# Mathematical formulation of the relationship between photosynthesis and light for phytoplankton

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1 Mathematical formulation of the relationship between photosynthesis and light for phytoplankton Alan D. Jassbyl and Tremor Platt Fisheries and Marine Service, Marine Ecology Laboratory, Bedford Institute of Oceanography, Dartmouth, Nova Scotia Abstract Eight different mathematical formulations of the photosynthesis-light curve for phytoplankton (up to and including light saturation) were recast in terms of the same two parameters : the initial slope CY, and the assimilation number P,fiD. Each equation was tested for its ability to describe empirical data from natural populations of marine phytoplankton: the results of 188 light-saturation experiments at three coastal locations in Nova Scotia over a 2-year period. The most consistently useful mathematical representation of the data was found to be the hyperbolic tangent function. Three factors motivate the search for mathematical formulations of the relationship between photosynthesis and light for phytoplankton (the light-saturation curve). First, the simulation and prediction of changes in phytoplankton biomass by means of mechanistic models require a quantitative expression for photosynthesis. Second, if in situ productivity could be estimated with sufficient precision from more easily measured variables (irradiance, chlorophyll a, nutrients, temperature, etc. ) we could collect productivity data with greater spatial and temporal resolution for the same amount of effort. Third, a mathematical formulation enables us to detcrmine the physiological parameters of photosynthesis operationally by choosing those parameter values which produce the best fit between the equation and the experimental data. Of the equations that have been proposed to describe the utilization of light by phytoplankton, a few are based on grossly oversimplified models of photosynthesis; the rest are completely empirical. This being the case, the only valid criterion for judging the relative merits of the different formulations is their ability to describe data with the minimum number of well chosen parameters. As discussed at some length in Platt et al. (1975), by well Present address : Lawrence Berkeley Laboratory, Energy and Environment Division, University of California, Berkeley chosen we mean parameters which are, as far as possible, mututally independent, and which have an informative physical interpretation, that is a significance transcending their role as operators in a particular mathematical expression. Here we compare the merits of eight different representations of the photosynthesis-light relation judged according to this criterion of ability to describe experimental data. We treat only that part of the curve up to and including light saturation. We shall not consider the range of light levels above the threshold of photoinhibition, where the introduction of at least one additional, independent parameter would be required for a complete description (Platt et al. 1975). The test data comprise measurements of 14C uptake and irradiance from 188 experiments made at three coastal locations in Nova Scotia over a period of 2 years. We thank B. Irwin, P. Dickie, and D. Rudder- ham for technical help and K. L. Denman for constructive discussion. T. T. Bannister and J. Steele made helpful comments on the manuscript. We are indebted to S. Ta- guchi for permission lished data. Basis for comparing to refer to his unpub- the equations All mathematical expressions relating primary productivity per unit chlorophyll biomass, P ( mg C[mg Chl n]-l h-l), to irradiance, I( W rns2), can be rewritten in terms of two common parameters : a ( mg LIMNOLOGY AND OCEANOGRAPIIY 540 JULY 1976, V. 21(4)

2 Photosynthesis-light equation 541 Table 1. The different formulations of the light-saturation curve tested. Equation number Equation Source* Blackman (1905) Baly (1935) Smith (1936) Steele (1962) Steele.(1962) modified't 6 Webb et a.1.(1974) 7 Platt et al.(1975) modifiedt 8 original? *The equations from various sources have been recast into common form in terms of the parameters a ( initial slope of the light-saturation curve ) and PB (specific production rate at optimal light intensity). In each case,,i ism the available light intensity and PB is the instantaneous production rate normalized to the chlorophyll biomass B. -IThese modifications are consistent with our decision not to study the range of light intensities where photoinhibition occurs. C [mg Chl al-l 11-l W-l m-2), the slope of the light-saturation curve at low light levels, i.e. in the linear range; and PmB( mg C [mg Chl al-l h-l), the specific photosynthetic rate at optimal illumination, i.e. the assimilation number. The influence of factors such as temperature, nutrients, and adaptation can be represented mathcmatitally through their effect on the basic parameters, a and Pm. Furthermore, only these two parameters are required for a sufficient representation of the light-saturation curve up to light levels where photoinhibition begins (Platt ct al. 1975). Models containing more than two parameters (Vollenweider 1965; Parker 1974) have therefore been excluded from consideration here. Eight two-parameter formulations of the light-saturation curve arc listed in Table 1. They have all been rewritten in terms of a and Pne. The list comprises several well known expressions from plant physiology as well as less widely used equations, including one that is used for the first time (as far as we are aware) in the context of photosynthesis. All eight equations are therefore cast in equivalent form and may legitimately be compared with respect to their success in fitting experimental data. Their general shapes are depicted in Fig. 1 for the same values of the parameters a and p2. The light-saturation curve usually refers to gross photosynthesis. Since our data are based on * C uptake, which corresponds

3 Jassby and Platt IO 12 Itarbitrary units) Fig. 1. The eight equations displayed for comparison. The same parameter values are used in each case: P, =lo; ~~12; Rn=l. approximately with net particulate productivity ( Eppley and Sloan 1965; Ryther and Menzel 1965), our curves of production versus light do not pass through the origin: 14C up take falls to zero while I is still positive. This is generally represented by assuming a constant, light-independent respiration loss RIZ( mg C [ mg Chl al-l h-l) and writing Plletn = PgrossB - R. We follow this convention, although it becomes increasingly untenable in the face of the accumulating evidence on photorespiration (Tolbert 1974). Our justification rests partly on pragmatism and partly on the principle of parsimony: the various formulations are, for the most part, empirical and the ultimate standard of validity is their ability to describe data. Methods Light-saturation experiments--light-saturation measurements were made using populations taken from three coastal stations during the period July 1973 to March 1975: Bedford Basin ( O W, N), a small fjordlike inlet with a heavily populated watershed; St. Margaret s Bay (64 02 W, N), a larger, more exposed bay in a relatively less populated area; and off Chebucto Head (63 3O W, N), a promontory directly south of IIalifax IIarbor. Two depths were sampled on each sampling day (either 1 and 5 m, or 5 and 10 m). Each experiment was made in duplicate. In all, 188 duplicate experiments were made.

5 544 Jc~~sby and Platt Table 2. Indices of goodness-of-fit to eight d&rent formulations of the light-saturation curve using experimental data from three stations. For each station, 62 is the mean scatter about the line fitted by the ith equation, and Nt is the numbor of times the ith equation gave the best fit according to an unweighted least-squares criterion. Equation Number Bedford Basin St. Margaret's Bay Chebucto Head 6; N. 2 6; Ni 5 zi ; 8 Total i : ; ; : : " Y 2.4% 1.82 i the jth experiment at the kth light intensity and fijjlcb is the value predicted by the ith equation for this data point, N is the total number of experiments made at that location, and rnj is the number of data points available for the jth experiment. We also calculated for each experiment the quantity WI I c k-l and added up the number of times Ni for which this quantity was lowest when the ith equation was used or, in other words, the number of times the ith equation gave the best fit according to this criterion. Results The raw data arc available as a technical report (Irwin et al. 1975). Table 2 gives the values of ai and Ni for each of the eight equations, calculated for the three stations. According to the index Q, Eq. 3 (Smith 1936) and 8 (hyperbolic tangent) were clearly superior to the rest at all three stations. The next best fit was given by the truncated parabola (Platt et al. 1975). The worst fits were given by Eq. 2 (rectangular hyperbola) and 4 (Steele 1962). The rank order of the equations with respect to the criterion Si2 was identical for all three stations. According to the index Ni, the hyperbolic tangent (Eq. 8) was the most successful formulation at each location. It gave the best fit in 38% of all the experiments. Smith s equation (Eq. 3) was the next most useful, but it fell substantially behind the hyperbolic tangent. Of the rest, the rectilinear representation (Eq. 1) gave the best fit in a surprisingly large proportion of cases. In any individual experiment, differences in the error variance between some of the curves may not have been significant. But the statistic Ni is based on the relative magnitude of the error variance in a large number of independent trials and should be a reliable guide to the best choice among the various empirical formulations of the light-saturation curve. We were not able to detect any effect of depth or season on the relative fit given by the different equations. A detailed analysis is the subjcct of another publication (Platt and Jassby 1976). Discussion The methodology described in this paper depends on the linearity of the lightsaturation function for small light fluxes. Reports of nonlinearity, even at very low irradianccs, appear from time to time in the literature. This nonlinearity has been ascribed to the interaction of mitochondrial respiration with photosynthetically produced ATP (Reid 1970). Nowhere in our data was there clear evidence of such a

7 546 Jassby and Platt Table 3. Values of 6? and Nt (see text) calculated from light-saturation experiments on seven species of marine diatom* in pure culture ( S. Taguchi unpublished), The cells were grown in seawater enrichment f/2-t at a temperature of 5-S% and an irradiance of about 10 W rnm2. Light-saturation measurements were made at various points in the log phase of growth, a total of 16 experiments. 2 : 5 ; 8 Total Ni *PlmeodactyZwn tricomutwn Bohlin, FragiZaria sp., SkeZetonema costatum (Greville) Cleve, Chaetoceros septentrionaze Oestrup, ThaZassiosira nordenskioz&i Cleve, Ditylum brightwell< Grunow, Coscinodiscus centratis Ehrenberg. obliged to use numerical integration. In those cases where it is considered essential to have an analytic solution for the integral over depth, we recommend the use of Smith s equation. Finally, we would also like to mention that a recent investigation in this laboratory of the nature of the light-saturation curve for seven different species of marine phytoplankton in culture (S. Taguchi unpublished) has also led to the conclusion that the hyperbolic tangent function affords the best description of the experimental data ( Table 3). References BALY, E. C. C The kinetics of photosynthesis. Proc. R. Sot. Lond. Ser. B 117: 21% 239. BEVINCTON, P. R Data reduction and error analysis IOr the physical sciences. McGraw-Hill. BLACKMAN, F. F Optima and limiting factors. Ann. Bot. 19: CONWAY, G. R., N. R. GLASS, AND J. C. WILCOX Fitting nonlinear models to biological data by Marquardt s algorithm. Ecology 51: EPPLEY, R. W., AND P. R. SLOAN Carbon balance experiments with marine phytoplankton. J. Fish. Res. Bd. Can. 22: HOLM-HANSEN, O., C. J. LORENZEN, R. W. HOLMES, AND J. D. H. STRICKLAND Fluorometric determination of chlorophyll. J. Cons., Cons. Int. Explor. Mer 30: IRWIN, B., T. PLATT, A. D. JASSBY, AND D. V. SUBBA RAO The relationship betwecn light intensity and photosynthesis by phytoplankton. Results of experiments at three stations in the coastal waters of Nova Scotia. Fish. Mar. Serv. Tech. Rep p. PARKER, R. A Empirical functions relating metabolic processes in aquatic systems to environmental variables. J. Fish. Res. Bd. Can. 31: PLATT, T., K. L. DENMAN, AND A. D. JASSBY The mathematical representation and prediction of phytoplankton productivity. Fish. Mar. Serv. Tech. Rep p. AND A. D. JASSBY The relationbetween photosynthesis and light for ship natural assemblages of coastal marine phytoplankton. J. Phycol. In press. RJED, A Energetic aspects of the interaction between photosynthesis and respiration, p In I. Setlik [ea.], Prediction and measurement of photosynthetic productivity. Center Agr. Publ. DOC., Wageningcn. RYTKER, J. H., AND D. W. MENZEL Comparison of the 14C-technique with direct measurement of photosynthetic carbon fixation. Limnol. Oceanogr. 10 : SMITEI, E. L Photosynthesis in relation to light and carbon dioxide. Proc. Natl. Acad. Sci..22: STEELE, J. I-I Environmental control of photosynthesis in the sea. Limnol. Oceanogr. 7: STEEMANN NIELSEN, E A reversible inactivation of chlorophyll in vitro. Physiol. Plant..2: STRICKLAND, J. D. H., AND T. R. PARSONS A practical handbook of seawater analysis, 2nd ed. Bull. Fish. Res. Bd. Can TOLBERT, N. E Photorespiration, p In W. D. Stewart [ea.], Algal physiology and biochemistry. Univ. Calif. TOOMING, H Mathematical description of net photosynthesis and adaptation processes in the photosynthetic apparatus of plant communities, p, Zn I. Setlik red.], Prediction and measurement of photosynthetic productivity. Center Agr. Publ. DOC., Wagcningen. VOLIXNWEIDER, R. A Calculation models of photosynthesis-depth curves and some implications regarding day rate estimates in

8 Photosynthesis-light equation 547 primary production measurement, p. 425 YENTSCH, C. S., AND D. W. MENZEL A 457. In C. R. Goldman [ea.], Primary pro- method for the determination of phytoplankductivity in aquatic cnvironmcnts. Univ. Calif. ton chlorophyll and phaeophytin by fluores- WEBB, W. L., M. NEWTON, AND D. STARR cence. Deep-Sea Res. 10: Carbon dioxide exchange of Alnus rubra: A mathematical model. Occologia 17: Submitted: 27 June 1975 Accepted: 6 February 1976

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