspension of identical constant pathlength cells, each with volume v m^, projected area A m^, pathlength d m, natural logarithm optical density D, and fractional light absorption a. From equation (5) Since and = {Cvy Aa)nz cosqc a = I e"^ it follows that or, where /WPA = A{D-i+e~'')nz cosec p inp;, = na.f{d)z cosec j5 (6) fid) = D-i+e-"" When D is zero, f{d) is also zero. In addition, f'{d) = i-e-^ Thus for all values of D greater than zero, /'(D) is positive, and so f{d) is greater than zero and increases in value as D increases. From this, by reference to equation (6), it follows that P^ is greater than i.o, and when A is constant P^ increases in value as D (or a) is increased by raising the intracellular pigment concentration (increase in C) or changing the wavelength to one more strongly absorbed (increase in y). In our further consideration of the dependence of P^ on the parameters of the system the following lemma, which can readily be proven, is of use: that when G is any factor greater than zero, and Z) > o, f(gd)>gf{d) forg>i
and Attenuation of light by algae 15 f{gd)<gf{d) iovg<i We shall now consider the effect of decreasing the cell number, but keeping the total pigment and biomass in the system constant i.e. A and a increase together. cosec From the lemma it follows that cosec = naf{d) z cosec fi i.e. the penetration coefficient is increased. Another kind of change in canopy structure, is an increase in total biomass (resulting from an increase in n or v) at constant total pigment. If cell numbers are increased at constant cell size (^)new = A D "77 where//>i ew = Hn.A.f(^j z cosec From the lemma it follows that i.e. the penetration coefficient is decreased. If cell size is increased at constant cell number («)new = n. A.-^ z cosec fi = inp 1 where//>i = n. H^A.f{H-W) z cosec
i6 J. T. O. KIRK From the lemma it follows that < n. H^A. H'^fiD) z cosec ^ = inp i.e. the penetration coefficient is decreased. It can also be shown that for a given increase in biomass the increase in n causes a greater diminution in P^ than the corresponding increase in v. The conclusions so far have been derived for suspensions of cells of uniform size. A suspension of constant pathlength cells of varying sizes can be regarded as a mixture of a number of different suspensions, each containing cells of one size. From our earlier assumption that the optical density of the suspension is equal to the sum of the optical densities of the components it follows that 7^,^,^, (and hence Px) will vary in the same qualitative manner with changes in canopy structure in the mixed suspension as in the uniform suspension. (5) Cells of arbitrary shape and orientation Any cell, whatever its shape and orientation can be considered as an aggregate of A^ very narrow rods parallel to the light beam each of cross-sectional area, 5^ m^. We saw in section 2 that the optical density of the suspension due to the /th cell is Ajaj. But j j where a^ is the fractional light absorption of the rth rod. baa^ is also the optical density of the suspension due to the rth rod of the^th cell. Thus, the optical density of the suspension is equal to the sum of the optical densities due to all the rods of which the cells are composed. Fach of these rods can be regarded as a very narrow cell which approximates closely to the constant pathlength criterion. As each of the modifications (changes in C, y, a, A, n, v) applied in the previous section to the constant pathlength cells is applied to the arbitrary cells, there will be corresponding changes in the 'constant pathlength' rods of which these arbitrary cells are composed. Consequently these modifications in canopy structure will produce the same kind of changes in the light attenuation properties of suspensions of cells of arbitrary shape and random orientation as they do in suspensions of constant pathlength cells. The relationship between phytoplankton canopy structure and light attenuation properties can conveniently be summarized in the following propositions which express the consequences of changes in canopy characteristics in terms of the penetration coefficient, P^: (1) P;>^ (ii) At constant cell size (constant A), when a is increased (by raising the intracellular pigment concentration or altering the wavelength), P^ increases. (iii) At constant total pigment and biomass in the system, when both A and a are increased (by decreasing cell number and thus increasing cell size), P^ increases. (iv) At constant total pigment in the system, when biomass is increased (by increasing cell number or cellular volume), P^ decreases. (v) At constant A and a, when n or z are increased or js is decreased P^ increases. This last proposition can be seen immediately to follow from equation (5).
Attenuation of light by algae 17 (6) Calculation of spectral intensity distribution and vertical attenuation coefficients For calculation of I^^^ for model suspensions all the parameters in equation (3) can readily be evaluated with the exception of Aa. For any given orientation of a cell to the light beam the pathlength, d, of a thin pencil of light through the cell will vary, in a manner determined by the shape of the cell, according to where the pencil passes through. At any given point, specified by the x and y coordinates on a plane normal to the beam, where the pathlength is d^yy the absorption, a^y, for a pencil of light is given by The mean value of Aa averaged over all values of x and y will have to be found by appropriate integration procedures. In addition, both A and the various values of a^y, will vary according to the orientation, 0, of the particle to the light beam. Thus multiple integration will have to be carried out to determine the value of Aa averaged over all values of x, y and 6. It is clear that, except for the special case of spherical cells, the determination of Za will not be a simple task. Given that all the parameters in equation (3) can, in principle, be evaluated for a specific type of algal suspension illuminated with PAR of a specified spectral distribution, then it should be possible to calculate /.^^ for a series of different wavelengths and in this way determine the spectral distribution of light intensity at any depth, z m, in the suspension. The calculation can be carried out for intensity expressed in terms of energy, or in terms of quanta. Since the photosynthetic process is a photochemical one, there is now a consensus of opinion that light intensities which are intended to be related to primary production should be in terms of quanta cm"^ s" \ for unit wavelength or for a specified wavelength range: it is also agreed that the most useful measurements of total light available for photosynthesis are those of quanta cm"^ s"^ for the wavelength range 350-700 nm (UNESCO, 1966; Jerlov and Nygard, 1969a). As a corollary, it has been suggested that the most useful way of characterizing the attenuation properties of natural waters for photosynthetically active radiation is in terms of the vertical attenuation coefficient for the whole waveband between 350 and 700 nm (Smith, 1968; Jerlov and Nygard, 1969a). For monochromatic light the vertical attenuation coefficient will be the same whether intensities are measured in terms of energy or quanta. However, for a broad waveband the spectral distribution of energy will be somewhat different from the spectral distribution of quanta. Therefore, for any medium with spectrally non-uniform absorption the attenuation of light in the whole waveband, and consequently the values obtained for vertical attenuation coefficient, will be somewhat different according to whether intensities are measured in terms of quanta or energy. We may define the vertical attenuation coefficient, K, for total photosynthetically active radiation in terms of the equation. or, Q = Q/-"'-' (7) K = -ln^ (8) z O in which O and O^ are the total number of quanta between 350 and 700 nm, per unit area per second, from natural radiation, measured by a horizontal detector facing upwards, at depth ;^ m, and just below the surface, respectively. It is often more convenient to use 10 rather than e as the logarithmic base (Westlake,
i8 J. T. O. KIRK 1965), in which case a different vertical attenuation coefficient, E, is obtained Q = Qo^o-^^ (9) Both K and E have the units m" ^ and Z E = 0.4343/^ It will be appreciated that equations (7)-(io), since they apply to broad band radiation, only hold approximately. The value of K for a given water body will be a function both of solar elevation (Jerlov and Nygard, 1969a) and of the spectral quality of the sunlight. Thus, if we are to calculate values of K for a particular system then we must make specific assumptions about these parameters. Once a particular solar elevation has been assumed then the corresponding spectral distribution of quantum ffux may be calculated from published data for spectral energy distribution of direct solar radiation for the appropriate air mass. The apparent solar elevation {p) within the water is obtained by correcting for refraction at the water surface. The values for quantum fiux may be calculated for the centre of each of the thirty-five 10 nm bands in the photosynthetic range and Q and Q^ obtained by summation. Because of the changing spectral quality of the light as it penetrates deeper into the water, the value of the extinction coefficient can change with depth (Smith, 1968). It is therefore necessary to specify a particular depth interval e.g. 0-5 m, for which the average value of K is calculated. If Q^, and Q^^^ represent the downward fiux of photosynthetically active quanta, at depth z metres, for the water without algae and for the suspension of algae respectively, then K^, and K^^^^ are the corresponding values of the vertical attenuation coefficient, obtained from equation (8). The average increment in vertical attenuation coefficient per unit algal concentration, ri, is given by (10) ^ = ^ - " ^ - (II) c where c is the algal concentration, expressed in mg chlorophyll a m'^, in the suspension for a given value of K^^^^. rj, which has the units m^ i^g" \ is of particular ecological interest since it is a measure of the extent to which self-shading develops within a growing phytoplankton population. DISCUSSION With the help of the theoretical treatment outlined in this paper it should be possible to achieve a better understanding of light attenuation in natural phytoplankton canopies: to predict the consequences of changes in cell geometry, cell numbers, intracellular pigment concentration. Predictions will inevitably diverge somewhat from reality because of the simplifying assumptions that have been made. Diffuse radiation from the sky has been ignored and all the downward intensity has been assumed to be due to a parallel beam of direct sunlight at an angle determined by the solar elevation and the angular refraction at the water surface. Jerlov and Nygard (1969a, 1969b) found this to be a useful approximation down to a depth of about 15 m
Attenuation of light by algae 19 in the Gulf of California. However, it is desirable that the treatment given in the present paper should eventually be improved to take into account the contribution of diffuse sky light. Another major approximation has been the assumption that the cells do not scatter light. This assumption is not so very far from reality since particles of the order of several microns in diameter, such as algal cells, scatter light predominantly within a relatively small angle to the initial direction of the beam. For example, Latimer and TuUy (1968) found that for yeast cells, diameter 4.9 ^m, nearly all the scattered intensity was contained within a cone of half-angle 5. With each successive interaction with an algal cell a quantum will, on average, be scattered at an increasing angle to the direction of the incident sunlight. For any given photon the scattered pathlength, L, between the surface and the detector will generally not be equal to the unscattercd pathlength, z cosec /?, and on average will be somewhat greater. The value of L will vary from one photon to another. Equation (3) then becomes If the frequency distribution for the different values of L were known, then Z^^,, could be evaluated more accurately. Any attempt to estimate the effect of light scattering in these model systems will also have to take account of the fact that the total optical crosssection of a cell for light scattering may be substantially greater than its geometrical cross section (Bryant, Seiber and Latimer, 1969). The assumption that the cells are randomly distributed in the medium, and randomly oriented is, for most algal suspensions, a realistic one. This is in marked contrast to the problem of light penetration into terrestrial plant canopies, for which neither the spatial distribution, nor the orientation, of the light-absorbing units, can be assumed to be random (Anderson, 1966). The modification of the light attenuation properties of the system brought about by placing the pigment in discrete packages rather than uniformly distributing it, is sometimes referred to as the 'sieve effect'. However, this term has also been applied specifically to the phenomenon of photons passing through leaves or algal suspensions without traversing any chloroplasts (Rabinowitch, 1951), and it therefore seems better not to use it for the present purpose because of its too specific connotation. A better name for the phenomenon would be the 'package effect': Bannister (1974) recently used the term 'packaging effect', which is essentially the same. The sieve effect, in the sense of Rabinowitch (1951) is included in the overall package effect. The rules relating changes in canopy structure to changes in light attenuation, as formulated in section 5, are qualitative in nature. To characterize this relationship in quantitative terms, specific assumptions with respect to cell size, shape, number, pigment content etc. must be made and the appropriate calculations carried out. In the following paper (Kirk, 1975) such calculations are carried out for various model suspensions of spherical algal cells. ACKNOWLEDGMENTS I am indebted to Dr A. N. Stokes for assistance in abbreviating the mathematical treatment, and to Dr M. C. Anderson for discussions on light attenuation in the natural environment.
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