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Dynamic behaviour of a model for photosynthesis and photoinhibition
Ecological Modelling, 69 (1993) 113-133
113
Elsevier Science Publishers B.V., Amsterdam
Dynamic behaviour of a model for photosynthesis and photoinhibition P . H . C . E i l e r s a a n d J.C.H. P e e t e r s b a Rijnmond Central Environmental Agency, Schiedam, Netherlands b Rijkswaterstaat, Tidal Waters Diuision, Middelburg, Netherlands
(Received 29 October 1991; accepted 4 September 1992)
ABSTRACT Eilers, P.H.C. and Peeters, J.C.H., 1993. Dynamic behaviour of a model for photosynthesis and photoinhibition. Ecol. Modelling, 69: 113-133. The dynamic behaviour of a simple model for photosynthesis and photoinhibiton, which was published before in this journal, is analysed. The differential equations are simplified and characteristic parameters and time scale are introduced. It is shown that the gradual development of photoinhibition is very important for the interpretation of observations on primary production. The model is used to explain differences between short and long incubations, the effect of intermittent illumination, the influence of prior illumination and hysteresis-effects. Two suggestions for extensions of the model are presented: one for extra consumption of oxygen, coupled to photoinhibition, the other for saturation of production at lower temperatures.
1 INTRODUCTION P h o t o i n h i b i t i o n is a d y n a m i c process, with a time scale t h a t is v e r y r e l e v a n t to ecological m o d e l s o f p r i m a r y p r o d u c t i o n in aquatic e n v i r o n m e n t s . T h e classic p u b l i c a t i o n s o f M y e r s a n d B u r r (1940), K o k (1956), V o l l e n w e i d e r (1965), H a r r i s a n d Piccinin (1977) and P l a t t et al. (1980) m a d e it clear t h a t p h o t o i n h i b i t i o n d e v e l o p s gradually. Y e t m o s t m o d e l s for the r e l a t i o n s h i p b e t w e e n light intensity, p h o t o s y n t h e s i s a n d p h o t o i n h i b i t i o n are o f a static c h a r a c t e r . R e c e n t l y , i n t e r e s t in the ecological m o d e l l i n g o f d y n a m i c aspects o f p r i m a r y p r o d u c t i o n has grown, see e.g. Falkowski a n d
Correspondence to: P.H.C. Eilers, Rijnmond Central Environmental Agency, 's Graveland-
seweg 565, 3119 XT Schiedam, Netherlands. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
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P.H.C. E I L E R S A N D J.C.H. P E E T E R S
Wirick (1981), D e n m a n and Gargett (1983), Lewis et al. (1984), Gallegos and Platt (1985), Neale and Marra (1985), D e n m a n and Marra (1986), Neale and Richerson (1987). In this paper we show that a simple model for photoproduction and photoinhibition lends itself for interpretation and modelling of experimental results. Because of its simplicity, the model is suitable for inclusion in larger ecologic models and for the simulation of dynamic processes. The model has been described by Peeters and Eilers (1978) and Eilers and Peeters (1988). It is based on hypothetical "photosynthetic factories" that can be in three different states. Transitions between the states occur by photochemical and enzymatic processes. The behaviour of the model is described by a system of linear differential equations, with coefficients that depend on light intensity and temperature. The model has two time constants. One is much shorter than a second and is connected to the photosynthesis process. The second time constant is about a thousand seconds and is connected to the gradual development of photoinhibition at high and moderate light intensities. For ecological applications the details of the fast process are not relevant, so we simplified the model on the assumption that equilibrium is reached instantly. As a result, a linear differential equation with only one time constant, dependent on light intensity, is obtained. We introduced dimensionless quantities and applied the model to a number of situations, including a study of the effect of temperature. We also considered the influence of prior illumination, and the effects of intermittent light. By means of numerical simulation we showed that pronounced hysteresis can occur when the intensity changes gradually. The model described in section 2 and applied in section 3 can give a (semi)quantitative description of a number of dynamic aspects of photoinhibition that are encountered in vivo and in vitro and that have puzzled many investigators. Two extensions are needed to make the model more realistic: oxygen consumption, coupled to photoinhibition, and saturation of the production rate at lower temperatures (section 4). The main part of the paper is a mathematical analysis of the dynamic behaviour of the model; physiological interpretations are discussed in section 5. 2 THE MODEL, D I F F E R E N T I A L EQUATIONS AND SIMPLIFICATIONS
2.1 The model and the differential equations A detailed description of the model is given by Eilers and Peeters (1988). It is assumed that phytoplankton contain "photosynthetic factories" (PSF) that can be in three different states: (1) the resting or " o p e n " state; (2) the
115
MODEL FOR PHOTOSYNTHESISAND PHOTOINHIBITION
activated or "closed" state; (3) the inhibited state. The possible transitions between the states and the corresponding rate constants are shown in Fig. 1. Two rate constants ( a I and f l I ) are proportional to the light intensity I, two other rate constants (3' and 6) represent enzymatic processes and are i n d e p e n d e n t of the light intensity. The production of oxygen is assumed to be proportional to the n u m b e r of transitions from the open state to the closed state. Let x a, x 2 and x 3 be the fractions of the total n u m b e r of PSFs that are in the respective states 1, 2 and 3. A system of differential equations can be obtained from Fig. 1: 1 + y x 2 + (~x 3
(1)
dxl/dt
= -alx
dxa/dt
= alx 1 - (ill + y)x a
(2)
dx3//dt
= [ 3 I x 2 - (~x 3
(3)
These equations are linear and have constant coefficients if I is constant. A general solution can be found by standard methods, e.g. the Laplace transform (Hildebrand, 1962); however, the equations will be simplified before they will be used further. Yet it is instructive to solve the equation for x2(t) in the special case when Xl(0) = 1, x2(0) = 0 and x3(0) = 0. This corresponds to an incubation after a long period in the dark, when all PSFs are initially in the open state. In this case, =
x2(t)
-
°'[( -
1 +
e sit-
S1 - - S 2
(s t] 1 +
e '2t
+ --,
(4)
S1S 2
where s 1 and s 2 are the roots of the quadratic equation s 2 + ( a I + 131 + r + ~)s + a / 3 I 2 + , ~ I + t ~ t + r ~ = 0.
(5)
Graphs of x2(t), for different values of u = I / I o p t , where /opt is the
closed
open
5
inhibited
Fig. 1. The model. The three states are: (1) " o p e n " , in rest; (2) "closed", activated; (3) inhibited.
116
P.H.C. EILERSAND J.C.H. PEETERS
o
0.3
~
,=o2
o.o0.0~
015
~-~
li0
115
210
t
215
Fig. 2. Graphs of x2(t) with Xl(0) = 1 for several values of u = I / I o p t (/opt is the optimal intensity in the steady state, see Fig. 3). The time scale is arbitrary. For clarity, the ratio between fast and slow reactions is chosen much smaller than in reality. This is the solution of the differential equation of the complete model.
optimal light intensity in the steady state (see below), are plotted in Fig. 2; the time scale is arbitrary and a = 5/3 and y = 56. The graph shows the three terms in the equation for Xz(t): a constant (the steady state value), a slow, decaying, exponential function, and a fast, rising, exponential function that describes the transient from the origin. For clarity, the ratios a//3 and y / 6 have been kept unrealistically low in these graphs. From Jones and Kok (1966), R a d m e r and Kok (1977) and Vincent et al. (1984) one can derive that a >> 1000/3 and y >> 1000 6. The transitions from the open to the closed state and back are so fast that they can be considered instantaneous. This observation is the key to a very useful simplification of the differential equations.
2.2 Simplification of the differential equations For practical applications the time constants of the model are important. The values of aI and y are much higher than those of/31 and 6. Changes in x a and x z, caused by changes in I, reach a balance within much less than a second. The values of /31 and 6 are such that - - even at high intensities - - changes in x 3 can take up to one hour or more. Time constants of the order of one second or less are not very interesting from an ecological point of view. However, to model primary production u n d e r fluctuating light-conditions, time constants of the order of one hour can be relevant. This is also important for the interpretation of the results of experimental incubations.
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
117
When time constants differ as much as is the case here, the equations can be simplified by assuming that the fast processes reach a balance instantaneously. The simplification makes it easier to apply the model analytically and numerically. It is known that models with very different time constants can show unstable behaviour with standard methods for numerical integration of differential equations. Special algorithms for socalled stiff equations are needed without the simplification. To simplify the equations, 3x 3 and fllx 2 can be omitted in the Eqs. (1) and (2):
dxl/dt
=
-ollx
(6)
1 + ")IX 2
d x 2 / d t = a l x 1 - ")/x2
(7)
d x 3 / d t = [3Ix 2 - 6x 3
(8)
The first two equations describe the fast process, the last one describes the slow process. Let X 0 =X
1 +X
2 =
1 -x 3
(9)
be the non-inhibited fraction of the PSFs. If the equilibrium between x a and x 2 is reached rapidly, d x l / d t and d x z / d t will be practically zero after a very short time. Then it follows that
x,=yxo/(aI+7
),
(10)
x2 = a l x o / ( a I + 7).
(11)
These equations describe how the non-inhibited PSFs are distributed, d e p e n d e n t on I, between the resting state and the activated state. The differential equation for x 0 is:
dxo/dt
=
3
-
( afiI2 -a I- + + y6
) x 0.
(12)
Before solving this equation, it is necessary to connect the coefficients to the parameters of the steady state production curve, and to introduce dimensionless variables that are easy to manipulate and interpret.
2.3 The steady state curve and dimensionless variables The steady state solution of the Eqs. (6)-(8) is found by making the left-hand side equal to zero. For the steady state value of x I we find: a6 Ycl = a f l I 2 + a ~ I + y8 "
(13)
We postulate that the rate of oxygen production, p, is proportional to the
118
P.H.C. EILERS AND J.C.H. PEETERS
pm
P
Fig. 3. The steady state production curve and some characteristic parameters ( p is production rate, I is intensity).
number of transitions from state 1 to state 2 per unit of time, so: kay6I P = ka121 = a ~ I 2 + a 3 I + 7 6
(14)
The general shape of this curve and some characteristic parameters are shown in Fig. 3. The optimal intensity lopt is given by: Iop t
=
~-/OLj~
(15)
.
The peak of the curve can be more or less pronounced. A suitable parameter, w, to characterize the sharpness of the peak is given by w
=
af~-/~3 T .
(16)
The smaller w, the sharper the peak (Eilers and Peeters, 1988). The maximal production rate in the steady state, Pro, is given by Pm=ky/(1
+ 2w).
(17)
To analyse dynamic behaviour further, three dimensionless quantities are introduced: the relative intensity, u, the relative production rate, q, and the scaled time, z: u = I/Iopt;
q =P/Pm;
~" = 6t.
(18)
With these quantities, Eq. (12) changes into: dxo/dr=l-
- - + 1 wu+ 1
x o.
(19)
119
M O D E L F O R P H O T O S Y N T H E S I S AND P H O T O I N H I B I T I O N
2.4 Solution o f the dimensionless equation Equation (19) is a linear differential equation with, for constant I, constant coefficients. T h e general solution is: x0(r, u ) = r i 0 ( u ) + [Xo(0, u ) - r i o ( U ) ] e -s',
(20)
with ri0(u) =Xo(OO, u)
=
wu + 1 U2 "q- W U "l- 1
u2 S -- --WU+ 1 + 1 = 1/rio(U ).
(21)
Consider the case Xo(0, u) = 1; this corresponds to incubation at a constant relative light intensity, u, after a long period in the dark. T h e steady state level, ~o(U), decreases with increasing relative intensity u, and the speed at which ri0(u) is a p p r o a c h e d increases with u. Because equilibrium is reached rapidly, the distribution of x o into x~ and x 2 is i n d e p e n d e n t of r and can be expressed in terms of u and w as follows:
u)
x2(r, u ) =
wu + 1
'
x0( , u)
x l ( r , u)
wu + 1
(22)
T h e relative production rate, q, is given by kotIx1/Pm. F r o m (17) and (18) follows: Pm = k ~ / w / ( w +
2).
(23)
Consequently, (w + 2)u q(r, u)-
wu+l
w+ 2 Xo(r, u ) =
w
X z ( r , u).
(24)
We find that q(r, u) is given by q(r, u)=q(u)
+ [q(0, u ) - q ( u ) ]
e -s',
(25)
where the initial rate, q(0, u), and the steady state rate, ~(u) = q(m, u), are given by: q(O, u) =
(w + 2)u wu + 1 '•
(w + 2)u = u2 + w u + 1
"
(26)
T h e initial rate is a curve of the saturation type, the steady state rate is a p e a k e d curve, as in Fig. 3, with a peak at u = 1. T h e transient between the curves is an exponential function with rate s, as given by Eq. (21). T h e rate increases with u. 2.5 Some aspects o f the solution Essentially, the Eqs. (25) and (26) describe the behaviour of the model. We feel that it is useful to present some aspects of the solution in a graphic
120
P.H.C. EILERS A N D J.C.H. P E E T E R S
form. This serves two purposes: to give the reader a better understanding of the solution, and to make it easier to interpret experimental results io terms of the model. It is assumed that x(0, u ) = 1: all PSFs start in the open state, after a long stay in the dark. In Fig. 4 the scale for the dimensionless production rate, q, is logarithmic. One sees that for high relative light intensities the curves are nearly straight lines, starting at the same value for ~- = 0. From the left part of (26) follows that q(0, u) approaches (w + 2)/w for ~- = 0 when u is high. The exponential function dominates in (25) because (~(u) is very small; its logarithm is -s~-, where s = u/w + 1. Plotting these functions on a logarithmic scale yields a straight line. The curves in Fig. 4 are similar to results of Kok (1956) for incubations at very high light intensities and high temperatures. At lower temperatures Kok found a saturation effect, which is discussed in section 4.2. In many incubations the rate of production cannot be observed directly, but the cumulative production of oxygen. This can be modelled by the time integral of q('r, u). From (25) follows: Q(~', u ) = fo q(~'' u) d~" = O(u)r + ( ~ ( u ) - q ( 0 ,
u))(1 - e S ' ) / s .
(27)
This equation is shown in graphic form in Fig. 5. At ~-= 0 the slope is greatest; it gradually decreases, but remains positive. In real experiments at high light intensities the slope can become negative after some time. In section 4.1 the model is extended to incorporate consumption of oxygen that is coupled to photoinhibition. To emphasize again that time and light intensity should be incorporated in studies of photosynthesis and photoinhibition, Fig. 6 gives a three-dimensional plot for incubations at constant illumination, starting with all
iii iu = 10 u = 20
0.1-
u = 50 u = 100 u = 200
0.01-
u = 500 u = 1000
0.0010 Fig.
4.
logaritmic
011 The
relative
scale
0'.2 production
( w = 1).
0'.3 rate,
q(~-,
014 u),
for
v
015
different
values
of
u =
I/loot
on
a
121
MODEL FOR PHOTOSYNTHESIS AND PHOTOINH1B1TION 3.0~ u=l u 2
Q
u
0.5
u
5
u u
0.2 10
i
0.1
2.0-
1.0-
0"00
.
.
.
.
.
Fig. 5. Integrated relative production rates, Q(r, u), for several values of u. The value of w is 1.
PSFs in the open state. From the plot one can see how a saturation curve gradually changes with time into an inhibition curve.
2.6 The influence of temperature The way in which temperature influences the values of the characteristic parameters of the steady state production curve has been discussed by Eilers and Peeters (1988). A similar analysis can be applied to the dynamic behaviour. The parameters a and/3 stand for the rates of biophysical processes and are independent of temperature. The parameters y and 6 represent
3~
q 2.
1.
000
0.5
1.0
15
4
6
8
U ~10
Fig. 6. Three-dimensional representation of the dependence of the relative production rate, q(~', u), on time, ~', and light intensity, u; w = 1.
P.H.C. EILERSAND J.C.H. PEETERS
122
enzymatic processes. We assume that they are influenced in the same way by temperature, 0:
7=yof(O);
a=aof(O ).
(28)
Generally f(O) will, over a limited range of 0, increase with increasing 0. If /opt(0) and Pm(0) denote the optimal light intensity and maximal production rate at temperature 0 respectively, we find from (28), (15) and (17): Pm(02) loot(02) f(02) Pm(01----~ = iopt(01~ -- f(01 ~ .
(29)
The parameter w is independent of temperature. For the dimensionless variables q, u and r we find: u(01) r(02) f(02) /g(02-------~ = g(01~---~ = f(O1-----'y.
(30)
The interpretation of these relations is that at higher temperatures maximal production rates are higher, photoinhibition is less pronounced and the steady state is reached faster. 3 THEORETICAL APPLICATIONS In this section we use the model to investigate phenomena that are caused by changes in light intensity. We first study the influence of prior illumination on production curves. We then look at rapidly switching a light-source on and off. Finally we study, using numerical simulations, how hysteresis occurs with gradually changing light intensities.
3.1 Prior illumination and short incubations In short incubations a saturation curve is found for any value of the light intensity at which prior illumination took place, as the following analysis will show. Let g = x0(0, u) be the proportion of uninhibited PSFs at the start of the incubation. If prior illumination, at light intensity flu, was long enough to reach the steady state, then w~+l g
K2+wK+ 1
(31)
During the short incubation x 0 does not change. From (24) it follows that
q(r,u)=
Xo(r, u)(w + 2)u g(w + 2)u = wu + 1 wu + 1
(32)
123
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
As Xo(Z, u) is essentially constant, so is q(~-, u); it is a saturation curve in u, with maximum g ( w + 2 ) / w . These results agree with experimental observations like those of Kok (1956) and Steeman Nielsen (1962). 3.2 Intermittent illumination
Let the source of light, with relative intensity u, be switched on and off repeatedly. Let the length of one cycle be T, expressed in relative units of time like z, let the light be on for a time r T and off for a time (1 - r ) T . The switching is repeated for a long time. While the light is on, the proportion of uninhibited PSFs, x3, will increase towards the steady state value, Y3(u), given by U2
(33)
3(u) = u 2 + w u + 1 '
and the time course is given by x3(z,u)=Y3(u)+[x;(u)-Y,3(u)]
e -s"
for
O<~z<~rT.
(34)
Here x; = x3(0, u), the value of x 3 at the start of the light phase and s is given by (21). When the light is off, x 3 decreases exponentially: x3('r , u ) = x ; ' ( u )
e -('-rT)
for
r T <~z <~ T.
(35)
Here x'3'(u)=x3(r T, u). If the sequence of light and dark phases is repeated long enough, each period will be identical. Then x3(T, u) has equal x3(0, u). This leads to X~r(U) =X3(U) + X;(U) e - s r T - .~3(u) e -srT
(36)
x ~ ( u ) =x~'(u) e - ( v - ' r )
(37)
If the light is switched on and off so fast that T is much smaller than 1, then the following approximations are accurate: e -s~-- 1 - s z ;
e -(~-'r) = 1 - z + rT.
(38)
If these approximations are introduced into the Eqs. (36) and (37), then x;'(u) - x;(u) + x;(u)srT= ~3(u)srT,
(39)
x ; ( u ) - x ; ' ( u ) + x ; ' ( u ) ( T - r T ) = 0.
(40)
Addition and simplification of these equations gives x;(u)lx3(u)
St" = sr + (1 - r)"
(41)
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P.H.C. EILERS AND J.C.H. PEETERS
2.5q 2.0= 0.02 1.50.05 1.0-
0.10 0.20
o.s-
o.o
t
~
~
a
u
g
Fig. 7. T h e relative p r o d u c t i o n rate, q'(r, u), during t h e light p h a s e of i n t e r m i t t e n t light, as a function of the relative light intensity, u, for several values of the fraction of time, r, d u r i n g which the light is on. T h e value of w is 1.
It is easy to see that this fraction is less than or equal to 1, for any value of r between 0 and 1. The inhibition is always less than in the case of continuous illumination. This is expected, as there is less time for inhibition to develop. F r o m Eq. (40) it follows that x~(u) and x~'(u) differ very little if T is small. The relative production rate is given by (w + 2)u
q'(u)
=p'(u)//pm
-
wu+ l
(1 - x ; ( u ) )
(w + 2)u
wu + 1
wu 4- 1
ru 2 q_ wig --1-1
(42)
H e r e it is assumed that the distribution of x 0 into x 1 and x 2 is established in a much shorter time than T. Graphs of q'(u) for a n u m b e r of values of r are plotted in Fig. 7. The steady state curve for continuous light corresponds to r = 1. The model predicts that as r becomes smaller, production rates become higher and there is less photoinhibition. Note that we are talking about the production rate in the light phase: there is no production in the dark phase. The average production rate is given by rq'(u). For any value of r, the average production rate is less than the steady state production rate. We therefore conclude that switching between light and dark increases the efficiency of production during the light phase, but cannot lead to a higher averaged production at any intensity.
125
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
3.3 Slowly changing intensities and hysteresis In this section we present a small numerical study of the behaviour of the model when the light intensity changes slowly. The differential equation (19) completely describes the dynamics of the model. It is nonlinear in u; for constant u it is a linear differential equation with constant coefficients, for which an analytical solution exists, as section 2.4 showed. When u changes on a time scale comparable to that for inhibition, an analytic solution generally cannot be found and a numerical approximation is necessary. As we are not interested in precise results, we use the Euler method of integration: for small time steps At, the changes Ax 0 are
2.57
a
1.5 1.0 0.5
O'°o.bO
O.b2
o.b4
o,b6
o,b8
r
0.i0
071
0:2
013
014
r
015
2.5--
q 2.01.51.00.5000
c
2.
1.0 0.5 00010
0~4
018
112
116
r
210
~
4
6
8
r
1"0
2.5-
q 2.01.5-
0.50.0
Fig. 8. Computed curves of the relative production rate, q(r), in the simulation of hysteresis for several values of T, the period of u ( r ) : (a) T = 0.1; (b) T = 0.5; (c) T = 2; (d) T = 10.
126
P.H.C. E1LERS A N D J.C.H. P E E T E R S
computed as
(dxo/dr)Az. The
value of Ar was chosen to be 0.1 or less.
To describe u(~-) the following function is used:
U(T)= //max[1-- cos(2"rrT/T)]/2,
(43)
because it is simple to compute and gives a reasonable approximation of the variation of sunlight on a bright day. The period T is important, as will become clear, and was changed in large steps between 0.1 and 10. Note that T is on the same relative time scale as ~-= ~t, see Eq. (18). Figure 8 shows q(~-) for a number of values of Umax (1, 2, 3, 4 and 5) and T. For small values of T, i.e. if the intensity changes rapidly compared to the reaction time of the organism, the curves of q(~-) are relatively a
2,5-
2.5-
q
q
2.0-
2.0-
1.5-
1.5-
1.0"
1.0-
0.5"
0.5'
0.0
0.0
211
c
2.o1
2.o-
1.5- ~
1.5-
1.0-i 0.5-
1.00.5-
0 "00
i
~
~
4 ~ 5
0m00
i
2
3
4 ~ m5
Fig. 9. Hysteresis curves: plots of q ( c , u ) in Fig. 8 a g a i n s t u ( 7 ) in Fig. 8; (a) T = 0.1; (b) T = 0.5; (c) T = 2; (d) T = 10.
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
127
symmetric and photoinhibition hardly occurs. For values of T about 1, photoinhibiton occurs and the curves of q(r) are strongly asymmetric. For T = 10, the curve of q(~-) is more symmetric and there is marked photoinhibition. Photoinhibition is most pronounced when u reaches its peak; the model shows an "afternoon depression". These curves should be compared to those for the results of Marra (1978). Another way to present these results is shown in Fig. 9. We have plotted q(~-) against u(~-) for all the curves in Fig. 8. Strong hysteresis is visible. We think that more than any other equation or graph in this paper this figure illustrates the dynamic aspects of photoinhibition. Similar experimental hysteresis curves were reported by Harris and Lott (1973) and Falkowski and Owens (1978). These simulations were done to illustrate the dynamics of the model and its ability to explain empirical results such as the "afternoon depression" and hysteresis. The simulations were kept as simple as possible for clarity. Obviously, the model lends itself to more complicated simulations, in which algae move up and down a water column, according to a deterministic or stochastic regime, such as the Markov process in Gallegos and Platt (1982). Also the light regime might be modelled more realistically, or be derived from experimental data. 4 TWO EXTENSIONS OF THE MODEL In the preceding sections the model for gross photosynthesis was analysed and applied to some typical problems of empirical relevance. We chose to make the model as simple as possible to keep the analysis uncomplicated and to emphasize the essential points. However, there is ample empirical evidence that extensions are needed to make the model more realistic. In this section we introduce two extensions: the incorporation of extra oxygen consumption and temperature-dependent saturation.
4.1 Oxygen consumption coupled to photoinhibition Figure 10 shows data on oxygen production during incubation, from a paper by Myers and Burr (1940). The datapoints were read from figure 2 in that paper, the lines were fitted with the model
cj(t)=ajlt +aiz(1-exp(-a~3t)). Here cj(t) stands for the oxygen
(44)
concentration at time t in incubation j, and a jl, a j2, and a j3 are estimated parameters. The functional form of (44) follows from (27). It is clear that the fitted lines agree well with the data points, but that there is a problem. In Fig. 10 the dark respiration is represented by the
128
P.H.C. E I L E R S A N D J.C.H. P E E T E R S
I00-5° ~ ~
.,.....--~4000 0
1
0
25-0
0 9700
v
~--
z -
-
~
0
0
o
- 12900
I -25~
5ol 0
-
,
,
10
20
dark l
i
30
40
18400
~
27700
50 t[min] 60
Fig. 10. Integrated oxygen production as measured by Myers and Burr, with fitted curves.
slope of the line marked "dark". We see that at high light intensities the increase in the concentration of oxygen rapidly reverses into a decrease at even a higher rate than in the dark. Our model cannot decribe this consumption of oxygen: only positive production rates can occur. One way to introduce the additional consumption into the model is to make it proportional to the proportion of inhibited PSFs, x 3. It is well known that oxygen consumption takes time to develop, as does photoinhibition. Moreover, the consumption of oxygen decays gradually, once the organism is placed in the dark (Harris and Piccinin, 1977). With consumption proportional to x3, we find:
(w + 2)u
u)- wu+-T Xo(, u)
(w + 2)r/ W
[1-Xo(~', u)]
(45)
where Xo(r, u) is given by (20) and -,7 is a new parameter; its value must determined experimentally.
4.2 Temperature-dependent saturation In Fig. 4, q(t) of the basic model at high light intensities is plotted on a logarithmic scale. Approximately, all curves start at the same point (w + 2)/w at z = 0 as straight lines. Kok (1956) shows very similar experimental data for incubations at higher temperatures. For lower temperatures, however, he found that the initial rates were appreciably lower, as they are in Fig. 11. These figures were constructed by assuming that it is not q(z) that is observed but q'(~-), the output of an intermediate enzymatic process with a
MODEL
FOR PHOTOSYNTHESIS
129
AND PHOTOINHIBITION
1m
01 q
~
u
=
5
0
u = 100 u = 200
i
u = 500 u = 1000
0. 0 . 0
.
.
.
0.b4
r
0.b5
Fig. 11. R e l a t i v e p r o d u c t i o n rates, q'(~-, u), with s a t u r a t i o n at low t e m p e r a t u r e s .
saturation curve that has q(~-) as input: q(~') q'(~') = 1 + q ( ~ ' ) / m ( O ) '
(46)
where re(O) is the saturation level that depends on the t e m p e r a t u r e 0; m(O) = 1 and w = 1 were used. It is interesting to note that this saturation leads to the same functional model for the steady state production curve for continuous illumination. If the curve is given by I P
-
(47)
aI 2 -t- bI + c
without saturation, and if the saturation is modelled as p'=
P
1 +p/m
,
(48)
then we find that I p' =
aI 2 + (b + l / m ) I
+c
(49)
By referring to (15)-(17) we can see that optimal intensity and initial slope do not change, but that the height of the peak and the shape parameter, w, do: the steady state curve is flattened. 5 DISCUSSION
We initially studied the simple model presented by Eilers and Peeters (1988) and found that its dynamic behaviour is simple enough for analysis,
130
P.H.C. EILERS A N D J.C,H. P E E T E R S
but complicated enough to model a number of experimental results. In its basic form our model is relatively small and completely described by four parameters: characteristic time scale (1/3), optimal light intensity (/opt), maximal production rate (Pro) and influence of photoinhibition (w). We found, however, that two more parameters are needed to incorporate oxygen consumption coupled to photoinhibition, and saturation at lower temperatures. Six is a large number of parameters to fit to experimental data. Dynamic measurements are necessary, probably at several temperatures and with judicious switching between light and dark. Prior illumination should also be taken into account. At the m o m e n t we cannot give much advice on incubation strategies, which are an interesting area for further research. Our main finding is that static interpretations of production curves are incomplete and confusing. We feel that many of the controversies in the literature about the existence and importance of photoinhibiton can be explained in this way. Without a good description of the dynamics of photoinhibition, it is not appropriate to search for sophisticated theoretical production curves, except perhaps for steady state conditions. But steady states are rare in nature. The need to incorporate time in the modelling of photosynthesis and photoinhibiton was noted by Platt and Gallegos (1981). Several authors have approached this problem in essentially two different ways: as ad-hoc time-dependent modifications of the parameters of a steady state production curve, or on the base of a two-state model with modifications. Neal and Marra (1985) use a decaying exponential function, that diminishes the Pm of a production curve, proportional to the light intensity. With (approximate) convolution the effect of varying light intensities can be computed. D e n m a n and Marra (1986) assume a transition along an exponential function from a non-inhibited to an inhibited production curve. The time constant of the exponential function is fixed. This approach shows some similarity to Fig. 9, where the transition from the instantaneous production curve to the steady state production curve is along exponential functions; however, in our model the time constant depends on the light intensity. Pahl-Wostl and Imboden (1990) also modify a steady state curve with a dynamic inhibition curve. Baumert (1976) and Gross (1982) use models with only two states: open and closed. The former uses it to analyse the effect of intermittent light. Megard et al. (1984) present a model that is identical to ours. They do not analyse its dynamic behaviour. Fasham and Platt (1983) presented a modified two-state model (unfortunately, this reference came to our attention only after publication of Eilers and Peeters, 1988). They modify a rate constant that corresponds to our a with a function that decreases exponen-
M O D E L F O R P H O T O S Y N T H E S I S AND P H O T O I N H I B I T I O N
131
tially with increasing light intensity. They do not analyse the dynamic behaviour. In our opinion models without an explicit inhibited state cannot give an adequate description of the gradual development of, and recovery from, photoinhibition. The photosynthetic factory (PSF), a concept introduced by Crill (1977), is the heart of our model. It is a mechanistic construct. We chose to describe the behaviour of the photosynthetic organism strictly as a consequence of the dynamics of the PSF. It seems appropriate to suggest here some physiological interpretations. A PSF models the photosystems I and II and the enzymes of dark reactions. Photoinhibition is in fact the inactivation of photosystem II (Kok et al., 1965; Raven and Beardall, 1981; Powles, 1984). After long illumination with strong light (section 3.1), both Pm and the quantum efficiency (the initial slope of the instantaneous production curve), are reduced. This response has been observed experimentally by Kok (1956), Steeman-Nielsen (1962) and others. The oxygen consumption coupled to photoinhibition (section 4.1) cannot be explained as photorespiration, as proposed by Harris and Lott (1973) and Harris and Piccinin (1977). This process and the Mehler reaction require photosynthetic electron transport. The transport is inhibited, proportionally to the number of inhibited PS II reaction centres. The consumption of oxygen develops gradually and it can be 3 to 5 times higher then dark respiration in the final stages of photoinhibition (Myers and Burr, 1940; Kok, 1956; Kandler and Sironval, 1959). It is therefore highly unlikely that this oxygen consumption is photorespiration. Kyle (1987) proposes that damage to the QB protein in photosystem II is accompanied by oxygen uptake. A similar mechanism was proposed by Sayre and H o m a n n (1979) for the inhibition of photosynthesis by the so-called A D R Y reagents, which have an effect very similar to photoinhibition. Kandler and Sironval (1959) assumed that the increased oxygen uptake is caused by dark respiration processes. So there are several candidate processes, coupled to photoinhibiton, that consume oxygen. ACKNOWLEDGMENTS We wish to thank the following persons: Thomas Bannister, Jacco Kromkamp, Jan Snel, Marcel van der Tol; they gave us useful comments and spotted a number of errors. We also thank Ms. J. Burroughs for many improvements on our use of the English language. REFERENCES Baumert, H., 1976. Mathematical model for the explanation of the increased phytoplankton production caused by intermittent light. Int. Rev. Gesamten Hydrobiol., 61: 517-527.
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P.H.C. E I L E R S A N D J.C.H. P E E T E R S
Crill, P.A., 1977. The photosynthesis-light curve: a simple analog model. J. Theor. Biol., 6: 503-516. Denman, K.L. and Gargett, A.E., 1983. Time and space scales of vertical mixing and advection of phytoplankton in the upper ocean. Limnol. Oceanogr., 28: 801-815. Denman, K.L. and Marra, J., 1986. Modelling the time dependent photoadaptation of phytoplankton to fluctuating light. In: J.C.J. Nihoul (Editor), Marine Interfaces Ecohydrodynamics. Elsevier, Amsterdam, pp. 341-349. Eilers, P.H.C. and Peeters, J.C.H., 1988. A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton. Ecol. Modelling, 42: 199-215. Falkowski, P.G. and Owens, T.G., 1978. Effects of light intensity on photosynthesis and dark respiration in six species of marine phytoplankton. Mar. Biol., 45: 289-295. Falkowski, P.G. and Wirick, C.D., 1981. A simulation study of the effects of vertical mixing on primary productivity. Mar. Biol., 65: 69-75. Fasham, M.J.R. and Platt, T., 1983. Photosynthetic response of phytoplankton to light: a physiological model. Proc. R. Soc. Lond. B, 219: 355-370. Gallegos, C.L. and Platt, T., 1982. Phytoplankton production and water motion in surface mixed layers. Deep-Sea Res., 29: 65-76. Gallegos, C.L. and Platt, T., 1985. Vertical advection of phytoplankton and productivity estimates: a dimensional analysis. Mar. Ecol. Prog. Ser., 26: 125-134. Gross, L.J., 1982. Photosynthetic dynamics in varying light environments: a model and its application to whole leave carbon gain. Ecology, 63: 84-93. Harris, G.P. and Lott, J.N.A., 1973. Light intensity and photosynthetic rates in phytoplankton. J. Fish. Res. Board Can., 30: 1771-1778. Harris, G.P. and Piccinin, B.B., 1977. Photosynthesis by natural phytoplankton populations. Arch. Hydrobiol., 59: 405-457. Hildebrand, F.B., 1962. Advanced Calculus for Applications. Prentice/Hall Englewood Cliffs, NJ. Jones, L.W. and Kok, B., 1966. Photoinhibition of chloroplast reactions I. Kinetics and action spectra. Plant Physiol., 41: 1037-1043. Kandler, O. and Sironval, C., 1959. Photooxidation processes in normal green chlorella cells II. Effects on metabolism. Biochim. Biophys. Acta, 33: 207-215. Kok, B., 1956. On the inhibition of photosynthesis by intense light. Biochim. Biophys. Acta, 21: 234-244. Kok, B., Gassner, E.B. and Rursinski, H.J., 1965. Photoinhibition of chloroplast reactions. Photochem. Photobiol., 4: 215-227. Kyle, D.J., 1987. The biochemical basis for photoinhibition of photosystem II. In: D.J. Kyle, C.B. Osmond and C.J. Arntzen (Editors), Photoinhibition. Elsevier, Amsterdam, pp. 197-226. Lewis, M.R., Cullen, J.J. and Platt, T., 1984. Relationships between vertical mixing and photoadaptation of phytoplankton: similarity criteria. Mar. Ecol. Prog. Ser., 15: 141-149. Marra, J., 1978. Effect of short-term variations in light intensity on photosynthesis of a marine phytoplankter: a laboratory simulation study. Mar. Biol., 46: 191-202. Megard, R.O., Tonkyn, D.W. and Senft, W.H., 1984. Kinetics of oxygenic photosynthesis in planktonic algae. J. Plankton Res., 6: 325-337. Myers, J. and Burr, G.O., 1940. Studies on photosynthesis. Some effects of light of high intensity on chlorella. J. Gen. Physiol., 24: 45-67. Neale, P.J. and Marra, J., 1985. Short-term variation of Pma~ under natural irradiance conditions: a model and its implications. Mar. Ecol. Prog. Ser., 26: 113-124. Neale, P.J. and Richerson, P.J., 1987. Photoinhibition and diurnal variation of phytoplank-
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FOR PHOTOSYNTHESIS
AND PHOTOINHIBIT1ON
133
ton photosynthesis - - I. Development of a photosynthesis irradiance model from studies of in situ responses. J. Plankton Res., 9: 167-193. Pahl-Wostl, C. and Imboden, D.M., 1990. D Y P H O R A - - a dynamic model for the rate of photosynthesis of algae. J. Plankton Res., 12: 1207-1221. Peeters, J.C.H. and Eilers, P., 1978. The relationship between light intensity and photosynthesis. A simple mathematical model. Hydrobiol. Bull., 12: 134-136. Platt, T. and Gallegos, C.L., 1981. Modelling primary production. In: P.G. Falkowski (Editor), Primary Productivity in the Sea. Plenum Press, New York, pp. 339-362. Platt, T., Gallegos, C.L. and Harrison, W.G., 1980. Photoinhibiton and photosynthesis in natural assemblages of marine phytoplankton. J. Mar. Res., 38: 687-701. Powles, S.B., 1984. Photoinhibition of photosynthesis by visible light. Annu. Rev. Plant Physiol., 35: 15-44. Radmer, R.J. and Kok, B., 1977. Light conversion efficiency in photosynthesis. In: A. Trebst and M. Avron (Editors), Encyclopedia of Plant Physiology: Photosynthesis I. Springer, New York, pp. 125-134. Raven, J.A. and Beardall, J., 1981. Respiration and photorespiration. In: T. Platt (Editor), Physiological Bases of Phytoplankton Ecology. Department of Fisheries and Oceans, Ottawa, pp. 55-82. Sayre, R.T. and Homann, P.H., 1979. A light-dependent oxygen consumption induced by photosystem II of isolated chloroplasts. Arch. Biochem. Biophys., 196: 525-533. Steeman-Nielsen, E., 1962. Inactivation of the photochemical mechanism in photosynthesis as a means to protect cells against too high light intensities. Physiol. Plant., 15: 161-171. Vincent, W.F., Neale, P.J. and Richerson, P.J., 1984. Photoinhibition: algal responses to bright light during diel stratification and mixing in a tropical alpine lake. J. Phycol., 20: 201-211. Vollenweider, R.A., 1965. Calculation models of photosynthesis - - depth curves and some implications regarding daily rate estimates in primary production measurements. Mem. Ist. Ital. Idrobiol., 18 Suppl.: 425-427.
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Elsevier Science Publishers B.V., Amsterdam
Dynamic behaviour of a model for photosynthesis and photoinhibition P . H . C . E i l e r s a a n d J.C.H. P e e t e r s b a Rijnmond Central Environmental Agency, Schiedam, Netherlands b Rijkswaterstaat, Tidal Waters Diuision, Middelburg, Netherlands
(Received 29 October 1991; accepted 4 September 1992)
ABSTRACT Eilers, P.H.C. and Peeters, J.C.H., 1993. Dynamic behaviour of a model for photosynthesis and photoinhibition. Ecol. Modelling, 69: 113-133. The dynamic behaviour of a simple model for photosynthesis and photoinhibiton, which was published before in this journal, is analysed. The differential equations are simplified and characteristic parameters and time scale are introduced. It is shown that the gradual development of photoinhibition is very important for the interpretation of observations on primary production. The model is used to explain differences between short and long incubations, the effect of intermittent illumination, the influence of prior illumination and hysteresis-effects. Two suggestions for extensions of the model are presented: one for extra consumption of oxygen, coupled to photoinhibition, the other for saturation of production at lower temperatures.
1 INTRODUCTION P h o t o i n h i b i t i o n is a d y n a m i c process, with a time scale t h a t is v e r y r e l e v a n t to ecological m o d e l s o f p r i m a r y p r o d u c t i o n in aquatic e n v i r o n m e n t s . T h e classic p u b l i c a t i o n s o f M y e r s a n d B u r r (1940), K o k (1956), V o l l e n w e i d e r (1965), H a r r i s a n d Piccinin (1977) and P l a t t et al. (1980) m a d e it clear t h a t p h o t o i n h i b i t i o n d e v e l o p s gradually. Y e t m o s t m o d e l s for the r e l a t i o n s h i p b e t w e e n light intensity, p h o t o s y n t h e s i s a n d p h o t o i n h i b i t i o n are o f a static c h a r a c t e r . R e c e n t l y , i n t e r e s t in the ecological m o d e l l i n g o f d y n a m i c aspects o f p r i m a r y p r o d u c t i o n has grown, see e.g. Falkowski a n d
Correspondence to: P.H.C. Eilers, Rijnmond Central Environmental Agency, 's Graveland-
seweg 565, 3119 XT Schiedam, Netherlands. 0304-3800/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
114
P.H.C. E I L E R S A N D J.C.H. P E E T E R S
Wirick (1981), D e n m a n and Gargett (1983), Lewis et al. (1984), Gallegos and Platt (1985), Neale and Marra (1985), D e n m a n and Marra (1986), Neale and Richerson (1987). In this paper we show that a simple model for photoproduction and photoinhibition lends itself for interpretation and modelling of experimental results. Because of its simplicity, the model is suitable for inclusion in larger ecologic models and for the simulation of dynamic processes. The model has been described by Peeters and Eilers (1978) and Eilers and Peeters (1988). It is based on hypothetical "photosynthetic factories" that can be in three different states. Transitions between the states occur by photochemical and enzymatic processes. The behaviour of the model is described by a system of linear differential equations, with coefficients that depend on light intensity and temperature. The model has two time constants. One is much shorter than a second and is connected to the photosynthesis process. The second time constant is about a thousand seconds and is connected to the gradual development of photoinhibition at high and moderate light intensities. For ecological applications the details of the fast process are not relevant, so we simplified the model on the assumption that equilibrium is reached instantly. As a result, a linear differential equation with only one time constant, dependent on light intensity, is obtained. We introduced dimensionless quantities and applied the model to a number of situations, including a study of the effect of temperature. We also considered the influence of prior illumination, and the effects of intermittent light. By means of numerical simulation we showed that pronounced hysteresis can occur when the intensity changes gradually. The model described in section 2 and applied in section 3 can give a (semi)quantitative description of a number of dynamic aspects of photoinhibition that are encountered in vivo and in vitro and that have puzzled many investigators. Two extensions are needed to make the model more realistic: oxygen consumption, coupled to photoinhibition, and saturation of the production rate at lower temperatures (section 4). The main part of the paper is a mathematical analysis of the dynamic behaviour of the model; physiological interpretations are discussed in section 5. 2 THE MODEL, D I F F E R E N T I A L EQUATIONS AND SIMPLIFICATIONS
2.1 The model and the differential equations A detailed description of the model is given by Eilers and Peeters (1988). It is assumed that phytoplankton contain "photosynthetic factories" (PSF) that can be in three different states: (1) the resting or " o p e n " state; (2) the
115
MODEL FOR PHOTOSYNTHESISAND PHOTOINHIBITION
activated or "closed" state; (3) the inhibited state. The possible transitions between the states and the corresponding rate constants are shown in Fig. 1. Two rate constants ( a I and f l I ) are proportional to the light intensity I, two other rate constants (3' and 6) represent enzymatic processes and are i n d e p e n d e n t of the light intensity. The production of oxygen is assumed to be proportional to the n u m b e r of transitions from the open state to the closed state. Let x a, x 2 and x 3 be the fractions of the total n u m b e r of PSFs that are in the respective states 1, 2 and 3. A system of differential equations can be obtained from Fig. 1: 1 + y x 2 + (~x 3
(1)
dxl/dt
= -alx
dxa/dt
= alx 1 - (ill + y)x a
(2)
dx3//dt
= [ 3 I x 2 - (~x 3
(3)
These equations are linear and have constant coefficients if I is constant. A general solution can be found by standard methods, e.g. the Laplace transform (Hildebrand, 1962); however, the equations will be simplified before they will be used further. Yet it is instructive to solve the equation for x2(t) in the special case when Xl(0) = 1, x2(0) = 0 and x3(0) = 0. This corresponds to an incubation after a long period in the dark, when all PSFs are initially in the open state. In this case, =
x2(t)
-
°'[( -
1 +
e sit-
S1 - - S 2
(s t] 1 +
e '2t
+ --,
(4)
S1S 2
where s 1 and s 2 are the roots of the quadratic equation s 2 + ( a I + 131 + r + ~)s + a / 3 I 2 + , ~ I + t ~ t + r ~ = 0.
(5)
Graphs of x2(t), for different values of u = I / I o p t , where /opt is the
closed
open
5
inhibited
Fig. 1. The model. The three states are: (1) " o p e n " , in rest; (2) "closed", activated; (3) inhibited.
116
P.H.C. EILERSAND J.C.H. PEETERS
o
0.3
~
,=o2
o.o0.0~
015
~-~
li0
115
210
t
215
Fig. 2. Graphs of x2(t) with Xl(0) = 1 for several values of u = I / I o p t (/opt is the optimal intensity in the steady state, see Fig. 3). The time scale is arbitrary. For clarity, the ratio between fast and slow reactions is chosen much smaller than in reality. This is the solution of the differential equation of the complete model.
optimal light intensity in the steady state (see below), are plotted in Fig. 2; the time scale is arbitrary and a = 5/3 and y = 56. The graph shows the three terms in the equation for Xz(t): a constant (the steady state value), a slow, decaying, exponential function, and a fast, rising, exponential function that describes the transient from the origin. For clarity, the ratios a//3 and y / 6 have been kept unrealistically low in these graphs. From Jones and Kok (1966), R a d m e r and Kok (1977) and Vincent et al. (1984) one can derive that a >> 1000/3 and y >> 1000 6. The transitions from the open to the closed state and back are so fast that they can be considered instantaneous. This observation is the key to a very useful simplification of the differential equations.
2.2 Simplification of the differential equations For practical applications the time constants of the model are important. The values of aI and y are much higher than those of/31 and 6. Changes in x a and x z, caused by changes in I, reach a balance within much less than a second. The values of /31 and 6 are such that - - even at high intensities - - changes in x 3 can take up to one hour or more. Time constants of the order of one second or less are not very interesting from an ecological point of view. However, to model primary production u n d e r fluctuating light-conditions, time constants of the order of one hour can be relevant. This is also important for the interpretation of the results of experimental incubations.
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
117
When time constants differ as much as is the case here, the equations can be simplified by assuming that the fast processes reach a balance instantaneously. The simplification makes it easier to apply the model analytically and numerically. It is known that models with very different time constants can show unstable behaviour with standard methods for numerical integration of differential equations. Special algorithms for socalled stiff equations are needed without the simplification. To simplify the equations, 3x 3 and fllx 2 can be omitted in the Eqs. (1) and (2):
dxl/dt
=
-ollx
(6)
1 + ")IX 2
d x 2 / d t = a l x 1 - ")/x2
(7)
d x 3 / d t = [3Ix 2 - 6x 3
(8)
The first two equations describe the fast process, the last one describes the slow process. Let X 0 =X
1 +X
2 =
1 -x 3
(9)
be the non-inhibited fraction of the PSFs. If the equilibrium between x a and x 2 is reached rapidly, d x l / d t and d x z / d t will be practically zero after a very short time. Then it follows that
x,=yxo/(aI+7
),
(10)
x2 = a l x o / ( a I + 7).
(11)
These equations describe how the non-inhibited PSFs are distributed, d e p e n d e n t on I, between the resting state and the activated state. The differential equation for x 0 is:
dxo/dt
=
3
-
( afiI2 -a I- + + y6
) x 0.
(12)
Before solving this equation, it is necessary to connect the coefficients to the parameters of the steady state production curve, and to introduce dimensionless variables that are easy to manipulate and interpret.
2.3 The steady state curve and dimensionless variables The steady state solution of the Eqs. (6)-(8) is found by making the left-hand side equal to zero. For the steady state value of x I we find: a6 Ycl = a f l I 2 + a ~ I + y8 "
(13)
We postulate that the rate of oxygen production, p, is proportional to the
118
P.H.C. EILERS AND J.C.H. PEETERS
pm
P
Fig. 3. The steady state production curve and some characteristic parameters ( p is production rate, I is intensity).
number of transitions from state 1 to state 2 per unit of time, so: kay6I P = ka121 = a ~ I 2 + a 3 I + 7 6
(14)
The general shape of this curve and some characteristic parameters are shown in Fig. 3. The optimal intensity lopt is given by: Iop t
=
~-/OLj~
(15)
.
The peak of the curve can be more or less pronounced. A suitable parameter, w, to characterize the sharpness of the peak is given by w
=
af~-/~3 T .
(16)
The smaller w, the sharper the peak (Eilers and Peeters, 1988). The maximal production rate in the steady state, Pro, is given by Pm=ky/(1
+ 2w).
(17)
To analyse dynamic behaviour further, three dimensionless quantities are introduced: the relative intensity, u, the relative production rate, q, and the scaled time, z: u = I/Iopt;
q =P/Pm;
~" = 6t.
(18)
With these quantities, Eq. (12) changes into: dxo/dr=l-
- - + 1 wu+ 1
x o.
(19)
119
M O D E L F O R P H O T O S Y N T H E S I S AND P H O T O I N H I B I T I O N
2.4 Solution o f the dimensionless equation Equation (19) is a linear differential equation with, for constant I, constant coefficients. T h e general solution is: x0(r, u ) = r i 0 ( u ) + [Xo(0, u ) - r i o ( U ) ] e -s',
(20)
with ri0(u) =Xo(OO, u)
=
wu + 1 U2 "q- W U "l- 1
u2 S -- --WU+ 1 + 1 = 1/rio(U ).
(21)
Consider the case Xo(0, u) = 1; this corresponds to incubation at a constant relative light intensity, u, after a long period in the dark. T h e steady state level, ~o(U), decreases with increasing relative intensity u, and the speed at which ri0(u) is a p p r o a c h e d increases with u. Because equilibrium is reached rapidly, the distribution of x o into x~ and x 2 is i n d e p e n d e n t of r and can be expressed in terms of u and w as follows:
u)
x2(r, u ) =
wu + 1
'
x0( , u)
x l ( r , u)
wu + 1
(22)
T h e relative production rate, q, is given by kotIx1/Pm. F r o m (17) and (18) follows: Pm = k ~ / w / ( w +
2).
(23)
Consequently, (w + 2)u q(r, u)-
wu+l
w+ 2 Xo(r, u ) =
w
X z ( r , u).
(24)
We find that q(r, u) is given by q(r, u)=q(u)
+ [q(0, u ) - q ( u ) ]
e -s',
(25)
where the initial rate, q(0, u), and the steady state rate, ~(u) = q(m, u), are given by: q(O, u) =
(w + 2)u wu + 1 '•
(w + 2)u = u2 + w u + 1
"
(26)
T h e initial rate is a curve of the saturation type, the steady state rate is a p e a k e d curve, as in Fig. 3, with a peak at u = 1. T h e transient between the curves is an exponential function with rate s, as given by Eq. (21). T h e rate increases with u. 2.5 Some aspects o f the solution Essentially, the Eqs. (25) and (26) describe the behaviour of the model. We feel that it is useful to present some aspects of the solution in a graphic
120
P.H.C. EILERS A N D J.C.H. P E E T E R S
form. This serves two purposes: to give the reader a better understanding of the solution, and to make it easier to interpret experimental results io terms of the model. It is assumed that x(0, u ) = 1: all PSFs start in the open state, after a long stay in the dark. In Fig. 4 the scale for the dimensionless production rate, q, is logarithmic. One sees that for high relative light intensities the curves are nearly straight lines, starting at the same value for ~- = 0. From the left part of (26) follows that q(0, u) approaches (w + 2)/w for ~- = 0 when u is high. The exponential function dominates in (25) because (~(u) is very small; its logarithm is -s~-, where s = u/w + 1. Plotting these functions on a logarithmic scale yields a straight line. The curves in Fig. 4 are similar to results of Kok (1956) for incubations at very high light intensities and high temperatures. At lower temperatures Kok found a saturation effect, which is discussed in section 4.2. In many incubations the rate of production cannot be observed directly, but the cumulative production of oxygen. This can be modelled by the time integral of q('r, u). From (25) follows: Q(~', u ) = fo q(~'' u) d~" = O(u)r + ( ~ ( u ) - q ( 0 ,
u))(1 - e S ' ) / s .
(27)
This equation is shown in graphic form in Fig. 5. At ~-= 0 the slope is greatest; it gradually decreases, but remains positive. In real experiments at high light intensities the slope can become negative after some time. In section 4.1 the model is extended to incorporate consumption of oxygen that is coupled to photoinhibition. To emphasize again that time and light intensity should be incorporated in studies of photosynthesis and photoinhibition, Fig. 6 gives a three-dimensional plot for incubations at constant illumination, starting with all
iii iu = 10 u = 20
0.1-
u = 50 u = 100 u = 200
0.01-
u = 500 u = 1000
0.0010 Fig.
4.
logaritmic
011 The
relative
scale
0'.2 production
( w = 1).
0'.3 rate,
q(~-,
014 u),
for
v
015
different
values
of
u =
I/loot
on
a
121
MODEL FOR PHOTOSYNTHESIS AND PHOTOINH1B1TION 3.0~ u=l u 2
Q
u
0.5
u
5
u u
0.2 10
i
0.1
2.0-
1.0-
0"00
.
.
.
.
.
Fig. 5. Integrated relative production rates, Q(r, u), for several values of u. The value of w is 1.
PSFs in the open state. From the plot one can see how a saturation curve gradually changes with time into an inhibition curve.
2.6 The influence of temperature The way in which temperature influences the values of the characteristic parameters of the steady state production curve has been discussed by Eilers and Peeters (1988). A similar analysis can be applied to the dynamic behaviour. The parameters a and/3 stand for the rates of biophysical processes and are independent of temperature. The parameters y and 6 represent
3~
q 2.
1.
000
0.5
1.0
15
4
6
8
U ~10
Fig. 6. Three-dimensional representation of the dependence of the relative production rate, q(~', u), on time, ~', and light intensity, u; w = 1.
P.H.C. EILERSAND J.C.H. PEETERS
122
enzymatic processes. We assume that they are influenced in the same way by temperature, 0:
7=yof(O);
a=aof(O ).
(28)
Generally f(O) will, over a limited range of 0, increase with increasing 0. If /opt(0) and Pm(0) denote the optimal light intensity and maximal production rate at temperature 0 respectively, we find from (28), (15) and (17): Pm(02) loot(02) f(02) Pm(01----~ = iopt(01~ -- f(01 ~ .
(29)
The parameter w is independent of temperature. For the dimensionless variables q, u and r we find: u(01) r(02) f(02) /g(02-------~ = g(01~---~ = f(O1-----'y.
(30)
The interpretation of these relations is that at higher temperatures maximal production rates are higher, photoinhibition is less pronounced and the steady state is reached faster. 3 THEORETICAL APPLICATIONS In this section we use the model to investigate phenomena that are caused by changes in light intensity. We first study the influence of prior illumination on production curves. We then look at rapidly switching a light-source on and off. Finally we study, using numerical simulations, how hysteresis occurs with gradually changing light intensities.
3.1 Prior illumination and short incubations In short incubations a saturation curve is found for any value of the light intensity at which prior illumination took place, as the following analysis will show. Let g = x0(0, u) be the proportion of uninhibited PSFs at the start of the incubation. If prior illumination, at light intensity flu, was long enough to reach the steady state, then w~+l g
K2+wK+ 1
(31)
During the short incubation x 0 does not change. From (24) it follows that
q(r,u)=
Xo(r, u)(w + 2)u g(w + 2)u = wu + 1 wu + 1
(32)
123
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
As Xo(Z, u) is essentially constant, so is q(~-, u); it is a saturation curve in u, with maximum g ( w + 2 ) / w . These results agree with experimental observations like those of Kok (1956) and Steeman Nielsen (1962). 3.2 Intermittent illumination
Let the source of light, with relative intensity u, be switched on and off repeatedly. Let the length of one cycle be T, expressed in relative units of time like z, let the light be on for a time r T and off for a time (1 - r ) T . The switching is repeated for a long time. While the light is on, the proportion of uninhibited PSFs, x3, will increase towards the steady state value, Y3(u), given by U2
(33)
3(u) = u 2 + w u + 1 '
and the time course is given by x3(z,u)=Y3(u)+[x;(u)-Y,3(u)]
e -s"
for
O<~z<~rT.
(34)
Here x; = x3(0, u), the value of x 3 at the start of the light phase and s is given by (21). When the light is off, x 3 decreases exponentially: x3('r , u ) = x ; ' ( u )
e -('-rT)
for
r T <~z <~ T.
(35)
Here x'3'(u)=x3(r T, u). If the sequence of light and dark phases is repeated long enough, each period will be identical. Then x3(T, u) has equal x3(0, u). This leads to X~r(U) =X3(U) + X;(U) e - s r T - .~3(u) e -srT
(36)
x ~ ( u ) =x~'(u) e - ( v - ' r )
(37)
If the light is switched on and off so fast that T is much smaller than 1, then the following approximations are accurate: e -s~-- 1 - s z ;
e -(~-'r) = 1 - z + rT.
(38)
If these approximations are introduced into the Eqs. (36) and (37), then x;'(u) - x;(u) + x;(u)srT= ~3(u)srT,
(39)
x ; ( u ) - x ; ' ( u ) + x ; ' ( u ) ( T - r T ) = 0.
(40)
Addition and simplification of these equations gives x;(u)lx3(u)
St" = sr + (1 - r)"
(41)
124
P.H.C. EILERS AND J.C.H. PEETERS
2.5q 2.0= 0.02 1.50.05 1.0-
0.10 0.20
o.s-
o.o
t
~
~
a
u
g
Fig. 7. T h e relative p r o d u c t i o n rate, q'(r, u), during t h e light p h a s e of i n t e r m i t t e n t light, as a function of the relative light intensity, u, for several values of the fraction of time, r, d u r i n g which the light is on. T h e value of w is 1.
It is easy to see that this fraction is less than or equal to 1, for any value of r between 0 and 1. The inhibition is always less than in the case of continuous illumination. This is expected, as there is less time for inhibition to develop. F r o m Eq. (40) it follows that x~(u) and x~'(u) differ very little if T is small. The relative production rate is given by (w + 2)u
q'(u)
=p'(u)//pm
-
wu+ l
(1 - x ; ( u ) )
(w + 2)u
wu + 1
wu 4- 1
ru 2 q_ wig --1-1
(42)
H e r e it is assumed that the distribution of x 0 into x 1 and x 2 is established in a much shorter time than T. Graphs of q'(u) for a n u m b e r of values of r are plotted in Fig. 7. The steady state curve for continuous light corresponds to r = 1. The model predicts that as r becomes smaller, production rates become higher and there is less photoinhibition. Note that we are talking about the production rate in the light phase: there is no production in the dark phase. The average production rate is given by rq'(u). For any value of r, the average production rate is less than the steady state production rate. We therefore conclude that switching between light and dark increases the efficiency of production during the light phase, but cannot lead to a higher averaged production at any intensity.
125
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
3.3 Slowly changing intensities and hysteresis In this section we present a small numerical study of the behaviour of the model when the light intensity changes slowly. The differential equation (19) completely describes the dynamics of the model. It is nonlinear in u; for constant u it is a linear differential equation with constant coefficients, for which an analytical solution exists, as section 2.4 showed. When u changes on a time scale comparable to that for inhibition, an analytic solution generally cannot be found and a numerical approximation is necessary. As we are not interested in precise results, we use the Euler method of integration: for small time steps At, the changes Ax 0 are
2.57
a
1.5 1.0 0.5
O'°o.bO
O.b2
o.b4
o,b6
o,b8
r
0.i0
071
0:2
013
014
r
015
2.5--
q 2.01.51.00.5000
c
2.
1.0 0.5 00010
0~4
018
112
116
r
210
~
4
6
8
r
1"0
2.5-
q 2.01.5-
0.50.0
Fig. 8. Computed curves of the relative production rate, q(r), in the simulation of hysteresis for several values of T, the period of u ( r ) : (a) T = 0.1; (b) T = 0.5; (c) T = 2; (d) T = 10.
126
P.H.C. E1LERS A N D J.C.H. P E E T E R S
computed as
(dxo/dr)Az. The
value of Ar was chosen to be 0.1 or less.
To describe u(~-) the following function is used:
U(T)= //max[1-- cos(2"rrT/T)]/2,
(43)
because it is simple to compute and gives a reasonable approximation of the variation of sunlight on a bright day. The period T is important, as will become clear, and was changed in large steps between 0.1 and 10. Note that T is on the same relative time scale as ~-= ~t, see Eq. (18). Figure 8 shows q(~-) for a number of values of Umax (1, 2, 3, 4 and 5) and T. For small values of T, i.e. if the intensity changes rapidly compared to the reaction time of the organism, the curves of q(~-) are relatively a
2,5-
2.5-
q
q
2.0-
2.0-
1.5-
1.5-
1.0"
1.0-
0.5"
0.5'
0.0
0.0
211
c
2.o1
2.o-
1.5- ~
1.5-
1.0-i 0.5-
1.00.5-
0 "00
i
~
~
4 ~ 5
0m00
i
2
3
4 ~ m5
Fig. 9. Hysteresis curves: plots of q ( c , u ) in Fig. 8 a g a i n s t u ( 7 ) in Fig. 8; (a) T = 0.1; (b) T = 0.5; (c) T = 2; (d) T = 10.
MODEL FOR PHOTOSYNTHESIS AND PHOTOINHIBITION
127
symmetric and photoinhibition hardly occurs. For values of T about 1, photoinhibiton occurs and the curves of q(r) are strongly asymmetric. For T = 10, the curve of q(~-) is more symmetric and there is marked photoinhibition. Photoinhibition is most pronounced when u reaches its peak; the model shows an "afternoon depression". These curves should be compared to those for the results of Marra (1978). Another way to present these results is shown in Fig. 9. We have plotted q(~-) against u(~-) for all the curves in Fig. 8. Strong hysteresis is visible. We think that more than any other equation or graph in this paper this figure illustrates the dynamic aspects of photoinhibition. Similar experimental hysteresis curves were reported by Harris and Lott (1973) and Falkowski and Owens (1978). These simulations were done to illustrate the dynamics of the model and its ability to explain empirical results such as the "afternoon depression" and hysteresis. The simulations were kept as simple as possible for clarity. Obviously, the model lends itself to more complicated simulations, in which algae move up and down a water column, according to a deterministic or stochastic regime, such as the Markov process in Gallegos and Platt (1982). Also the light regime might be modelled more realistically, or be derived from experimental data. 4 TWO EXTENSIONS OF THE MODEL In the preceding sections the model for gross photosynthesis was analysed and applied to some typical problems of empirical relevance. We chose to make the model as simple as possible to keep the analysis uncomplicated and to emphasize the essential points. However, there is ample empirical evidence that extensions are needed to make the model more realistic. In this section we introduce two extensions: the incorporation of extra oxygen consumption and temperature-dependent saturation.
4.1 Oxygen consumption coupled to photoinhibition Figure 10 shows data on oxygen production during incubation, from a paper by Myers and Burr (1940). The datapoints were read from figure 2 in that paper, the lines were fitted with the model
cj(t)=ajlt +aiz(1-exp(-a~3t)). Here cj(t) stands for the oxygen
(44)
concentration at time t in incubation j, and a jl, a j2, and a j3 are estimated parameters. The functional form of (44) follows from (27). It is clear that the fitted lines agree well with the data points, but that there is a problem. In Fig. 10 the dark respiration is represented by the
128
P.H.C. E I L E R S A N D J.C.H. P E E T E R S
I00-5° ~ ~
.,.....--~4000 0
1
0
25-0
0 9700
v
~--
z -
-
~
0
0
o
- 12900
I -25~
5ol 0
-
,
,
10
20
dark l
i
30
40
18400
~
27700
50 t[min] 60
Fig. 10. Integrated oxygen production as measured by Myers and Burr, with fitted curves.
slope of the line marked "dark". We see that at high light intensities the increase in the concentration of oxygen rapidly reverses into a decrease at even a higher rate than in the dark. Our model cannot decribe this consumption of oxygen: only positive production rates can occur. One way to introduce the additional consumption into the model is to make it proportional to the proportion of inhibited PSFs, x 3. It is well known that oxygen consumption takes time to develop, as does photoinhibition. Moreover, the consumption of oxygen decays gradually, once the organism is placed in the dark (Harris and Piccinin, 1977). With consumption proportional to x3, we find:
(w + 2)u
u)- wu+-T Xo(, u)
(w + 2)r/ W
[1-Xo(~', u)]
(45)
where Xo(r, u) is given by (20) and -,7 is a new parameter; its value must determined experimentally.
4.2 Temperature-dependent saturation In Fig. 4, q(t) of the basic model at high light intensities is plotted on a logarithmic scale. Approximately, all curves start at the same point (w + 2)/w at z = 0 as straight lines. Kok (1956) shows very similar experimental data for incubations at higher temperatures. For lower temperatures, however, he found that the initial rates were appreciably lower, as they are in Fig. 11. These figures were constructed by assuming that it is not q(z) that is observed but q'(~-), the output of an intermediate enzymatic process with a
MODEL
FOR PHOTOSYNTHESIS
129
AND PHOTOINHIBITION
1m
01 q
~
u
=
5
0
u = 100 u = 200
i
u = 500 u = 1000
0. 0 . 0
.
.
.
0.b4
r
0.b5
Fig. 11. R e l a t i v e p r o d u c t i o n rates, q'(~-, u), with s a t u r a t i o n at low t e m p e r a t u r e s .
saturation curve that has q(~-) as input: q(~') q'(~') = 1 + q ( ~ ' ) / m ( O ) '
(46)
where re(O) is the saturation level that depends on the t e m p e r a t u r e 0; m(O) = 1 and w = 1 were used. It is interesting to note that this saturation leads to the same functional model for the steady state production curve for continuous illumination. If the curve is given by I P
-
(47)
aI 2 -t- bI + c
without saturation, and if the saturation is modelled as p'=
P
1 +p/m
,
(48)
then we find that I p' =
aI 2 + (b + l / m ) I
+c
(49)
By referring to (15)-(17) we can see that optimal intensity and initial slope do not change, but that the height of the peak and the shape parameter, w, do: the steady state curve is flattened. 5 DISCUSSION
We initially studied the simple model presented by Eilers and Peeters (1988) and found that its dynamic behaviour is simple enough for analysis,
130
P.H.C. EILERS A N D J.C,H. P E E T E R S
but complicated enough to model a number of experimental results. In its basic form our model is relatively small and completely described by four parameters: characteristic time scale (1/3), optimal light intensity (/opt), maximal production rate (Pro) and influence of photoinhibition (w). We found, however, that two more parameters are needed to incorporate oxygen consumption coupled to photoinhibition, and saturation at lower temperatures. Six is a large number of parameters to fit to experimental data. Dynamic measurements are necessary, probably at several temperatures and with judicious switching between light and dark. Prior illumination should also be taken into account. At the m o m e n t we cannot give much advice on incubation strategies, which are an interesting area for further research. Our main finding is that static interpretations of production curves are incomplete and confusing. We feel that many of the controversies in the literature about the existence and importance of photoinhibiton can be explained in this way. Without a good description of the dynamics of photoinhibition, it is not appropriate to search for sophisticated theoretical production curves, except perhaps for steady state conditions. But steady states are rare in nature. The need to incorporate time in the modelling of photosynthesis and photoinhibiton was noted by Platt and Gallegos (1981). Several authors have approached this problem in essentially two different ways: as ad-hoc time-dependent modifications of the parameters of a steady state production curve, or on the base of a two-state model with modifications. Neal and Marra (1985) use a decaying exponential function, that diminishes the Pm of a production curve, proportional to the light intensity. With (approximate) convolution the effect of varying light intensities can be computed. D e n m a n and Marra (1986) assume a transition along an exponential function from a non-inhibited to an inhibited production curve. The time constant of the exponential function is fixed. This approach shows some similarity to Fig. 9, where the transition from the instantaneous production curve to the steady state production curve is along exponential functions; however, in our model the time constant depends on the light intensity. Pahl-Wostl and Imboden (1990) also modify a steady state curve with a dynamic inhibition curve. Baumert (1976) and Gross (1982) use models with only two states: open and closed. The former uses it to analyse the effect of intermittent light. Megard et al. (1984) present a model that is identical to ours. They do not analyse its dynamic behaviour. Fasham and Platt (1983) presented a modified two-state model (unfortunately, this reference came to our attention only after publication of Eilers and Peeters, 1988). They modify a rate constant that corresponds to our a with a function that decreases exponen-
M O D E L F O R P H O T O S Y N T H E S I S AND P H O T O I N H I B I T I O N
131
tially with increasing light intensity. They do not analyse the dynamic behaviour. In our opinion models without an explicit inhibited state cannot give an adequate description of the gradual development of, and recovery from, photoinhibition. The photosynthetic factory (PSF), a concept introduced by Crill (1977), is the heart of our model. It is a mechanistic construct. We chose to describe the behaviour of the photosynthetic organism strictly as a consequence of the dynamics of the PSF. It seems appropriate to suggest here some physiological interpretations. A PSF models the photosystems I and II and the enzymes of dark reactions. Photoinhibition is in fact the inactivation of photosystem II (Kok et al., 1965; Raven and Beardall, 1981; Powles, 1984). After long illumination with strong light (section 3.1), both Pm and the quantum efficiency (the initial slope of the instantaneous production curve), are reduced. This response has been observed experimentally by Kok (1956), Steeman-Nielsen (1962) and others. The oxygen consumption coupled to photoinhibition (section 4.1) cannot be explained as photorespiration, as proposed by Harris and Lott (1973) and Harris and Piccinin (1977). This process and the Mehler reaction require photosynthetic electron transport. The transport is inhibited, proportionally to the number of inhibited PS II reaction centres. The consumption of oxygen develops gradually and it can be 3 to 5 times higher then dark respiration in the final stages of photoinhibition (Myers and Burr, 1940; Kok, 1956; Kandler and Sironval, 1959). It is therefore highly unlikely that this oxygen consumption is photorespiration. Kyle (1987) proposes that damage to the QB protein in photosystem II is accompanied by oxygen uptake. A similar mechanism was proposed by Sayre and H o m a n n (1979) for the inhibition of photosynthesis by the so-called A D R Y reagents, which have an effect very similar to photoinhibition. Kandler and Sironval (1959) assumed that the increased oxygen uptake is caused by dark respiration processes. So there are several candidate processes, coupled to photoinhibiton, that consume oxygen. ACKNOWLEDGMENTS We wish to thank the following persons: Thomas Bannister, Jacco Kromkamp, Jan Snel, Marcel van der Tol; they gave us useful comments and spotted a number of errors. We also thank Ms. J. Burroughs for many improvements on our use of the English language. REFERENCES Baumert, H., 1976. Mathematical model for the explanation of the increased phytoplankton production caused by intermittent light. Int. Rev. Gesamten Hydrobiol., 61: 517-527.
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Crill, P.A., 1977. The photosynthesis-light curve: a simple analog model. J. Theor. Biol., 6: 503-516. Denman, K.L. and Gargett, A.E., 1983. Time and space scales of vertical mixing and advection of phytoplankton in the upper ocean. Limnol. Oceanogr., 28: 801-815. Denman, K.L. and Marra, J., 1986. Modelling the time dependent photoadaptation of phytoplankton to fluctuating light. In: J.C.J. Nihoul (Editor), Marine Interfaces Ecohydrodynamics. Elsevier, Amsterdam, pp. 341-349. Eilers, P.H.C. and Peeters, J.C.H., 1988. A model for the relationship between light intensity and the rate of photosynthesis in phytoplankton. Ecol. Modelling, 42: 199-215. Falkowski, P.G. and Owens, T.G., 1978. Effects of light intensity on photosynthesis and dark respiration in six species of marine phytoplankton. Mar. Biol., 45: 289-295. Falkowski, P.G. and Wirick, C.D., 1981. A simulation study of the effects of vertical mixing on primary productivity. Mar. Biol., 65: 69-75. Fasham, M.J.R. and Platt, T., 1983. Photosynthetic response of phytoplankton to light: a physiological model. Proc. R. Soc. Lond. B, 219: 355-370. Gallegos, C.L. and Platt, T., 1982. Phytoplankton production and water motion in surface mixed layers. Deep-Sea Res., 29: 65-76. Gallegos, C.L. and Platt, T., 1985. Vertical advection of phytoplankton and productivity estimates: a dimensional analysis. Mar. Ecol. Prog. Ser., 26: 125-134. Gross, L.J., 1982. Photosynthetic dynamics in varying light environments: a model and its application to whole leave carbon gain. Ecology, 63: 84-93. Harris, G.P. and Lott, J.N.A., 1973. Light intensity and photosynthetic rates in phytoplankton. J. Fish. Res. Board Can., 30: 1771-1778. Harris, G.P. and Piccinin, B.B., 1977. Photosynthesis by natural phytoplankton populations. Arch. Hydrobiol., 59: 405-457. Hildebrand, F.B., 1962. Advanced Calculus for Applications. Prentice/Hall Englewood Cliffs, NJ. Jones, L.W. and Kok, B., 1966. Photoinhibition of chloroplast reactions I. Kinetics and action spectra. Plant Physiol., 41: 1037-1043. Kandler, O. and Sironval, C., 1959. Photooxidation processes in normal green chlorella cells II. Effects on metabolism. Biochim. Biophys. Acta, 33: 207-215. Kok, B., 1956. On the inhibition of photosynthesis by intense light. Biochim. Biophys. Acta, 21: 234-244. Kok, B., Gassner, E.B. and Rursinski, H.J., 1965. Photoinhibition of chloroplast reactions. Photochem. Photobiol., 4: 215-227. Kyle, D.J., 1987. The biochemical basis for photoinhibition of photosystem II. In: D.J. Kyle, C.B. Osmond and C.J. Arntzen (Editors), Photoinhibition. Elsevier, Amsterdam, pp. 197-226. Lewis, M.R., Cullen, J.J. and Platt, T., 1984. Relationships between vertical mixing and photoadaptation of phytoplankton: similarity criteria. Mar. Ecol. Prog. Ser., 15: 141-149. Marra, J., 1978. Effect of short-term variations in light intensity on photosynthesis of a marine phytoplankter: a laboratory simulation study. Mar. Biol., 46: 191-202. Megard, R.O., Tonkyn, D.W. and Senft, W.H., 1984. Kinetics of oxygenic photosynthesis in planktonic algae. J. Plankton Res., 6: 325-337. Myers, J. and Burr, G.O., 1940. Studies on photosynthesis. Some effects of light of high intensity on chlorella. J. Gen. Physiol., 24: 45-67. Neale, P.J. and Marra, J., 1985. Short-term variation of Pma~ under natural irradiance conditions: a model and its implications. Mar. Ecol. Prog. Ser., 26: 113-124. Neale, P.J. and Richerson, P.J., 1987. Photoinhibition and diurnal variation of phytoplank-
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AND PHOTOINHIBIT1ON
133
ton photosynthesis - - I. Development of a photosynthesis irradiance model from studies of in situ responses. J. Plankton Res., 9: 167-193. Pahl-Wostl, C. and Imboden, D.M., 1990. D Y P H O R A - - a dynamic model for the rate of photosynthesis of algae. J. Plankton Res., 12: 1207-1221. Peeters, J.C.H. and Eilers, P., 1978. The relationship between light intensity and photosynthesis. A simple mathematical model. Hydrobiol. Bull., 12: 134-136. Platt, T. and Gallegos, C.L., 1981. Modelling primary production. In: P.G. Falkowski (Editor), Primary Productivity in the Sea. Plenum Press, New York, pp. 339-362. Platt, T., Gallegos, C.L. and Harrison, W.G., 1980. Photoinhibiton and photosynthesis in natural assemblages of marine phytoplankton. J. Mar. Res., 38: 687-701. Powles, S.B., 1984. Photoinhibition of photosynthesis by visible light. Annu. Rev. Plant Physiol., 35: 15-44. Radmer, R.J. and Kok, B., 1977. Light conversion efficiency in photosynthesis. In: A. Trebst and M. Avron (Editors), Encyclopedia of Plant Physiology: Photosynthesis I. Springer, New York, pp. 125-134. Raven, J.A. and Beardall, J., 1981. Respiration and photorespiration. In: T. Platt (Editor), Physiological Bases of Phytoplankton Ecology. Department of Fisheries and Oceans, Ottawa, pp. 55-82. Sayre, R.T. and Homann, P.H., 1979. A light-dependent oxygen consumption induced by photosystem II of isolated chloroplasts. Arch. Biochem. Biophys., 196: 525-533. Steeman-Nielsen, E., 1962. Inactivation of the photochemical mechanism in photosynthesis as a means to protect cells against too high light intensities. Physiol. Plant., 15: 161-171. Vincent, W.F., Neale, P.J. and Richerson, P.J., 1984. Photoinhibition: algal responses to bright light during diel stratification and mixing in a tropical alpine lake. J. Phycol., 20: 201-211. Vollenweider, R.A., 1965. Calculation models of photosynthesis - - depth curves and some implications regarding daily rate estimates in primary production measurements. Mem. Ist. Ital. Idrobiol., 18 Suppl.: 425-427.