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The photosynthesis-light curve: A simple analog model
J. theor. Biol. (1977) 64, 503-516
The Photosynthesis-Light Curve : A Simple Analog Model PETER A. CRILL Scripps Institution
of Oceanography, La Jolla, Calfornia U.S.A.
92037,
(Received 20 June 1975, and in revised form, 18 March 1976)
An analog model of the photosynthesis-light curve is presented and applied to published data. Previous equations have been constructed with parameters having geometric rather than biologic definition. The present study defines each parameter as an analog of a hypothetical photosynthetic system. The model is constructed of two stages, a light stage composed of photosynthetic factories, N, with probability P of intercepting a given amount of light, and a dark stage approximated by the Michaelis-Menten equation. An index of the speed of the dark stage, relative to the potential speed of the light stage is provided. The model describes a family of photosynthesis-light curves and can be applied to data to interpret changes in the photosynthetic system from changes in the shape of the curve.
1. Introduction Adaptation
of the algal photosynthetic
mechanism
to environmental
light
and temperature has an important effect on the ultimate rate of photosynthesis and growth. The most apparent changes are the increase in chlorophyll at low light intensities and the increase in enzyme concentration at low temperatures,
and these changes are clearly evidenced in changes in the shape
of the respective photosynthesis-light curves (Tailing, 1957; Ryther & Menzel, 1959; Steemann Nielsen & Hansen, 1959, 1961; Steemann Nielsen & Jsrgensen, 1968; Jorgensen, 1968, 1969; Morris & Farrel, 1971). A number of equations have been proposed to fit photosynthesis-light curves, most notably
those of Steele (1962, 1965) and Vollenweider
(1965, 1970). Other
authors such as Fee (1969) and Bannister (1974) have extended the application of these models. Photosynthesis-light
equations
have been principally
used to estimate integral primary production, and neither Steele’s nor Vollenweider’s model makes any assumptions about the underlying physiologic mechanisms producing the curves. The parameters of these models have geometric, rather than biologic definition, and are therefore not easily assessable in terms of algal physiology. so3
504
P.
A.
CRILL
This effort attempts to define a model which at a crude scale is an analog of algal photosynthesis. In the model each parameter has a parallel in the biologic system, and consequently may provide a measure of that part of the system. An advantage of this type of model is that it permits the interpretation of the parameter values in terms of the photosynthetic analog, and may indicate changes in the algae from changes in the curve. Since the complexity of the model will be limited to the information content of the photosynthesis-light curve, it is instructive to review its attributes. At low light intensities the curve is almost linear and indicates limiting photochemical reactions; at light saturation a ceiling is reached, and the speed of the enzymatic reactions is limiting; at still higher intensities photo-inhibition occurs and the overall rate of photosynthesis declines. In order to allow estimation of the model parameters from the relatively simple photosynthesis-light curve the number of parameters must be kept as few as possible. Hence, though photosynthesis is presently understood to incorporate a number of light and dark reactions, the model will be generalized to two stages: a light-dependent photochemical stage, and a light-independent enzymatic stage, with the parameters of each describing the net effect of the more complex system, but invisible to the observer at this scale.
2. The Light Stage Model The light stage is described by a simple analog model consisting of two parameters: photosynthetic factories and hit probability. The factory is defined as the light trapping and reaction centers and associated apparatus which are activated by a given amount of light energy to produce a given amount of photoproduct. Furthermore, factory operation is cancelled by any further light energy received prior to completion of the reaction. Functionally, the proposed factory parameter is comprised of both constructive and destructive phases. The constructive phase may be considered roughly analogous to Emerson’s photosynthetic unit, and the destructive phase may be associated with photorespiration or other effects. Physically the factory encompasses the photoreactions of both photosystems I and II, factory output indicating the net throughput of the light stage. For purposes of the model the amount of light necessary for factory operation is termed a “unit” of light, and the amount of photoproduct produced is termed a “unit” of photoproduct. Given one factory and light intensity I, with probability p that any given light unit will strike the factory,
THE
the probability theorem :
PHOTOSYNTHESIS-LIGHT
CURVE
505
of one and only one unit in time At is given by the binomial
where q = l-p. in time At is:
Prob. (I, 1, p) = _1 !(&j
P’F
If there are Nindependent
factory systems, the mean output
= IPP?
P = NIpq’-‘.
(2)
Equation (2) can be shown to be identical to Steele’s (1962, 1965) equation if it is expressed in terms of P,, and I,,,, maximal photosynthesis, and optimal light intensity, respectively, rather than N and p. At ( = I,,,, the first derivative of equation (2) with respect to I, is equal to zero: dP rtrx- ‘(1 + I,, log, q), -iiT = O = Npq and from equation (3): q=exp(-l/Z,). Substituting
(4
equation (4) into equation (2) at P = P,,, and solving for N: 0)
Substituting
equations (5) and (4) into equation (2) gives Steele’s equation : P = P,,III,
exp (I -1/I,).
(6)
Since Steele’s equation can be derived from the analog model described by equation (2), changes in geometric parameters P,,, and I,,, may now be used to obtain the relative changes in the hypothetical analogs factories, N, and hit probability, p. This light stage model is applicable to algal photosynthesis if the rate of the dark reactions are fast relative to the rate of the light reactions. In these cases, increased photosynthetic efficiency is achieved by increasing the output of the light stage. In the analog this might be accomplished in two ways: by increasing the number of factories or by varying the hit probability. Increasing the number of factories increases the rate of photosynthesis over all light intensities, and increasing the hit probability shifts the curve to the left, reducing the light intensity required for maximal photosynthesis (Fig. 1).
506
P. A. GRILL
xii I.80 IGO-
‘j g
0,45-
Light
intensity
FIG. 1. Effect of varying the light stage parameters. Increasing the number of factories, N, stretches the curve in the verticaI and increases the rate of photosynthesis over all light intensities by a factor equal to the increase in N. Increasing the hit probability, p, compresses the curve in the horizontal and decreases optimal lighht intensity. Maximal photosynthesis also increases slightly.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
507
3. The Light and Dark Stage Model The dark stage is approximated first-order enzymatic reaction :
by the Michaelis-Menten
Q - ‘md K,+S’
formula for a
(7)
where Q is the reaction rate, V,,, is the maximum reaction rate, S is the substrate concentration, and KS is the Michaelis constant. Once a factory has processed a light unit and some part of the factory is tied up in the photoproduct, the entire factory must be removed from the system. If S is equivalent to the number of factories removed in this way, then equation (2) becomes: P = (N-S)zpq’-‘.
(8)
At steady state P = Q and from equations (7) and (8) the combined lightand dark-stage model is obtained: Q = 3{Vmax+XN+XK,-[(I/,,,+XN+XK,)‘-4~~,,XN]~),
(9)
where X = Zpq’-‘, and Q is the rate of photosynthesis for the combined light and dark stages. This yields a family of curves (Fig. 2). At very low light intensities the slope of the curve is nearly linear and represents the maximum photosynthetic yield. The maximum yield, bO, is given by the first derivative of equation (9) at Z = 0.0: bo = Nplq, (10) and is identical to the value of the first derivative of equation (2), the light stage model, at 1 = 0.0. A measure of the speed of the dark stage, relative to the potential rate of the light stage is given by the energy transfer index, defined as: (11) Tm = Qmlpm, where Q, is given by equation (9) at Z = I,,,, and P,,, is given by equation (2) at Z = I,,,. The transfer. index is a function of substrate reservoir S. From equation (7) the substrate reservoir at Z = I,,, is: (12) S, = Q,K/V'm - QnA This substrate reservoir approaches zero if the dark stage is very fast relative to the light stage and approaches N is the dark stage is very slow relative to the light stage. An index of the relative substrate backlog is given by: B, = S,IN.
(13)
508
I’.
A.
Hypothetical
GRILL family
of P/lcurves
I
I
I
I
I
Light
I
I
I
I
I
intensity
FIG. 2. Family of photosynthesis-light curves produced by the model describedin equation (9). Curve A (Vmax = ~0 , K, = 0.0) describes a system which is light limited over its entire range, increasing linearly at low light intensities and decreasing exponentially at high light intensities. Curve B (V,,, = 5.0, K, = 54)) and curve 12 (V,,, = 2.0, KS = 6tl) describe systems in which the potential rate of output of the light stage (curve A) is limited by the speed of the dark reactions.
From equations (11) and (13) and the appropriate and transfer indices can be related: T, = I-B,,.
substitutions
the backlog (14)
4. Sensitivity Analysis
The model described by equation (9) must be fitted numerically to photosynthesis-light data. Once initial estimates for three of the four parameters (N, p, V,,,,, and KS) are selected, each parameter in turn, is systematically varied through a range of values to find the value which minimizes the sum of squares deviation of the estimated and observed rates of photosynthesis. This procedure is continued until the rate of convergence to a constant value for each parameter is sufficiently close to zero. When the model is fitted to artificial data previously generated by equation (9), variables N, p, and T, can be estimated with very little error. There are many combinations of em,, and KS, however, which will very closely approximate the same curve.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
509
Therefore, only the relative speed of the dark reactions, as given by the backlog or transfer indices can be extracted from the curve in this way; the relative contribution of V,, and K, cannot be adequately so determined.
5. Fitting the Model to Data (A) LIGHT
ADAPTATION
The model was fitted to the data of Steemann Nielsen (1962) for ChloreZlu the analog (Fig. 3, Table 1) shows that the hit probability for the algae grown at 3 klx is 2.2 times that of the algae grown at 30 kIx. This corresponds to the reported two-fold increase in chlorophyll content. Factories decreased 26x, and the transfer index decreased 13 %, together accounting for a 34% drop in Q, when cultured at the lower intensity. When the 3 klx population was transferred to 30 klx for 3 h the curve shifted so that the hit probability became intermediate between that of the 3 klx population and the 30 klx population. The number of factories decreased 65x, and this could explain the increase in the relative speed of the dark reactions as indicated by the transfer index. oulgaris cultured at light intensities of 3 klx and 30 klx. Fitting
‘“*oo-
Light
intensity
(klx)
FIG. 3. Chlorella vulgaris at conditioned light intensity. Points are from Steemann Nielsen (1962). Cultures were grown at 30 klx and 3 klx before the experiment. The lower curve illustrates the effect of transferring algae previously grown at 3 klx to 30 klx for 3 h. Curves were drawn from the model.
510
P. A. CKILL
TABLE 1 Modelparametersjtted to the datu of SteemarmNie1.w (1962). Photosynthesis is given per unit volume of algae. In the tables S.E. represents the standard error of the observed and predicted rates of photosynthesis, N is factory wit concentration, p is hit probability, I, is optimal light intensitJ1,and T, is the index of the speed of the dark reactions reluticr to the potentiul speedof t/w light stage. Since N, I,, and to someextent, p, are depender,ltupon the wits of measurement,ort1.vthe relative changesin theseparameters, and not thei, absolute values are consideredin the text
30 klx Population 3 klx Population 3 klx Population 3 h at 30 klx
0.363 0.155 0.086
4% 3% 3O’ /”
17 70 7
-0.034 0.075 0.053
19 13 18
8.5 5.6 2.6
ss:,; 74 7; 100:;
Steemann Nielsen suggestedthat the transfer of the 3 klx algae to 30 klx for 3 h must have inhibited both the photochemical and the enzymatic processessince both the slope and the ceiling of the curve are depressed. He observed that if the initial slope of the transferred population is made identical to the slope of the 30 klx population the curves would be practically the same, and therefore the algae must have adjusted their photochemical and enzymatic partial processesto the same mutual relationship as if they had been growing at 30 klx for a long time. Application of the analog, however, suggeststhat since T,, is relatively high in both instances both ceilings might be attributed to photo-inhibition rather than a limiting dark reaction. If this is the case, making the initial slopes the same is shown by equation (10) to be equivalent to adjusting the systems to the same values of N and p. Yentsch & Lee (1966) determined the photosynthesis-light curves of the marine flagellate, Nunnochloris atomus, cultured at 10 klx and after 41 h and 65 h in the dark. The model approximated the data well (Table 2) and shows both p and T, increasing marginally with time spent in darkness. The changes in N and Q, are not assessedsince rates are given per unit chlorophyll. The author’s note that if each of the three curves is drawn to a maximum of 100% then the curves are almost identical. They conclude that prolonged darkness equally affects both the light and dark stages. However, high 7;, values again indicate that the curves can be described by the light stage model alone, and that the ceilings may be mostly due to
THE
PHOTOSYNTHESIS-LIGHT TABLE
511
CURVE
2
Model parameters fitted to the data of Yentsch t Lee (1966). Photosynthesis is per unit chlorophyll se.
S.E.
N
a,
Initial population 41 h in dark 65 h in dark 25 klx pop. 6 klx pop. 2 klx pop.
0.137 0.055 OWL5 0.277 0.092 o-062
4% 3% 7% 6% 7% 14%
9-s 4.9 2.7 12.7 3.7 3-l
P 0.037 0.039 0.043 0.037 0.055 0063
Im
27 25 23 27 18 15
Qm
Till
3.0 1.6 * :.; A.4” *
87% 89% 100% 10% 100% 39%
photo-inhibition. If this is the case, then it is primarily the light stage which is being affected by the prolonged darkness. In another experiment the authors conditioned Nannochloris to different light intensities for 24 h before measuring their photosynthesis-light curves. The analog indicates that hit probability increased with conditioning to lower intensities. T, remained at 100% for both the 25 klx and the 6 klx populations, but dropped to 39% for the 2 klx population, indicating a substantial decrease in the speed of the dark reactions when preconditioned to a very low intensity (Table 2). (B)
TEMPERATURE
ADAPTATION
Aruga (1965) reported to the photosynthesis-light curves of algae at various unadapted temperatures. The family of curves for each species exhibited much similarity to that of Chlorella ellipsoidea (Table 3). Except at the highest temperature for some of the algae, Q, increased with temperature. There was no trend to the fitted values of N or p, although some fluctuation was evident due to the long ceilings. In most cases the combination of N and p produced similar P,,, values, regardless of the temperature, and the increase in Q, was primarily due to an increase in the dark stage velocity index, T,. The relative changes in N and Q, may be considered valid because the algae tested at different temperatures were similarly adapted and presumably had comparable chlorophyll content. Jsrgensen & Steemann Nielsen (1965) reported photosynthesis-light curves for Skeletonema costatum at temperatures to which the lgae had been previously adapted. The fitted parameters N, p and Q, are relatively stable at all but the lowest adapted temperature, where a decrease in N and Q,, and a rise in p occurs. With pre-adaptation there is no noticeable trend to the values of T, (Table 4).
512
P.
A.
GRILL
3
TABLE
Modelparametersjtted
S.E. --___. 5°C lOT 15’C 20°C 25°C 3O’C
test test test test test test
0.066 0.036 0444
to the data of Aruga (1965) for Chlorella Photosynthesis is per unit chlorophyll S.E. Qm o ;$ 22
ellipsoidea.
N
P
Ll
Qm
TIII
Pt7l
37 46 28 43 46 28
0,030 0.010 0.017 0.011 0.012 0.021
33 97 58 88 85 48
I.0 1.7 2.2 3.0 4.6 5.5
7% IO% 21 % 19% 27% 53%
14 17 10 16 17 10
TABLE
4
Model parameters fitted to the data of Jorgensen & Steemann Nielsen (1965). Photosynthesis is per unit cell number S.E. 2°C pop. at “C pop. at 14°C pop. at 20°C pop. at
2°C 8°C 14°C 20°C
0.032 0.077 O-052 0.036
fi* Qm
N
P
Im
Qm
T,
3% 4%
3.0 5.6
OG61 0.050 0440 0.042
16 i;
0.9 1.9 1.8 2.1
83% 91% 65%
23
82%
6. Discussion Fitting the analog to a number of published light adaptation experiments indicates that adaptation to lower intensities is primarily observed as an increase in hit probability, p, while the relative speed of the dark reactions, T,, is relatively unaffected. Fitting the model to published temperature experiments indicates that a temperature change is observed as a change in the dark stage reaction index, T,,, while factory units, N, and hit probability, p, are essentially unaffected. When the algae are given sufficient time for acclimation the transfer index, T,, and maximal photosynthesis, Q,, are stable over a range of temperatures. These changes are consistent with observed changes occurring in the physiologic system. Increasing T, with increasing temperature reflects the temperature dependence of enzymatic reactions, and its stability after a period of adaptation reflects the cell’s adjustment of cellular enzyme concentration (Steemann Nielsen & Hansen, 1959; Steemann Nielsen & Jorgensen, 1968 ; Morris & Farrel, 1971).
THE
PHOTOSYNTHESIS-LIGHT
513
CURVE
Adaptation to lower light intensities is achieved primarily by increasing the size of the chlorophyll antenna, and thereby decreasing the light intensity required for saturation (Jorgensen, 1969; Sheridan, 1972u,b) and increasing the hit probability. Other factors may also be observed as changes in hit probability. For example, there are indications that the orientation of the chlorophyll molecules may be an important factor in the trapping efficiency of the antenna (Mayer, 1971; Seely, 1973; Hoff, 1974). Furthermore, Seely suggests that different spectral varieties of chlorophyll exist in the photosynthetic unit in order to accelerate the transfer of excitation energy to the reaction center. Changes in the relative abundances of different spectral varieties might therefore be expected to occur during the process of adaptation, or relative changes in photosystem I and II may have important effects. An example of this may be indicated by the selective formation of photosystem I under low light intensity, observed for two types of isolated wheat chloroplasts (Ogawa & Shibata, 1973). Other contributors to the effective hit probability may be indicated by any number of light intensity effects on algal ultrastructure (Findley et al., 1970). Kiefer (1973) observed that under light stress the chloroplasts of some of the centric diatoms contract and move to the valvar ends of the cell. Some of the filamentous algae, such as Mougeotia change the orientation of their chloroplast (Mayer, 1971). This alga has a platelike chloroplast which is Direction
of illumination
I I
/Side
I Side
Axis
2
of rototion
FIG. 4. Hit probability and chloroplast orientation. Some of the filamentous algae, such as Mougeofiu, changethe orientationof their chloroplastin response to changinglight intensity.If the chloroplastis assumed to bea thme-diional box free to rotate about one axis, then the angleof rotation, 0, can be relatedto the hit probability,p. 34 T.B.
514
P. A. CRILI.
oriented face-on to low light and edge-on to high light. The angle of rotation of the chloroplast, 0, and the hit probability, p, can be directly related if the chloroplast is assumed to be a three dimensional box free to rotate about one axis (Fig. 4). Given the areas of the two receptive sides A, and A,;f, andf,, the efficiency or effective or light trapping fractions of sides 1 and 2 respectively; (AT +A$)* = 1.0, so that the hit probability will not exceed unity; then : p = A,f, cos Q+ A2fZ sin H. (15)
Light
intensity
Fro. 5. Photosynthesis-light curve of the photosynthetic sulfur bacteria Chvomati~r~ compared to the curve drawn from equation (2) by adding the output of two independent photosystems, each with a different hit probability.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
515
Traditionally, the slope of the photosynthesis-light curve has been related to the speed of the light stage, and the ceiling of the curve to the speed of the dark reactions. Some high T, values, especially for algae adapted to low intensities, suggests that light reactions can be limiting at all intensities. Application of the analog may then suggest whether the ceiling of the curve is sufficiently long and flattened to be attributed to dark stage limitation, whether it has the shape one would expect from photo-inhibition, or whether there are insufficient data to resolve the difference. An advantage of an analog-type model over a geometric-type is that it may be of some interest even when it fails, since it may reveal cases where the inadequacy of the underlying assumptions becomes critical, cases which might otherwise be considered quite ordinary. For example, curves of some photosynthetic sulfur bacteria reported by Takahashi & Ichimura (1970) could not be fitted with the model because the exponential decline portion of the curve approached a value significantly greater than zero. It is interesting that the sum of two independent photosystems, each with a different hit probability, can produce a curve very similar to the curve produced by the bacteria (Fig. 5). This suggests either the presence of two distinct populations or some basic inadequacy of the model to describe the photosynthesis-light curve of some bacteria. In summary, the proposed simple four-parameter analog model of the photosynthesis-light curve not only provides a close approximation for many observed curves, but also presents the possibility for a meaningful interpretation of the parameters. Although in the present model, parameter values are only relative and dependent upon the measured units of photosynthesis and light, they present advantages over strictly geometric ones, and the possibility for future refinement and extension. REFERENCES ARUGA, Y. (1965). Bot. Mug. Tokyo. 78, 360. BANNISTER, T. T. (1974). Limnol. Oceanogr. 19, 1, FEE, E. J. (1969). Limnol. Oceanogr. 14, 906. FINDLEY, (1970). J. Phycol. 6, 182. HOFF, A. J. (1974). Photochem. Photobiof. 19, 51. JBRGENSEN, E. G., (1968). Physiol. Pl. 21, 423. JPIRGENSEN, E. G. (1969). Physiol. PI. 22, 1307. E. G. & STEEMANN NIELSEN, E. (1965). In Primary Productivity in Aquatic Environments (C. R. Goldman, ed.), p. 37. Berkeley: University of California Press. KIEFER, D. A. (1973). Mar. Biol. 23, 39. MAYER, F. (1971). In Structure and Function of Chloroplasts (M. Gibbs, ed.), p. 35. New York: Springer-Verlag. MORRJS, I. & FARREL, K. (1971). Physiol. PI. 25, 372. OGAWA, T. & SHIBATA, K. (1973). Physiol. PI. 29, 112. RYTHER, J. H. & MENZEL, D. W. (1959). Limnol. Oceanogr. 4, 492. JORGENSEN,
516
I’.
A.
C’Kll.1.
SEELY, G. R. (1973). J. rheor. Biol. 40, 173. SHERIDAN, R. P. (1972a). J. Phycof. 8, 166. SHERIDAN, R. P. (19726). PI. physiof. 50, 355. STEELE, J. H. (1962). Linmol. Oceurzogr. 7, 137. STEELE, J. H. (1965). Memoir 1st Ital. Oa’robiol. 18 (suppl.), 353. STEEMANN NIELSEN, E. (1962). Physiol. PI. 15, 161. STEEMANN NIELSEN, E. & HANSEN, V. K. (1959). Physiol. PI. 12, 353. STEEMANN NIELSEN, E. & HANSEN, V. K. (1961). Physiol. PI. 14, 595. STEEMANN NIELSEN, E. & JORGENSEN, E. G. (1968). Phydiol. PI. 21, 401. TAKAHASHI, M. & ICHIMURA, S. (1970). Limnol. Ocemogr. 15, 929. TALLING, J. F. (1957). New Phytol. 56, 133. VOLLENWEIDER, R. A. (1965). Memoir 1st Ital. Idrobiol. 18 (suppl.), 425. VOLLENWEIDER, R. A. (1970). In Proc. IBP/PP Tech. Meeting, Trebon, p. 455. Wageningen: Center for Agricultural Publications and Documentation. YENTSCH, C. S. & LEE, R. W. (1966). J. Mar. Res. 24, 319.
The Photosynthesis-Light Curve : A Simple Analog Model PETER A. CRILL Scripps Institution
of Oceanography, La Jolla, Calfornia U.S.A.
92037,
(Received 20 June 1975, and in revised form, 18 March 1976)
An analog model of the photosynthesis-light curve is presented and applied to published data. Previous equations have been constructed with parameters having geometric rather than biologic definition. The present study defines each parameter as an analog of a hypothetical photosynthetic system. The model is constructed of two stages, a light stage composed of photosynthetic factories, N, with probability P of intercepting a given amount of light, and a dark stage approximated by the Michaelis-Menten equation. An index of the speed of the dark stage, relative to the potential speed of the light stage is provided. The model describes a family of photosynthesis-light curves and can be applied to data to interpret changes in the photosynthetic system from changes in the shape of the curve.
1. Introduction Adaptation
of the algal photosynthetic
mechanism
to environmental
light
and temperature has an important effect on the ultimate rate of photosynthesis and growth. The most apparent changes are the increase in chlorophyll at low light intensities and the increase in enzyme concentration at low temperatures,
and these changes are clearly evidenced in changes in the shape
of the respective photosynthesis-light curves (Tailing, 1957; Ryther & Menzel, 1959; Steemann Nielsen & Hansen, 1959, 1961; Steemann Nielsen & Jsrgensen, 1968; Jorgensen, 1968, 1969; Morris & Farrel, 1971). A number of equations have been proposed to fit photosynthesis-light curves, most notably
those of Steele (1962, 1965) and Vollenweider
(1965, 1970). Other
authors such as Fee (1969) and Bannister (1974) have extended the application of these models. Photosynthesis-light
equations
have been principally
used to estimate integral primary production, and neither Steele’s nor Vollenweider’s model makes any assumptions about the underlying physiologic mechanisms producing the curves. The parameters of these models have geometric, rather than biologic definition, and are therefore not easily assessable in terms of algal physiology. so3
504
P.
A.
CRILL
This effort attempts to define a model which at a crude scale is an analog of algal photosynthesis. In the model each parameter has a parallel in the biologic system, and consequently may provide a measure of that part of the system. An advantage of this type of model is that it permits the interpretation of the parameter values in terms of the photosynthetic analog, and may indicate changes in the algae from changes in the curve. Since the complexity of the model will be limited to the information content of the photosynthesis-light curve, it is instructive to review its attributes. At low light intensities the curve is almost linear and indicates limiting photochemical reactions; at light saturation a ceiling is reached, and the speed of the enzymatic reactions is limiting; at still higher intensities photo-inhibition occurs and the overall rate of photosynthesis declines. In order to allow estimation of the model parameters from the relatively simple photosynthesis-light curve the number of parameters must be kept as few as possible. Hence, though photosynthesis is presently understood to incorporate a number of light and dark reactions, the model will be generalized to two stages: a light-dependent photochemical stage, and a light-independent enzymatic stage, with the parameters of each describing the net effect of the more complex system, but invisible to the observer at this scale.
2. The Light Stage Model The light stage is described by a simple analog model consisting of two parameters: photosynthetic factories and hit probability. The factory is defined as the light trapping and reaction centers and associated apparatus which are activated by a given amount of light energy to produce a given amount of photoproduct. Furthermore, factory operation is cancelled by any further light energy received prior to completion of the reaction. Functionally, the proposed factory parameter is comprised of both constructive and destructive phases. The constructive phase may be considered roughly analogous to Emerson’s photosynthetic unit, and the destructive phase may be associated with photorespiration or other effects. Physically the factory encompasses the photoreactions of both photosystems I and II, factory output indicating the net throughput of the light stage. For purposes of the model the amount of light necessary for factory operation is termed a “unit” of light, and the amount of photoproduct produced is termed a “unit” of photoproduct. Given one factory and light intensity I, with probability p that any given light unit will strike the factory,
THE
the probability theorem :
PHOTOSYNTHESIS-LIGHT
CURVE
505
of one and only one unit in time At is given by the binomial
where q = l-p. in time At is:
Prob. (I, 1, p) = _1 !(&j
P’F
If there are Nindependent
factory systems, the mean output
= IPP?
P = NIpq’-‘.
(2)
Equation (2) can be shown to be identical to Steele’s (1962, 1965) equation if it is expressed in terms of P,, and I,,,, maximal photosynthesis, and optimal light intensity, respectively, rather than N and p. At ( = I,,,, the first derivative of equation (2) with respect to I, is equal to zero: dP rtrx- ‘(1 + I,, log, q), -iiT = O = Npq and from equation (3): q=exp(-l/Z,). Substituting
(4
equation (4) into equation (2) at P = P,,, and solving for N: 0)
Substituting
equations (5) and (4) into equation (2) gives Steele’s equation : P = P,,III,
exp (I -1/I,).
(6)
Since Steele’s equation can be derived from the analog model described by equation (2), changes in geometric parameters P,,, and I,,, may now be used to obtain the relative changes in the hypothetical analogs factories, N, and hit probability, p. This light stage model is applicable to algal photosynthesis if the rate of the dark reactions are fast relative to the rate of the light reactions. In these cases, increased photosynthetic efficiency is achieved by increasing the output of the light stage. In the analog this might be accomplished in two ways: by increasing the number of factories or by varying the hit probability. Increasing the number of factories increases the rate of photosynthesis over all light intensities, and increasing the hit probability shifts the curve to the left, reducing the light intensity required for maximal photosynthesis (Fig. 1).
506
P. A. GRILL
xii I.80 IGO-
‘j g
0,45-
Light
intensity
FIG. 1. Effect of varying the light stage parameters. Increasing the number of factories, N, stretches the curve in the verticaI and increases the rate of photosynthesis over all light intensities by a factor equal to the increase in N. Increasing the hit probability, p, compresses the curve in the horizontal and decreases optimal lighht intensity. Maximal photosynthesis also increases slightly.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
507
3. The Light and Dark Stage Model The dark stage is approximated first-order enzymatic reaction :
by the Michaelis-Menten
Q - ‘md K,+S’
formula for a
(7)
where Q is the reaction rate, V,,, is the maximum reaction rate, S is the substrate concentration, and KS is the Michaelis constant. Once a factory has processed a light unit and some part of the factory is tied up in the photoproduct, the entire factory must be removed from the system. If S is equivalent to the number of factories removed in this way, then equation (2) becomes: P = (N-S)zpq’-‘.
(8)
At steady state P = Q and from equations (7) and (8) the combined lightand dark-stage model is obtained: Q = 3{Vmax+XN+XK,-[(I/,,,+XN+XK,)‘-4~~,,XN]~),
(9)
where X = Zpq’-‘, and Q is the rate of photosynthesis for the combined light and dark stages. This yields a family of curves (Fig. 2). At very low light intensities the slope of the curve is nearly linear and represents the maximum photosynthetic yield. The maximum yield, bO, is given by the first derivative of equation (9) at Z = 0.0: bo = Nplq, (10) and is identical to the value of the first derivative of equation (2), the light stage model, at 1 = 0.0. A measure of the speed of the dark stage, relative to the potential rate of the light stage is given by the energy transfer index, defined as: (11) Tm = Qmlpm, where Q, is given by equation (9) at Z = I,,,, and P,,, is given by equation (2) at Z = I,,,. The transfer. index is a function of substrate reservoir S. From equation (7) the substrate reservoir at Z = I,,, is: (12) S, = Q,K/V'm - QnA This substrate reservoir approaches zero if the dark stage is very fast relative to the light stage and approaches N is the dark stage is very slow relative to the light stage. An index of the relative substrate backlog is given by: B, = S,IN.
(13)
508
I’.
A.
Hypothetical
GRILL family
of P/lcurves
I
I
I
I
I
Light
I
I
I
I
I
intensity
FIG. 2. Family of photosynthesis-light curves produced by the model describedin equation (9). Curve A (Vmax = ~0 , K, = 0.0) describes a system which is light limited over its entire range, increasing linearly at low light intensities and decreasing exponentially at high light intensities. Curve B (V,,, = 5.0, K, = 54)) and curve 12 (V,,, = 2.0, KS = 6tl) describe systems in which the potential rate of output of the light stage (curve A) is limited by the speed of the dark reactions.
From equations (11) and (13) and the appropriate and transfer indices can be related: T, = I-B,,.
substitutions
the backlog (14)
4. Sensitivity Analysis
The model described by equation (9) must be fitted numerically to photosynthesis-light data. Once initial estimates for three of the four parameters (N, p, V,,,,, and KS) are selected, each parameter in turn, is systematically varied through a range of values to find the value which minimizes the sum of squares deviation of the estimated and observed rates of photosynthesis. This procedure is continued until the rate of convergence to a constant value for each parameter is sufficiently close to zero. When the model is fitted to artificial data previously generated by equation (9), variables N, p, and T, can be estimated with very little error. There are many combinations of em,, and KS, however, which will very closely approximate the same curve.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
509
Therefore, only the relative speed of the dark reactions, as given by the backlog or transfer indices can be extracted from the curve in this way; the relative contribution of V,, and K, cannot be adequately so determined.
5. Fitting the Model to Data (A) LIGHT
ADAPTATION
The model was fitted to the data of Steemann Nielsen (1962) for ChloreZlu the analog (Fig. 3, Table 1) shows that the hit probability for the algae grown at 3 klx is 2.2 times that of the algae grown at 30 kIx. This corresponds to the reported two-fold increase in chlorophyll content. Factories decreased 26x, and the transfer index decreased 13 %, together accounting for a 34% drop in Q, when cultured at the lower intensity. When the 3 klx population was transferred to 30 klx for 3 h the curve shifted so that the hit probability became intermediate between that of the 3 klx population and the 30 klx population. The number of factories decreased 65x, and this could explain the increase in the relative speed of the dark reactions as indicated by the transfer index. oulgaris cultured at light intensities of 3 klx and 30 klx. Fitting
‘“*oo-
Light
intensity
(klx)
FIG. 3. Chlorella vulgaris at conditioned light intensity. Points are from Steemann Nielsen (1962). Cultures were grown at 30 klx and 3 klx before the experiment. The lower curve illustrates the effect of transferring algae previously grown at 3 klx to 30 klx for 3 h. Curves were drawn from the model.
510
P. A. CKILL
TABLE 1 Modelparametersjtted to the datu of SteemarmNie1.w (1962). Photosynthesis is given per unit volume of algae. In the tables S.E. represents the standard error of the observed and predicted rates of photosynthesis, N is factory wit concentration, p is hit probability, I, is optimal light intensitJ1,and T, is the index of the speed of the dark reactions reluticr to the potentiul speedof t/w light stage. Since N, I,, and to someextent, p, are depender,ltupon the wits of measurement,ort1.vthe relative changesin theseparameters, and not thei, absolute values are consideredin the text
30 klx Population 3 klx Population 3 klx Population 3 h at 30 klx
0.363 0.155 0.086
4% 3% 3O’ /”
17 70 7
-0.034 0.075 0.053
19 13 18
8.5 5.6 2.6
ss:,; 74 7; 100:;
Steemann Nielsen suggestedthat the transfer of the 3 klx algae to 30 klx for 3 h must have inhibited both the photochemical and the enzymatic processessince both the slope and the ceiling of the curve are depressed. He observed that if the initial slope of the transferred population is made identical to the slope of the 30 klx population the curves would be practically the same, and therefore the algae must have adjusted their photochemical and enzymatic partial processesto the same mutual relationship as if they had been growing at 30 klx for a long time. Application of the analog, however, suggeststhat since T,, is relatively high in both instances both ceilings might be attributed to photo-inhibition rather than a limiting dark reaction. If this is the case, making the initial slopes the same is shown by equation (10) to be equivalent to adjusting the systems to the same values of N and p. Yentsch & Lee (1966) determined the photosynthesis-light curves of the marine flagellate, Nunnochloris atomus, cultured at 10 klx and after 41 h and 65 h in the dark. The model approximated the data well (Table 2) and shows both p and T, increasing marginally with time spent in darkness. The changes in N and Q, are not assessedsince rates are given per unit chlorophyll. The author’s note that if each of the three curves is drawn to a maximum of 100% then the curves are almost identical. They conclude that prolonged darkness equally affects both the light and dark stages. However, high 7;, values again indicate that the curves can be described by the light stage model alone, and that the ceilings may be mostly due to
THE
PHOTOSYNTHESIS-LIGHT TABLE
511
CURVE
2
Model parameters fitted to the data of Yentsch t Lee (1966). Photosynthesis is per unit chlorophyll se.
S.E.
N
a,
Initial population 41 h in dark 65 h in dark 25 klx pop. 6 klx pop. 2 klx pop.
0.137 0.055 OWL5 0.277 0.092 o-062
4% 3% 7% 6% 7% 14%
9-s 4.9 2.7 12.7 3.7 3-l
P 0.037 0.039 0.043 0.037 0.055 0063
Im
27 25 23 27 18 15
Qm
Till
3.0 1.6 * :.; A.4” *
87% 89% 100% 10% 100% 39%
photo-inhibition. If this is the case, then it is primarily the light stage which is being affected by the prolonged darkness. In another experiment the authors conditioned Nannochloris to different light intensities for 24 h before measuring their photosynthesis-light curves. The analog indicates that hit probability increased with conditioning to lower intensities. T, remained at 100% for both the 25 klx and the 6 klx populations, but dropped to 39% for the 2 klx population, indicating a substantial decrease in the speed of the dark reactions when preconditioned to a very low intensity (Table 2). (B)
TEMPERATURE
ADAPTATION
Aruga (1965) reported to the photosynthesis-light curves of algae at various unadapted temperatures. The family of curves for each species exhibited much similarity to that of Chlorella ellipsoidea (Table 3). Except at the highest temperature for some of the algae, Q, increased with temperature. There was no trend to the fitted values of N or p, although some fluctuation was evident due to the long ceilings. In most cases the combination of N and p produced similar P,,, values, regardless of the temperature, and the increase in Q, was primarily due to an increase in the dark stage velocity index, T,. The relative changes in N and Q, may be considered valid because the algae tested at different temperatures were similarly adapted and presumably had comparable chlorophyll content. Jsrgensen & Steemann Nielsen (1965) reported photosynthesis-light curves for Skeletonema costatum at temperatures to which the lgae had been previously adapted. The fitted parameters N, p and Q, are relatively stable at all but the lowest adapted temperature, where a decrease in N and Q,, and a rise in p occurs. With pre-adaptation there is no noticeable trend to the values of T, (Table 4).
512
P.
A.
GRILL
3
TABLE
Modelparametersjtted
S.E. --___. 5°C lOT 15’C 20°C 25°C 3O’C
test test test test test test
0.066 0.036 0444
to the data of Aruga (1965) for Chlorella Photosynthesis is per unit chlorophyll S.E. Qm o ;$ 22
ellipsoidea.
N
P
Ll
Qm
TIII
Pt7l
37 46 28 43 46 28
0,030 0.010 0.017 0.011 0.012 0.021
33 97 58 88 85 48
I.0 1.7 2.2 3.0 4.6 5.5
7% IO% 21 % 19% 27% 53%
14 17 10 16 17 10
TABLE
4
Model parameters fitted to the data of Jorgensen & Steemann Nielsen (1965). Photosynthesis is per unit cell number S.E. 2°C pop. at “C pop. at 14°C pop. at 20°C pop. at
2°C 8°C 14°C 20°C
0.032 0.077 O-052 0.036
fi* Qm
N
P
Im
Qm
T,
3% 4%
3.0 5.6
OG61 0.050 0440 0.042
16 i;
0.9 1.9 1.8 2.1
83% 91% 65%
23
82%
6. Discussion Fitting the analog to a number of published light adaptation experiments indicates that adaptation to lower intensities is primarily observed as an increase in hit probability, p, while the relative speed of the dark reactions, T,, is relatively unaffected. Fitting the model to published temperature experiments indicates that a temperature change is observed as a change in the dark stage reaction index, T,,, while factory units, N, and hit probability, p, are essentially unaffected. When the algae are given sufficient time for acclimation the transfer index, T,, and maximal photosynthesis, Q,, are stable over a range of temperatures. These changes are consistent with observed changes occurring in the physiologic system. Increasing T, with increasing temperature reflects the temperature dependence of enzymatic reactions, and its stability after a period of adaptation reflects the cell’s adjustment of cellular enzyme concentration (Steemann Nielsen & Hansen, 1959; Steemann Nielsen & Jorgensen, 1968 ; Morris & Farrel, 1971).
THE
PHOTOSYNTHESIS-LIGHT
513
CURVE
Adaptation to lower light intensities is achieved primarily by increasing the size of the chlorophyll antenna, and thereby decreasing the light intensity required for saturation (Jorgensen, 1969; Sheridan, 1972u,b) and increasing the hit probability. Other factors may also be observed as changes in hit probability. For example, there are indications that the orientation of the chlorophyll molecules may be an important factor in the trapping efficiency of the antenna (Mayer, 1971; Seely, 1973; Hoff, 1974). Furthermore, Seely suggests that different spectral varieties of chlorophyll exist in the photosynthetic unit in order to accelerate the transfer of excitation energy to the reaction center. Changes in the relative abundances of different spectral varieties might therefore be expected to occur during the process of adaptation, or relative changes in photosystem I and II may have important effects. An example of this may be indicated by the selective formation of photosystem I under low light intensity, observed for two types of isolated wheat chloroplasts (Ogawa & Shibata, 1973). Other contributors to the effective hit probability may be indicated by any number of light intensity effects on algal ultrastructure (Findley et al., 1970). Kiefer (1973) observed that under light stress the chloroplasts of some of the centric diatoms contract and move to the valvar ends of the cell. Some of the filamentous algae, such as Mougeotia change the orientation of their chloroplast (Mayer, 1971). This alga has a platelike chloroplast which is Direction
of illumination
I I
/Side
I Side
Axis
2
of rototion
FIG. 4. Hit probability and chloroplast orientation. Some of the filamentous algae, such as Mougeofiu, changethe orientationof their chloroplastin response to changinglight intensity.If the chloroplastis assumed to bea thme-diional box free to rotate about one axis, then the angleof rotation, 0, can be relatedto the hit probability,p. 34 T.B.
514
P. A. CRILI.
oriented face-on to low light and edge-on to high light. The angle of rotation of the chloroplast, 0, and the hit probability, p, can be directly related if the chloroplast is assumed to be a three dimensional box free to rotate about one axis (Fig. 4). Given the areas of the two receptive sides A, and A,;f, andf,, the efficiency or effective or light trapping fractions of sides 1 and 2 respectively; (AT +A$)* = 1.0, so that the hit probability will not exceed unity; then : p = A,f, cos Q+ A2fZ sin H. (15)
Light
intensity
Fro. 5. Photosynthesis-light curve of the photosynthetic sulfur bacteria Chvomati~r~ compared to the curve drawn from equation (2) by adding the output of two independent photosystems, each with a different hit probability.
THE
PHOTOSYNTHESIS-LIGHT
CURVE
515
Traditionally, the slope of the photosynthesis-light curve has been related to the speed of the light stage, and the ceiling of the curve to the speed of the dark reactions. Some high T, values, especially for algae adapted to low intensities, suggests that light reactions can be limiting at all intensities. Application of the analog may then suggest whether the ceiling of the curve is sufficiently long and flattened to be attributed to dark stage limitation, whether it has the shape one would expect from photo-inhibition, or whether there are insufficient data to resolve the difference. An advantage of an analog-type model over a geometric-type is that it may be of some interest even when it fails, since it may reveal cases where the inadequacy of the underlying assumptions becomes critical, cases which might otherwise be considered quite ordinary. For example, curves of some photosynthetic sulfur bacteria reported by Takahashi & Ichimura (1970) could not be fitted with the model because the exponential decline portion of the curve approached a value significantly greater than zero. It is interesting that the sum of two independent photosystems, each with a different hit probability, can produce a curve very similar to the curve produced by the bacteria (Fig. 5). This suggests either the presence of two distinct populations or some basic inadequacy of the model to describe the photosynthesis-light curve of some bacteria. In summary, the proposed simple four-parameter analog model of the photosynthesis-light curve not only provides a close approximation for many observed curves, but also presents the possibility for a meaningful interpretation of the parameters. Although in the present model, parameter values are only relative and dependent upon the measured units of photosynthesis and light, they present advantages over strictly geometric ones, and the possibility for future refinement and extension. REFERENCES ARUGA, Y. (1965). Bot. Mug. Tokyo. 78, 360. BANNISTER, T. T. (1974). Limnol. Oceanogr. 19, 1, FEE, E. J. (1969). Limnol. Oceanogr. 14, 906. FINDLEY, (1970). J. Phycol. 6, 182. HOFF, A. J. (1974). Photochem. Photobiof. 19, 51. JBRGENSEN, E. G., (1968). Physiol. Pl. 21, 423. JPIRGENSEN, E. G. (1969). Physiol. PI. 22, 1307. E. G. & STEEMANN NIELSEN, E. (1965). In Primary Productivity in Aquatic Environments (C. R. Goldman, ed.), p. 37. Berkeley: University of California Press. KIEFER, D. A. (1973). Mar. Biol. 23, 39. MAYER, F. (1971). In Structure and Function of Chloroplasts (M. Gibbs, ed.), p. 35. New York: Springer-Verlag. MORRJS, I. & FARREL, K. (1971). Physiol. PI. 25, 372. OGAWA, T. & SHIBATA, K. (1973). Physiol. PI. 29, 112. RYTHER, J. H. & MENZEL, D. W. (1959). Limnol. Oceanogr. 4, 492. JORGENSEN,
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SEELY, G. R. (1973). J. rheor. Biol. 40, 173. SHERIDAN, R. P. (1972a). J. Phycof. 8, 166. SHERIDAN, R. P. (19726). PI. physiof. 50, 355. STEELE, J. H. (1962). Linmol. Oceurzogr. 7, 137. STEELE, J. H. (1965). Memoir 1st Ital. Oa’robiol. 18 (suppl.), 353. STEEMANN NIELSEN, E. (1962). Physiol. PI. 15, 161. STEEMANN NIELSEN, E. & HANSEN, V. K. (1959). Physiol. PI. 12, 353. STEEMANN NIELSEN, E. & HANSEN, V. K. (1961). Physiol. PI. 14, 595. STEEMANN NIELSEN, E. & JORGENSEN, E. G. (1968). Phydiol. PI. 21, 401. TAKAHASHI, M. & ICHIMURA, S. (1970). Limnol. Ocemogr. 15, 929. TALLING, J. F. (1957). New Phytol. 56, 133. VOLLENWEIDER, R. A. (1965). Memoir 1st Ital. Idrobiol. 18 (suppl.), 425. VOLLENWEIDER, R. A. (1970). In Proc. IBP/PP Tech. Meeting, Trebon, p. 455. Wageningen: Center for Agricultural Publications and Documentation. YENTSCH, C. S. & LEE, R. W. (1966). J. Mar. Res. 24, 319.