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Frequency response analysis of oxygen evolution by algae
Journal
4 (1986)
of Biotechnology.
125-142
125
Elsevier JBT 00203
Frequency response analysis of oxygen evolution by algae H.L.Y. Lam * and H.R. Bungay Department
(Received
of Chemical
10 October
and Environmental Troy, NY
1985;
revised
Engineering, 12180, U.S.A.
21 January
Rensselaer
Polytechnic
1986; accepted
24 January
Institute.
1986)
Suinmary Common photosynthetic organisms were excited with steps, pulses, or sine waves of light as dissolved oxygen was measured with a microelectrode. Frequency response analysis revealed two fundamental time constants of 4 and 16 s. These time constants are assigned provisionally to processes for mass transfer and to the biochemical reactions of reduction of carbon dioxide. photosynthesis, oxygen evolution, algae, cyanobacteria, electrode, dissolved oxygen, control
time constants,
micro-
Introduction Microbial photosynthesis plays key roles in natural waters and in the biological treatment of wastes. Also of engineering importance is the formation in cooling towers of slimes that include photosynthetic organisms. Future production of fuels and chemicals may employ light-powered fermentation processes (Mitsui, 1976). Although the pathways of photosynthesis are reasonably well known, little of the fundamental information is useful for engineering applications. We are using engineering techniques for research on photosynthesis with the goal of uncovering rate coefficients applicable to analysis and design of practical devices. Engineering aspects of microbial photosynthesis are reviewed elsewhere (Lam et al., 1986). The shortest naturally occurring fluctuation in light takes about 0.1-l s; this is
* Present
address:
Pfizer
Central
Research,
Eastern
Point
Road,
Groton,
CT 06340,
U.S.A.
126
the time scale of the so-called ‘flicker’ effect due to the alternate focusing and defocusing of light by waves at the air-water interface (Dera and Gordon, 1968; Walsh and Legendre, 1983). Similar phenomena can arise from turbulence (Powell et al., 1965). Variations in light having periods up to many seconds are created by passing of clouds across the sun. Understanding the effects of intermittent illumination is crucial to the design and control of photobiological reactors. Efficiency of light utilization is markedly increased by exposing the cells to alternating periods of light and dark (Emerson and Arnold, 1932a.b; Rieke and Graffron, 1943; Weller and Franck, 1941). Marra (1978) reported increases of up to 87% in photosynthesis by simply modulating the light intensity of algal cells on a time scale ranging from minutes to hours. Seibert and Lavorel (1982) reported increases for periods’in the approximate range of 0.1-10 s. Walsh and Legendre (1982) found that fluctuating light could result in lower, equal, or higher photosynthetic capacity or efficiency than for a stable light regime, depending on the frequency of the fluctuation. Frechette and Legendre (1978) noted that adaptation to fluctuating bright light regimes resulted in a daily trend of increased efficiency. There are countless publications on the responses of oxygen exchange to a step change in light intensity. Responses with a time scale ranging from milliseconds to minutes have been reported. Unfortunately, most information obtained thus far has not been suitable for systems engineering. For a highly non-linear photosynthetic system, sinusoidal forcing develops periodic responses that are distorted sine waves. The gain and phase shift can be estimated from a reconstructed perfect sine wave based on the data. If the sine wave is very distorted, it is an indication that linearized analysis has questionable value. It is often possible to restrict the range of input so that departures from linearity in the response are not great, and the data are usable. Frequency response data can also be obtained from pulse-testing experiments by using Fourier transformations (Bergland, 1968). Clements and Snelle (1963) evaluated various pulse shapes and discussed criteria for evaluating experimental frequency response data obtained from pulse testing. Nonlinearities in biological systems have not been investigated thoroughly. There are a few attempts to define some special cases detected mainly in biomedical systems (Clynes, 1961, 1962, 1969; DeVoe, 1967). Identification of nonlinear behavior must often come from an examination of the response pattern rather from a consideration of the process itself. Nonlinear analysis is difficult, and the simplified linear approach used in our research provides a reasonable beginning for engineering of photosynthetic systems.
Materials and Methods In all experiments, the input signal was light intensity with variations that were step, pulse, or sinusoidal. The light originated from a Carl Zeiss Model 39-25-24 variable microscope lamp with the bulb mounted in a vented tube fitted with glass lenses to attenuate infrared radiation. Sinusoidal variation was caused by inserting
127
two polarizers in the light beam and rotating one of them. With a synchronous motor of constant speed of 135 rpm, pulleys and belts produced sine periods ranging from 1 to 500 s. A Lambda Model Ll-185A quantum meter adjacent to the microbial sample sent signals to a computer and to a strip-chart recorder. Oxygen microelectrodes were supplied by Dr. Nair of Louisiana State University, Dr. Lee of Drexel University, or Diamond Electra Inc. of Ann Arbor, Michigan. At the approximate breakdown voltage of oxygen of 0.6-0.8 V, current is directly proportional to the dissolved oxygen activity in the solution as measured with a Keithley Model 416 Picoammeter. Its rise time within the current ranges of lo-’ to 10-l” A is approximately 30 ms. The 2-j-pm tip diameter of the microprobe yields very fast response times of about 50 ms. The apparatus was placed in a foil-lined box on a stainless steel surface to minimize electrical noise. Connecting wires were firmly taped down to minimize movement that may induce sharp peaks in the current. Humidity had to be kept low in order to prevent erratic current leaks in the picoammeter. The electrode system was calibrated in 8% saline that was either aerated or stripped with nitrogen. Samples were cross-checked with a commercial dissolved oxygen meter (Yellow Springs Inst. Co.). Three experimental designs were used for oxygen measurements: agar flow chamber, perfusion cell, and suspension cell. Specimens of biofilms were cut to fit into the flow chamber and were surrounded by modeling clay to present a flat surface to the flow of nutrient medium. A micromanipulator positioned the oxygen microelectrode at selected locations in the biofilm. The perfusion cell had two small chambers separated by a membrane (Fig. 1). Algal cultures were centrifuged to give Oxyqen Microelectrode
Aq /AqCI Reference Electrode /
Cellophone Mernbrone
Screw
O-ring A
\Algoe
Gloss/ Cover Slide I Light Source
Fig.
1. Experimental
system
for algal film.
Film
128 oxygen nlcroelectrade
Lamp
Waler lor thermal C.SDrSCltl,
a Algal SUSDWWIOII
Fig. 2. Experimental
system
for algal suspension.
a dense suspension. About 0.5 ml of the suspension was introduced to the chamber above the membrane, and 20 n-tin of filtering formed a film of algae. The membrane (Nuclepore N-003-CPR-062-00) pore size of about 0.22 pm allows ions and molecules to pass from the lower chamber to the algae. The cell shown in Fig. 2 has one side of plexiglass containing a 22 mm diameter window closed by two pieces of thin polystyrene to form a closed chamber 4 mm thick. Algal suspensions of 10’ to lOa cells per ml were used. Holes allowed insertion of the microprobe, reference probe, and capillary tube required for gas sparging. The other chamber of the cell provides thermal mass with passage of constant temperature water. The two main parts of the cell are clamped together. TABLE
1
Medium
for culturing
Cldorellu
uulgaris
*
Component
gl-’
MgSO, .7 H ,O KH2P0., GlCl~ KNO, EDTA Fe(SO,).7 H,O
1.oo 1.25 0.0835 1.25 0.5 0.0498 0.1142 0.0144 0.0882 0.0071 0.0157 0.0049
HIBOX
MnCI,.4 H,O ZnSO,.‘l H,O MOO, (85%) CuSO, .5 H 2O Co(NO,),.6 H,O * pH of the medium
is 6.8.
129
Suspensions of Chlorella uulgaris were grown at 25°C from inoculum supplied by Carolina Biological Supply. Culture medium is listed in Table 1. The cultures were illuminated by a 75 W Genera1 Electric Gro & Show lamp operated by a timer to produce cycles of 12 h light (375 PE me2 s-’ or 2000 footcandles) and 12 h dark. Anacystis nidulans from the Culture Collection of Algae at the University of Texas (Starr, 1978) was grown in batch culture at 37°C. A Scenedesmus culture (supplied by Dr. Rhee at the New York State Health Department) was grown in a defined inorganic medium (Rhee, 1974). The cultures were illuminated by circular fluorescent light bulbs (approximately 94 PE mm2 s-‘) and exposed to a 12 h light and 12 h dark cycle. All experiments were performed with young cells harvested about 3-6 h after the start of the light period. Cell counts were performed with a microscope and counting chamber. Previous experiments in our laboratory (Cullotta, 1983) used specific inhibitors for photosynthesis to show that the oxygen responses are not spurious artifacts of these experimental conditions. Results The response of a Chorella biofilm to step change in light intensity from darkness to about 1000 footcandles (187 PE m-’ s-‘) was essentially complete in about 1 1 mg I -’
Permd T (5)
I 144
-
303
/\
Fig. refers
3. Tracings 10 period
of typical of oscillation
responses
of dissolved
in seconds.
oxygen
to sinusoidal
forcing
of illumination.
Number
130
Frequency,w 0
001
(rod 01
Frequency,
s 1, 001
10
w (rod
s-1)
01
1.0
0
-10
-20
-30
0
.
b -,a0 x 2a -270
Fig. 4 (left).
Fig. 5 (right).
Bode
diagram
Bode diagram
of oxygen
response
for oxygen
lo,_
response
100 pE m-2
Fig. 6. Steady
state
characteristics
for Chlorello
for stream
150 s-1
of Scetredesmus.
film.
biofilm.
I
I I I lllll
I
I llllln
131
min. A biofilm left in the dark for more than 30 min had an induction period ranging from seconds to 5 min, and it was difficult to get reproducible results. However, step responses were useful in selecting the frequency range for the sinusoidal experiments. At each frequency, the light intensity was varied sinusoidally from 10 to 187 PE mm2 s- ‘. Although the typical responses shown in Fig. 3 are not perfect sine waves, gain and phase shift can be measured with reasonable confidence to get the Bode diagram shown in Fig. 4. The approach of phase angle towards 180” suggests a second-order system or two first-order systems in series. Corner frequencies of 0.3 and 0.07 rad s-’ were obtained by fitting asymptotes. The corresponding time constants are 3.3. and 14.3 s. Similar sinusoidal forcing experiments with stream biofilms obtained from Frear Park, Troy, New York gave the Bode diagram shown in Fig. 5. The corner frequencies are 0.3 and 0.06 rad s-l, with time constants of 3.3 and 16.7 s. respectively. Laboratory biofilms and stream biofilms have non-uniform characteristics. Moreover, growth is slow, and it is difficult to control the conditions. Suspended cultures were filtered to form mats that were placed in the perfusion cell. A set of step responses of Scenedesmus biofilm to different light intensities was obtained. Various wavelengths produced by inserting optical filters caused no qualitative differences, but the magnitudes of the responses relate to the light intensity. Figure 6 shows the dependence of steady-state oxygen evolution rate on
Frequency.
w
(rad
s-1)
a -101-H-
a
001
Fig. 7. Bode diagram
01
of oxygen
10
response
for Scenedesmus
film.
132
Light oil
I
lght on
Fig. 8. Step response
of Scenedemur
suspension
to light
intensity
as measured
by an oxygraph.
the light intensity. By working with a restricted range of light intensities, nonlinearity as in Fig. 6 did not distort our sinusoidal responses. A Bode diagram for the Scenedesnw film is shown in Fig. 7. The approach of phase angle towards 180’ again suggests a second-order system or two first-order systems in series with time constants of 3.3 and 16.7 s. Pulse data were also analyzed with a computer program that develops frequency responses. For three different light intensities, there was no significant difference in the time constants (18.5 and 3.8 s). At high frequencies, the Bode diagrams showed erratic behavior characteristic of this Fourier transform technique. Preliminary runs with a completely closed system (Gilson Model 5/6 oxygraph) showed that the oxygen evolution rate (slope of the trace in Fig. 8) responded with a delay of only a few seconds to a step change in light intensity. Comparison with our data for open systems suggests that the longer time constant was a result of the diffusional oxygen exchange process. When the light was left on for a long period of time (more than 10 min) the oxygen evolution rate decreased gradually. This can be explained by the gradual accumulation of oxygen and depletion of carbon dioxide, conditions that favor photorespiration. The effect of photorespiration is not very important under a low light intensity environment. Development of a control model A block diagram for the control scheme of photosynthesis is postulated (Fig. 9). It cannot be emphasized too strongly that this model features the bottleneck steps
133
02
reducllon (F) flUOreSCenCe
(to H' 1 light
.
04)
1‘
NAOPH
DhOlOsystems 1
cheln 1
e-
n* rlu0reSceOce (F)
+
Fig.
T
summlng
IunctIon
Oxygen production
4 . I?-
electron
9. Block
diagram
I--
or uptake
DrOduClIOn
rate
or uptake
rate
of photosynthesis.
while lumping the fast steps that have little effect on timing. It is a model of control, not a model of the fine details of photosynthesis. Simple models that make use of frequency response analysis in parameter estimations provide a semi-empirical
4 k8
. e-
+ u
Fig. 10. Structure
diagram
of photosynthesis.
l w
T,:
l/k2
GZ(s) : k4 G3(s)
q
K3 z
K3
‘TX5 + I
GS(s) : __
KS
Tgs
Fig.
11. Block
l
h7k4(Ro-Rs)
K5 I I/k9
I
TJ =
k4 Is + k6 ICO21
k4 Is + k6 ICO21
Tgz
l/k9
I
diagram
of transfer
functions.
treatment of the effects of light fluctuations in photosynthesis in a ‘biological’ rather than ‘biochemical’ approach. The block diagram is translated into a structure diagram, i.e., it is described mathematically. For simulations the structure diagram is expressed as a system of differential equations and transfer functions. For the synthesis of the structure diagram, numerous assumptions and simplification have to be made. They are: (i) All chemical reactions are considered to be first order. At constant enzyme concentration and at low substrate concentrations, the Michaelis-Menten equation gives rates that are about proportional to substrate concentration. (ii) Sequences of fast reactions that have minor importance for the timing of the overall process are lumped. (iii) All reactions are assumed irreversible. (iv) The only input variables are light intensity and the ambient CO, concentration. Based on the simplifying assumptions, a structure diagram is shown in Fig. 10. The diagram is then rearranged and grouped into five major blocks representing five subsystems (Fig. 11). Each of the subsystems will be analysed individually. Photosynthetic oxygen production (G,) Photosynthetic oxygen production occurs primarily in Photosystem II (PS II). Joliot (1966) proposed that a thermal reaction occurring between the photochemical act and the formation of oxygen is limiting. Assuming that the rate of oxygen production follows a first-order dependency with respect to an intermediate X*, i.e., P = d02/dt
= k,X*
(1)
135
where P is the production rate and k, is the first-order generation of X* follows the photochemical reaction: hv+X-+X*+X
kinetic constant. The
k2
and the rate equation for X is, dX*/dr
= I q(X,
- X*)
- k2X*
(2)
where I is the absorbed light intensity and q is the intrinsic quantum yield. The system is non-linear because of the second-order nature of the formation reaction. If the concentration of X* may be neglected in comparison to X,, (a condition which holds at low intensities, that is, when (I q) -=zk2), the linearized equation is: dX*/dt
= I q X, - k,X*
(3) Laplace transformed with respect to f, treating X* and I as deviation variables, and rearranging, one obtains:
(s + k,)X*(s)
= q X,,l(s)
(4)
= q X,,/(s + 4)
(5)
(4 X,/k,) = (l,k,)s + l
(6)
and thus, X*(s)/Z(s) or, x*(s)/l(s)
Similarly, P(s)/l(s)
= k,X*(s)/l(s)
= G,(s)
(7)
Hence,
G1(s) = ,1,;,;; + 1
(8)
The value of k, is about 680 SC’ (Joliot, 1966; Joliot et al., 1966). In the time scale of the frequency response experiment, G,(s) is approximately equal to q X, or a constant k,, and the oxygen production rate responds almost instantaneously to changes of light intensity. Electron
transport
system (G,)
Electron transfers occur at a time scale ranging from microseconds to milliseconds. The response of the rate of production of reduced electron carrier (A-) is almost instantaneous in comparison to other processes. For the purpose of this study, a constant gain of k, is assumed. NADPH
and CO, fixation
(G,)
In a simple model, the following assumptions are made: (1) The light flux I interacts with NADP+ according to: NADP+ + I + NADPH
136
to produce the reducing power NADPH. The rate of formation of NADPH is further assumed to be proportional to the product of I and [NADP+], with rate constant k,. (2) NADPH simply reacts with CO, to give sugar (CH,O),, and regenerates NADP+: NADPH
+ CO, + NADP+ + (CH,O),,
The rate of utilization of NADPH is assumed to be proportional to the product of INADPH] and [CO,] with rate constant k,. (3) The dynamics of regeneration of the electron carrier A, and thus the dynamics of the light reactions are determined by the dynamics of the concentration of NADP+ and NADPH. Hence the rate of removal of A- is assumed to be proportional to the rate of utilization of NADPH. From these assumptions, the following equations can be written: d[NADPH]/dl= -d[A-]/dt
k,l[NADP+]
- k,[COz][NADPH]
(9)
= k,k,[COz][NADPH]
(10)
where k, is a proportionality constant. It is further assumed that there is a maximum total concentration of NADP+ and NADPH, R,, and that it is constant for light intensities below saturation, i.e., R, = [NADP+] Introducing
+ [NADPH]
(11)
the notations R = [NADPH],
Eqns. (9) and (10) can be rewritten as:
dR/dt=k,(R,-R)I-k,[CO,]R
(12)
-AA-
(13)
= dr4-/dt
= k,k,[CO>]R
Eqn. (12) can be linearized by assuming that [CO,] remains relatively constant throughout the experiment. Define deviation variables _I and & as: _r=I--I,
(14)
and &=R-R,
05)
where R, and Z, are steady-state values. Substituting simplifying gives, d&/dt
= k,R,_I+k,R,Is
Applying
k,R,_I-
I and R in Eqn. (12) and
k41s& -k,[CO,]R
- k,R,I,
- k,[CO,]R,
- k,RI
(16)
steady-state conditions, and neglecting the B _I term,
d&/dr=k,(R,-R,)_I-
(k,ls+k,[C02])~
(17)
Taking Laplace transforms and rearranging gives, R(s)/l(s)
= k,(R,
- R,>/(s
+ (Ws
+ k&Ozl))
(18)
137
Using Eqn. (13), -AA-(s)/l(s)
=
k,k,(& - RSMWS+ kmZ1) = G Cs) S/(WS+ k[COZl)+ 1 3
(19)
It is interesting to note that the time constant of this block depends on both the steady-state light intensity and the CO, concentration. Direct oxygen reduction (G4) The photosynthetic systems have mechanisms to prevent over-reduction. Furthermore, excessive oxygen can be toxic to the system because of production of highly active superoxides. The direct reduction of oxygen by reduced electron carriers thus has dual purposes in protecting the system. Although the detailed mechanisms are not well defined, it can be assumed that the reactions must follow closely the change in environmental conditions. Hence it is assumed that the rate of oxygen reduction is proportional to the rate of over-reduction or rate of production of excess reduced electron carrier, i.e., net AA-. There is also the regulatory mechanism of photorespiration that helps to remove excess oxygen and produce carbon dioxide. Dissolved oxygen concentration and photorespiration The process by which oxygen is exchanged between the atmosphere and water is essentially diffusion across a thin water film. For a suspension system, oxygen mass balance gives, dC/dt=P-R+K(C,-C)-k,C
(20)
where C = dissolved oxygen concentration; P = net oxygen evolution rate; R = respiration rate; K = mass transfer coefficient; Cl = saturated oxygen concentration; k, = rate constant for photorespiration. Here the photorespiration rate is assumed proportional to the dissolved oxygen concentration. Assuming that R is constant throughout the experiment, and defining the deviation variables as follows: _c = c - c,,
(21)
f=(P-R)-(P,,-R)=P-P,,
(22)
Substituting the deviation variables into Eqn. (20) and simplifying: d_C/dt=_P-KC-k,g Laplace transforming K(s)
= P(s) -KC(s)
(23) both sides of the equation gives, - k,C(s)
(24)
rearranging: C(s)/(s)
= l/( s + K + k,)
or
l/(K+ k,) G,(s) = s/(K+k,)+l
(25)
13s
I
+
GI - [(G2-G3)G41
NET
)
1021
G5
62 G, _ ,(G2-G3)G41
where
= (Kl-kBW$
K3
2
It8 (k4-
K3)
k?k4(RwRs) k4
G5.
; KI-
IS
l
It6 ICO21
k5
~
TgS*
Fig. 12. Overall
I
block
diagram.
For biofilms from nature, the mass transfer coefficient, K, cannot be assumed constant throughout the film, as the concentration gradient within the film is substantial. Using a quadratic approximation of the concentration profile, the constant K can be approximated as follows: K=
-2D/(x’-
xx,,)
(27)
where D = diffusion coefficient; x = depth of the measuring point from the surface; x0 = total thickness of the film. Overall model The five subsystems (G, to G,) can further be grouped into two blocks. The first block represents the internal processes giving rise to the net oxygen evolution rate. The transfer function of the first block (Fig. 12) indicates a first-order lag and a first-order lead. However, if k, can be assumed to be equal to k,k,, a first-order lag results. This implies that when the CO, fixation is inoperative, the regulatory mechanisms ensure that the system is properly balanced and that the net oxygen evolution is zero. This is supported by the well-known induction phenomena in which the oxygen evolution rate may remain zero for periods ranging from seconds to minutes. There also exist some reports of an oxygen gush after long dark periods and special conditions. In most cases, the magnitude of the gush is very small; and can be explained by the leakages in the regulatory mechanisms. For the purpose of this study, the transfer function for the internal processes can be assumed to be a first order lag. This implies that the response of the net oxygen evolution rate is controlled by the rate of the carbon fixation cycle. The second block represents the oxygen exchange with the atmosphere and the effects of photorespiration. The transfer function is again a simple first order lag. This implies that the responses of dissolved oxygen are controlled by the configuration of the system and the extent of photorespiration.
139
Discussion From the results presented earlier, it is clear that there are two limiting processes contributing to the response of dissolved oxygen to light intensity. The two processes can be approximated as two first order processes with time constants of about 4 and 16 s. The control model also approximates the overall process as two first order processes in series. The remaining problem lies in the assignment of experimental time constants to the two processes. It is postulated that at low frequencies of light modulation, the principal limiting process is the oxygen exchange between the control volume close to the electrode and the surroundings (G, in Fig. 12). From the Bode plots, the time constant is about 16 s. When a partially closed system was used, a longer time constant of 58 s was obtained at low frequency, while the time constant at high frequency remained at about 5 s. Further evidence was obtained with a completely closed system in which the 0, evolution rate (slope of the trace in Fig. 8) responded rapidly to a steady value after a step change in light intensity. The rate decreased gradually only after more than 10 min indicating that the longer time constant was approaching infinity. The rate decrease is probably a result of photorespiration favored by accumulation of 0, and depletion of CO,. At higher frequencies of light modulation, the intracellular processes are limiting. The overall response of overall oxygen evolution rate is controlled by the rate of CO, reduction and it can be approximated as a first-order process. From the Bode plots, the time constant is about 3-5 s. The major drawback in this research was the inability to maintain truly constant concentrations of carbon dioxide. Moreover, there was no direct measure of CO, concentration. Even if the CO, concentration were measured by some means, there is still debate about the forms of CO, taken up by the cells. The CO, limitation factor has been overlooked by many workers in measuring oxygen evolution, and reported rates of oxygen evolution vary widely. Similar work has been done with pH and fluorescence responses to changes in light intensity. We were able to obtain sinusoidal fluorescence response when the photosynthetic system was subjected to sinusoidal light input. Although a time constant of 0.5 s was obtained, the value is too close to the response time of the measuring system. A faster response measuring system is required to give more meaningful estimation of time constants. Our pH responses are much slower than others reported in the literature. In most of the previous studies, chloroplasts with broken envelopes were used. Moreover, the response is complicated by the effects of the buffering capacity of the suspending solution. The work presented here illustrates ways in which the theory of linear system can be used to analyze the behavior of photosynthetic systems subject to time-varying light intensities. It is emphasized that the analysis of biological processes as linear systems is only a mathematical approximation. The value of the technique, as applied in this work, lies in its ability to suggest possible mathematical expressions to account for the observed time-dependent behavior, and also to relate to hypotheses about underlying mechanisms.
140
Conclusion Much of the research on photosynthesis addressed the fine details of pathways and of energy transfer. Although there are significant attempts to improve productivity of algal systems and to construct large, practical growth units, there are little data of use to engineers. In the work presented here, system engineering techniques have been applied to study the dynamic responses of photosynthesis to environmental changes. When light variations are not so extreme that nonlinearities cause distortions, the magnitude of oxygen response is sufficient for Bode diagram analysis over a range of frequencies of practical interest. Two time constants of 4 and 16 s have been determined by frequency response analysis with Bode diagrams that are in accord with two first-order systems in series. The proposed control model of algal photosynthesis allows fresh insights into photosynthesis and should be of value to engineers working with photosynthetic systems.
Acknowledgements Advice and encouragement from Dr. G.-Yull Rhee of the New York State Department of Health and from Dr. L.S. Clesceri were essential to this research. Financial support came from National Science Foundation Grant No. CPE82-17676.
Symbols
N-1 AAAAc c C(s) c, :o,l D Gi (S) hv I Is _I I(s) ki kr K
[NADP+]
Concentration of reduced electron carrier A Rate of change of [A-] Laplace transform of AADissolved oxygen concentration Deviation variable of C Laplace transform of _C Saturated dissolved oxygen concentration Steady-state of C Intracellular carbon dioxide concentration Diffusion coefficient Transfer functions One quantum Absorbed light intensity Steady-state of I Deviation variable of I Laplace transform of _I Kinetic constants Rate constant for photorespiration Mass transfer coefficient Concentration of nicotinamide-adenine dinucleotide phosphate
141
[NADPH]
iO,l
P E P(s) pss
i B R(s) Rc Rs s t X X0
WI KJ IX”1 x*w
Concentration of reduced NADP Intracellular oxygen concentration Photosynthetic oxygen production rate Deviation variable of P Laplace transform of _P Steady-state of P Intrinsic quantum yield [NADPH] Deviation variable of R Laplace transform of & Cellular respiration rate Steady-state of R A variable defined in the complex plane Independent time variable Depth of the measuring point from the surface Total thickness of film Concentration of an intermediate in PS II in normal state Total concentration of X and X* Concentration of an intermediate in PS II in excited state Laplace transform of [X*]
References Bergland, G.G. (1968) Fast Fourier transform for real-value series. Commun. ACM, ll(lO), 370-379. Clements, W.C. and Snelle, K.B. (1963) Pulse testing for dynamic analysis. Ind. Eng. Chem. Proc. Design Dev. 2, 94-102. Clynes, M. (1961) Unidirectional rate sensitivity: A biocybernetic Law of Reflex and Humoral Systems as Physiologic Channels of Control and Communication. Ann. N.Y. Acad. Sci. 92, 946-969. Clynes, M. (1962) The Non-Linear Biological Dynamics of Unidirectional Rate Sensitivity Illustrated by Analog Computer Analysis. Ann. N.Y. Acad. Sci. 112. 468-508. Clynes, M. (1969) Rein Control, or Unidirectional Rate Sensitivity, Fundamental Dynamic and Organizing Function in Biology. Ann. N.Y. Acad. Sci. 156, 627-668. Cullotta, L.G. (1983) Rate Limiting Steps in Microbial Photosynthesis. M.S. Thesis, Rensselaer Polytechnic Institute, Troy, NY. Dera, J. and Gordon, H.R. (1968) Light Field Fluctuations in the Photic Zone. Limnol. Oceanogr. 13. 697-707. DeVoe, R.D. (1967) Nonlinear Transient Responses from Light-Adapted Wolf Spider Eyes to Changes in Background Illumination. J. Gen. Physiol. 50, 961-991. Emerson, R. and Arnold, W. (1932a) A Separation of the Reactions in Photosynthesis by Means of Intermittent Light. J. Gen. Physiol. 15, 391-420. Emerson, R. and Arnold, W. (1932b) The Photochemical Reaction in Photosynthesis. J. Gen. Physiol. 16, 191-205. Frechette, M. and Legendre, L. (1978) Photoplankton Photosynthesis Response to Simple Stimulus Imitating Rapid Light Fluctuations Induced by Sea Waves. J. Exp. Mar. Biol. Ecol. 32, 15-20. Joliot, P. (1966) Oxygen Evolution in Algae Illuminated by Modulated Light. Brookhaven Symp. Biol. 19, 418-430. Joliot, P., Hoffnung, M. and Chabaud, R. (1966) Oxygen Release by Algae Maintained under Sinusoidally Modulated Light. J. Chim. Phys. 63(10), 1423-1442.
142 Lam.
H.L.Y., Bungay, H.R. and Culotta, L.C. (1986) An Engineer Looks at Photosynthesis. Appl. Biochem. Biotechnol.. in press. Marra. J. (1978) Phytoplankton Photosynthetic Response to Vertical Movement in a Mixed Layer. Mar. Biol. 46(3), 203-208. Mitsui. A. (1976) Long Range Concepts: Applications of Photosynthetic Hydrogen Production and Nitrogen Fixation Research, Paper presented in the Conference on Capturing the Sun Through Bioconversion. School of Marine and Atmospheric Sciences, University of Miami, Miami, Florida. Powell, C.K.. Chaddock, J.B. and Dixon, J.R. (1965) Motion of Algae in Turbulent Flow. Biotechnol. Bioeng. 7, 295-308. Rhee, G.-Y. (1974) Phosphate Uptake under Nitrate Limitation by Scenedesmus Species and Its Ecological Implications. J. Phycol. 10(4), 470. Ried, A. (1969) Studies on Light-Dark Transients in Chlorella. In: Progress in Photosynthetic Research, Vol. 1 (H. Metzner, ed,), pp. 512-530. Tubingen. Rieke, F.F. and Graffron, H. (1943) Flash Saturation and Reaction Periods in Photosynthesis. J. Phys. Chem. 47. 299-308. Seibert, M. and Lavorel, J. (1982) Oxygen-Evolution Patterns from Oxygen-Evolving Photosystem 11 Particles. In: SERI Biomass Program Principal Investigators Review Meeting. SERI. Washington, D.C.. p. 17. Starr, R.C. (1978) The Culture Collection of Algae at the University of Texas at Austin. J. Phycol. 14, 770-172. Walsh, P. and Legendre, L. (1982) Effets des Fluctuations Rapides de la Lumiere Sur la Photosynthese du Phytoplankton. J. Plankton Res. 4, 313-319. Walsh, P. and Legendre, L. (1983) Photosynthesis of Natural Phytoplankton under High Frequency Light Fluctuations Simulating those Induced by Sea Surface Waves. Limnol. Oceanogr. 28. 688-695. Weller, S. and Franck, J. (1941) Photosynthesis in Flashing Light. J. Phys. Chem. 45, 1359-1373.
4 (1986)
of Biotechnology.
125-142
125
Elsevier JBT 00203
Frequency response analysis of oxygen evolution by algae H.L.Y. Lam * and H.R. Bungay Department
(Received
of Chemical
10 October
and Environmental Troy, NY
1985;
revised
Engineering, 12180, U.S.A.
21 January
Rensselaer
Polytechnic
1986; accepted
24 January
Institute.
1986)
Suinmary Common photosynthetic organisms were excited with steps, pulses, or sine waves of light as dissolved oxygen was measured with a microelectrode. Frequency response analysis revealed two fundamental time constants of 4 and 16 s. These time constants are assigned provisionally to processes for mass transfer and to the biochemical reactions of reduction of carbon dioxide. photosynthesis, oxygen evolution, algae, cyanobacteria, electrode, dissolved oxygen, control
time constants,
micro-
Introduction Microbial photosynthesis plays key roles in natural waters and in the biological treatment of wastes. Also of engineering importance is the formation in cooling towers of slimes that include photosynthetic organisms. Future production of fuels and chemicals may employ light-powered fermentation processes (Mitsui, 1976). Although the pathways of photosynthesis are reasonably well known, little of the fundamental information is useful for engineering applications. We are using engineering techniques for research on photosynthesis with the goal of uncovering rate coefficients applicable to analysis and design of practical devices. Engineering aspects of microbial photosynthesis are reviewed elsewhere (Lam et al., 1986). The shortest naturally occurring fluctuation in light takes about 0.1-l s; this is
* Present
address:
Pfizer
Central
Research,
Eastern
Point
Road,
Groton,
CT 06340,
U.S.A.
126
the time scale of the so-called ‘flicker’ effect due to the alternate focusing and defocusing of light by waves at the air-water interface (Dera and Gordon, 1968; Walsh and Legendre, 1983). Similar phenomena can arise from turbulence (Powell et al., 1965). Variations in light having periods up to many seconds are created by passing of clouds across the sun. Understanding the effects of intermittent illumination is crucial to the design and control of photobiological reactors. Efficiency of light utilization is markedly increased by exposing the cells to alternating periods of light and dark (Emerson and Arnold, 1932a.b; Rieke and Graffron, 1943; Weller and Franck, 1941). Marra (1978) reported increases of up to 87% in photosynthesis by simply modulating the light intensity of algal cells on a time scale ranging from minutes to hours. Seibert and Lavorel (1982) reported increases for periods’in the approximate range of 0.1-10 s. Walsh and Legendre (1982) found that fluctuating light could result in lower, equal, or higher photosynthetic capacity or efficiency than for a stable light regime, depending on the frequency of the fluctuation. Frechette and Legendre (1978) noted that adaptation to fluctuating bright light regimes resulted in a daily trend of increased efficiency. There are countless publications on the responses of oxygen exchange to a step change in light intensity. Responses with a time scale ranging from milliseconds to minutes have been reported. Unfortunately, most information obtained thus far has not been suitable for systems engineering. For a highly non-linear photosynthetic system, sinusoidal forcing develops periodic responses that are distorted sine waves. The gain and phase shift can be estimated from a reconstructed perfect sine wave based on the data. If the sine wave is very distorted, it is an indication that linearized analysis has questionable value. It is often possible to restrict the range of input so that departures from linearity in the response are not great, and the data are usable. Frequency response data can also be obtained from pulse-testing experiments by using Fourier transformations (Bergland, 1968). Clements and Snelle (1963) evaluated various pulse shapes and discussed criteria for evaluating experimental frequency response data obtained from pulse testing. Nonlinearities in biological systems have not been investigated thoroughly. There are a few attempts to define some special cases detected mainly in biomedical systems (Clynes, 1961, 1962, 1969; DeVoe, 1967). Identification of nonlinear behavior must often come from an examination of the response pattern rather from a consideration of the process itself. Nonlinear analysis is difficult, and the simplified linear approach used in our research provides a reasonable beginning for engineering of photosynthetic systems.
Materials and Methods In all experiments, the input signal was light intensity with variations that were step, pulse, or sinusoidal. The light originated from a Carl Zeiss Model 39-25-24 variable microscope lamp with the bulb mounted in a vented tube fitted with glass lenses to attenuate infrared radiation. Sinusoidal variation was caused by inserting
127
two polarizers in the light beam and rotating one of them. With a synchronous motor of constant speed of 135 rpm, pulleys and belts produced sine periods ranging from 1 to 500 s. A Lambda Model Ll-185A quantum meter adjacent to the microbial sample sent signals to a computer and to a strip-chart recorder. Oxygen microelectrodes were supplied by Dr. Nair of Louisiana State University, Dr. Lee of Drexel University, or Diamond Electra Inc. of Ann Arbor, Michigan. At the approximate breakdown voltage of oxygen of 0.6-0.8 V, current is directly proportional to the dissolved oxygen activity in the solution as measured with a Keithley Model 416 Picoammeter. Its rise time within the current ranges of lo-’ to 10-l” A is approximately 30 ms. The 2-j-pm tip diameter of the microprobe yields very fast response times of about 50 ms. The apparatus was placed in a foil-lined box on a stainless steel surface to minimize electrical noise. Connecting wires were firmly taped down to minimize movement that may induce sharp peaks in the current. Humidity had to be kept low in order to prevent erratic current leaks in the picoammeter. The electrode system was calibrated in 8% saline that was either aerated or stripped with nitrogen. Samples were cross-checked with a commercial dissolved oxygen meter (Yellow Springs Inst. Co.). Three experimental designs were used for oxygen measurements: agar flow chamber, perfusion cell, and suspension cell. Specimens of biofilms were cut to fit into the flow chamber and were surrounded by modeling clay to present a flat surface to the flow of nutrient medium. A micromanipulator positioned the oxygen microelectrode at selected locations in the biofilm. The perfusion cell had two small chambers separated by a membrane (Fig. 1). Algal cultures were centrifuged to give Oxyqen Microelectrode
Aq /AqCI Reference Electrode /
Cellophone Mernbrone
Screw
O-ring A
\Algoe
Gloss/ Cover Slide I Light Source
Fig.
1. Experimental
system
for algal film.
Film
128 oxygen nlcroelectrade
Lamp
Waler lor thermal C.SDrSCltl,
a Algal SUSDWWIOII
Fig. 2. Experimental
system
for algal suspension.
a dense suspension. About 0.5 ml of the suspension was introduced to the chamber above the membrane, and 20 n-tin of filtering formed a film of algae. The membrane (Nuclepore N-003-CPR-062-00) pore size of about 0.22 pm allows ions and molecules to pass from the lower chamber to the algae. The cell shown in Fig. 2 has one side of plexiglass containing a 22 mm diameter window closed by two pieces of thin polystyrene to form a closed chamber 4 mm thick. Algal suspensions of 10’ to lOa cells per ml were used. Holes allowed insertion of the microprobe, reference probe, and capillary tube required for gas sparging. The other chamber of the cell provides thermal mass with passage of constant temperature water. The two main parts of the cell are clamped together. TABLE
1
Medium
for culturing
Cldorellu
uulgaris
*
Component
gl-’
MgSO, .7 H ,O KH2P0., GlCl~ KNO, EDTA Fe(SO,).7 H,O
1.oo 1.25 0.0835 1.25 0.5 0.0498 0.1142 0.0144 0.0882 0.0071 0.0157 0.0049
HIBOX
MnCI,.4 H,O ZnSO,.‘l H,O MOO, (85%) CuSO, .5 H 2O Co(NO,),.6 H,O * pH of the medium
is 6.8.
129
Suspensions of Chlorella uulgaris were grown at 25°C from inoculum supplied by Carolina Biological Supply. Culture medium is listed in Table 1. The cultures were illuminated by a 75 W Genera1 Electric Gro & Show lamp operated by a timer to produce cycles of 12 h light (375 PE me2 s-’ or 2000 footcandles) and 12 h dark. Anacystis nidulans from the Culture Collection of Algae at the University of Texas (Starr, 1978) was grown in batch culture at 37°C. A Scenedesmus culture (supplied by Dr. Rhee at the New York State Health Department) was grown in a defined inorganic medium (Rhee, 1974). The cultures were illuminated by circular fluorescent light bulbs (approximately 94 PE mm2 s-‘) and exposed to a 12 h light and 12 h dark cycle. All experiments were performed with young cells harvested about 3-6 h after the start of the light period. Cell counts were performed with a microscope and counting chamber. Previous experiments in our laboratory (Cullotta, 1983) used specific inhibitors for photosynthesis to show that the oxygen responses are not spurious artifacts of these experimental conditions. Results The response of a Chorella biofilm to step change in light intensity from darkness to about 1000 footcandles (187 PE m-’ s-‘) was essentially complete in about 1 1 mg I -’
Permd T (5)
I 144
-
303
/\
Fig. refers
3. Tracings 10 period
of typical of oscillation
responses
of dissolved
in seconds.
oxygen
to sinusoidal
forcing
of illumination.
Number
130
Frequency,w 0
001
(rod 01
Frequency,
s 1, 001
10
w (rod
s-1)
01
1.0
0
-10
-20
-30
0
.
b -,a0 x 2a -270
Fig. 4 (left).
Fig. 5 (right).
Bode
diagram
Bode diagram
of oxygen
response
for oxygen
lo,_
response
100 pE m-2
Fig. 6. Steady
state
characteristics
for Chlorello
for stream
150 s-1
of Scetredesmus.
film.
biofilm.
I
I I I lllll
I
I llllln
131
min. A biofilm left in the dark for more than 30 min had an induction period ranging from seconds to 5 min, and it was difficult to get reproducible results. However, step responses were useful in selecting the frequency range for the sinusoidal experiments. At each frequency, the light intensity was varied sinusoidally from 10 to 187 PE mm2 s- ‘. Although the typical responses shown in Fig. 3 are not perfect sine waves, gain and phase shift can be measured with reasonable confidence to get the Bode diagram shown in Fig. 4. The approach of phase angle towards 180” suggests a second-order system or two first-order systems in series. Corner frequencies of 0.3 and 0.07 rad s-’ were obtained by fitting asymptotes. The corresponding time constants are 3.3. and 14.3 s. Similar sinusoidal forcing experiments with stream biofilms obtained from Frear Park, Troy, New York gave the Bode diagram shown in Fig. 5. The corner frequencies are 0.3 and 0.06 rad s-l, with time constants of 3.3 and 16.7 s. respectively. Laboratory biofilms and stream biofilms have non-uniform characteristics. Moreover, growth is slow, and it is difficult to control the conditions. Suspended cultures were filtered to form mats that were placed in the perfusion cell. A set of step responses of Scenedesmus biofilm to different light intensities was obtained. Various wavelengths produced by inserting optical filters caused no qualitative differences, but the magnitudes of the responses relate to the light intensity. Figure 6 shows the dependence of steady-state oxygen evolution rate on
Frequency.
w
(rad
s-1)
a -101-H-
a
001
Fig. 7. Bode diagram
01
of oxygen
10
response
for Scenedesmus
film.
132
Light oil
I
lght on
Fig. 8. Step response
of Scenedemur
suspension
to light
intensity
as measured
by an oxygraph.
the light intensity. By working with a restricted range of light intensities, nonlinearity as in Fig. 6 did not distort our sinusoidal responses. A Bode diagram for the Scenedesnw film is shown in Fig. 7. The approach of phase angle towards 180’ again suggests a second-order system or two first-order systems in series with time constants of 3.3 and 16.7 s. Pulse data were also analyzed with a computer program that develops frequency responses. For three different light intensities, there was no significant difference in the time constants (18.5 and 3.8 s). At high frequencies, the Bode diagrams showed erratic behavior characteristic of this Fourier transform technique. Preliminary runs with a completely closed system (Gilson Model 5/6 oxygraph) showed that the oxygen evolution rate (slope of the trace in Fig. 8) responded with a delay of only a few seconds to a step change in light intensity. Comparison with our data for open systems suggests that the longer time constant was a result of the diffusional oxygen exchange process. When the light was left on for a long period of time (more than 10 min) the oxygen evolution rate decreased gradually. This can be explained by the gradual accumulation of oxygen and depletion of carbon dioxide, conditions that favor photorespiration. The effect of photorespiration is not very important under a low light intensity environment. Development of a control model A block diagram for the control scheme of photosynthesis is postulated (Fig. 9). It cannot be emphasized too strongly that this model features the bottleneck steps
133
02
reducllon (F) flUOreSCenCe
(to H' 1 light
.
04)
1‘
NAOPH
DhOlOsystems 1
cheln 1
e-
n* rlu0reSceOce (F)
+
Fig.
T
summlng
IunctIon
Oxygen production
4 . I?-
electron
9. Block
diagram
I--
or uptake
DrOduClIOn
rate
or uptake
rate
of photosynthesis.
while lumping the fast steps that have little effect on timing. It is a model of control, not a model of the fine details of photosynthesis. Simple models that make use of frequency response analysis in parameter estimations provide a semi-empirical
4 k8
. e-
+ u
Fig. 10. Structure
diagram
of photosynthesis.
l w
T,:
l/k2
GZ(s) : k4 G3(s)
q
K3 z
K3
‘TX5 + I
GS(s) : __
KS
Tgs
Fig.
11. Block
l
h7k4(Ro-Rs)
K5 I I/k9
I
TJ =
k4 Is + k6 ICO21
k4 Is + k6 ICO21
Tgz
l/k9
I
diagram
of transfer
functions.
treatment of the effects of light fluctuations in photosynthesis in a ‘biological’ rather than ‘biochemical’ approach. The block diagram is translated into a structure diagram, i.e., it is described mathematically. For simulations the structure diagram is expressed as a system of differential equations and transfer functions. For the synthesis of the structure diagram, numerous assumptions and simplification have to be made. They are: (i) All chemical reactions are considered to be first order. At constant enzyme concentration and at low substrate concentrations, the Michaelis-Menten equation gives rates that are about proportional to substrate concentration. (ii) Sequences of fast reactions that have minor importance for the timing of the overall process are lumped. (iii) All reactions are assumed irreversible. (iv) The only input variables are light intensity and the ambient CO, concentration. Based on the simplifying assumptions, a structure diagram is shown in Fig. 10. The diagram is then rearranged and grouped into five major blocks representing five subsystems (Fig. 11). Each of the subsystems will be analysed individually. Photosynthetic oxygen production (G,) Photosynthetic oxygen production occurs primarily in Photosystem II (PS II). Joliot (1966) proposed that a thermal reaction occurring between the photochemical act and the formation of oxygen is limiting. Assuming that the rate of oxygen production follows a first-order dependency with respect to an intermediate X*, i.e., P = d02/dt
= k,X*
(1)
135
where P is the production rate and k, is the first-order generation of X* follows the photochemical reaction: hv+X-+X*+X
kinetic constant. The
k2
and the rate equation for X is, dX*/dr
= I q(X,
- X*)
- k2X*
(2)
where I is the absorbed light intensity and q is the intrinsic quantum yield. The system is non-linear because of the second-order nature of the formation reaction. If the concentration of X* may be neglected in comparison to X,, (a condition which holds at low intensities, that is, when (I q) -=zk2), the linearized equation is: dX*/dt
= I q X, - k,X*
(3) Laplace transformed with respect to f, treating X* and I as deviation variables, and rearranging, one obtains:
(s + k,)X*(s)
= q X,,l(s)
(4)
= q X,,/(s + 4)
(5)
(4 X,/k,) = (l,k,)s + l
(6)
and thus, X*(s)/Z(s) or, x*(s)/l(s)
Similarly, P(s)/l(s)
= k,X*(s)/l(s)
= G,(s)
(7)
Hence,
G1(s) = ,1,;,;; + 1
(8)
The value of k, is about 680 SC’ (Joliot, 1966; Joliot et al., 1966). In the time scale of the frequency response experiment, G,(s) is approximately equal to q X, or a constant k,, and the oxygen production rate responds almost instantaneously to changes of light intensity. Electron
transport
system (G,)
Electron transfers occur at a time scale ranging from microseconds to milliseconds. The response of the rate of production of reduced electron carrier (A-) is almost instantaneous in comparison to other processes. For the purpose of this study, a constant gain of k, is assumed. NADPH
and CO, fixation
(G,)
In a simple model, the following assumptions are made: (1) The light flux I interacts with NADP+ according to: NADP+ + I + NADPH
136
to produce the reducing power NADPH. The rate of formation of NADPH is further assumed to be proportional to the product of I and [NADP+], with rate constant k,. (2) NADPH simply reacts with CO, to give sugar (CH,O),, and regenerates NADP+: NADPH
+ CO, + NADP+ + (CH,O),,
The rate of utilization of NADPH is assumed to be proportional to the product of INADPH] and [CO,] with rate constant k,. (3) The dynamics of regeneration of the electron carrier A, and thus the dynamics of the light reactions are determined by the dynamics of the concentration of NADP+ and NADPH. Hence the rate of removal of A- is assumed to be proportional to the rate of utilization of NADPH. From these assumptions, the following equations can be written: d[NADPH]/dl= -d[A-]/dt
k,l[NADP+]
- k,[COz][NADPH]
(9)
= k,k,[COz][NADPH]
(10)
where k, is a proportionality constant. It is further assumed that there is a maximum total concentration of NADP+ and NADPH, R,, and that it is constant for light intensities below saturation, i.e., R, = [NADP+] Introducing
+ [NADPH]
(11)
the notations R = [NADPH],
Eqns. (9) and (10) can be rewritten as:
dR/dt=k,(R,-R)I-k,[CO,]R
(12)
-AA-
(13)
= dr4-/dt
= k,k,[CO>]R
Eqn. (12) can be linearized by assuming that [CO,] remains relatively constant throughout the experiment. Define deviation variables _I and & as: _r=I--I,
(14)
and &=R-R,
05)
where R, and Z, are steady-state values. Substituting simplifying gives, d&/dt
= k,R,_I+k,R,Is
Applying
k,R,_I-
I and R in Eqn. (12) and
k41s& -k,[CO,]R
- k,R,I,
- k,[CO,]R,
- k,RI
(16)
steady-state conditions, and neglecting the B _I term,
d&/dr=k,(R,-R,)_I-
(k,ls+k,[C02])~
(17)
Taking Laplace transforms and rearranging gives, R(s)/l(s)
= k,(R,
- R,>/(s
+ (Ws
+ k&Ozl))
(18)
137
Using Eqn. (13), -AA-(s)/l(s)
=
k,k,(& - RSMWS+ kmZ1) = G Cs) S/(WS+ k[COZl)+ 1 3
(19)
It is interesting to note that the time constant of this block depends on both the steady-state light intensity and the CO, concentration. Direct oxygen reduction (G4) The photosynthetic systems have mechanisms to prevent over-reduction. Furthermore, excessive oxygen can be toxic to the system because of production of highly active superoxides. The direct reduction of oxygen by reduced electron carriers thus has dual purposes in protecting the system. Although the detailed mechanisms are not well defined, it can be assumed that the reactions must follow closely the change in environmental conditions. Hence it is assumed that the rate of oxygen reduction is proportional to the rate of over-reduction or rate of production of excess reduced electron carrier, i.e., net AA-. There is also the regulatory mechanism of photorespiration that helps to remove excess oxygen and produce carbon dioxide. Dissolved oxygen concentration and photorespiration The process by which oxygen is exchanged between the atmosphere and water is essentially diffusion across a thin water film. For a suspension system, oxygen mass balance gives, dC/dt=P-R+K(C,-C)-k,C
(20)
where C = dissolved oxygen concentration; P = net oxygen evolution rate; R = respiration rate; K = mass transfer coefficient; Cl = saturated oxygen concentration; k, = rate constant for photorespiration. Here the photorespiration rate is assumed proportional to the dissolved oxygen concentration. Assuming that R is constant throughout the experiment, and defining the deviation variables as follows: _c = c - c,,
(21)
f=(P-R)-(P,,-R)=P-P,,
(22)
Substituting the deviation variables into Eqn. (20) and simplifying: d_C/dt=_P-KC-k,g Laplace transforming K(s)
= P(s) -KC(s)
(23) both sides of the equation gives, - k,C(s)
(24)
rearranging: C(s)/(s)
= l/( s + K + k,)
or
l/(K+ k,) G,(s) = s/(K+k,)+l
(25)
13s
I
+
GI - [(G2-G3)G41
NET
)
1021
G5
62 G, _ ,(G2-G3)G41
where
= (Kl-kBW$
K3
2
It8 (k4-
K3)
k?k4(RwRs) k4
G5.
; KI-
IS
l
It6 ICO21
k5
~
TgS*
Fig. 12. Overall
I
block
diagram.
For biofilms from nature, the mass transfer coefficient, K, cannot be assumed constant throughout the film, as the concentration gradient within the film is substantial. Using a quadratic approximation of the concentration profile, the constant K can be approximated as follows: K=
-2D/(x’-
xx,,)
(27)
where D = diffusion coefficient; x = depth of the measuring point from the surface; x0 = total thickness of the film. Overall model The five subsystems (G, to G,) can further be grouped into two blocks. The first block represents the internal processes giving rise to the net oxygen evolution rate. The transfer function of the first block (Fig. 12) indicates a first-order lag and a first-order lead. However, if k, can be assumed to be equal to k,k,, a first-order lag results. This implies that when the CO, fixation is inoperative, the regulatory mechanisms ensure that the system is properly balanced and that the net oxygen evolution is zero. This is supported by the well-known induction phenomena in which the oxygen evolution rate may remain zero for periods ranging from seconds to minutes. There also exist some reports of an oxygen gush after long dark periods and special conditions. In most cases, the magnitude of the gush is very small; and can be explained by the leakages in the regulatory mechanisms. For the purpose of this study, the transfer function for the internal processes can be assumed to be a first order lag. This implies that the response of the net oxygen evolution rate is controlled by the rate of the carbon fixation cycle. The second block represents the oxygen exchange with the atmosphere and the effects of photorespiration. The transfer function is again a simple first order lag. This implies that the responses of dissolved oxygen are controlled by the configuration of the system and the extent of photorespiration.
139
Discussion From the results presented earlier, it is clear that there are two limiting processes contributing to the response of dissolved oxygen to light intensity. The two processes can be approximated as two first order processes with time constants of about 4 and 16 s. The control model also approximates the overall process as two first order processes in series. The remaining problem lies in the assignment of experimental time constants to the two processes. It is postulated that at low frequencies of light modulation, the principal limiting process is the oxygen exchange between the control volume close to the electrode and the surroundings (G, in Fig. 12). From the Bode plots, the time constant is about 16 s. When a partially closed system was used, a longer time constant of 58 s was obtained at low frequency, while the time constant at high frequency remained at about 5 s. Further evidence was obtained with a completely closed system in which the 0, evolution rate (slope of the trace in Fig. 8) responded rapidly to a steady value after a step change in light intensity. The rate decreased gradually only after more than 10 min indicating that the longer time constant was approaching infinity. The rate decrease is probably a result of photorespiration favored by accumulation of 0, and depletion of CO,. At higher frequencies of light modulation, the intracellular processes are limiting. The overall response of overall oxygen evolution rate is controlled by the rate of CO, reduction and it can be approximated as a first-order process. From the Bode plots, the time constant is about 3-5 s. The major drawback in this research was the inability to maintain truly constant concentrations of carbon dioxide. Moreover, there was no direct measure of CO, concentration. Even if the CO, concentration were measured by some means, there is still debate about the forms of CO, taken up by the cells. The CO, limitation factor has been overlooked by many workers in measuring oxygen evolution, and reported rates of oxygen evolution vary widely. Similar work has been done with pH and fluorescence responses to changes in light intensity. We were able to obtain sinusoidal fluorescence response when the photosynthetic system was subjected to sinusoidal light input. Although a time constant of 0.5 s was obtained, the value is too close to the response time of the measuring system. A faster response measuring system is required to give more meaningful estimation of time constants. Our pH responses are much slower than others reported in the literature. In most of the previous studies, chloroplasts with broken envelopes were used. Moreover, the response is complicated by the effects of the buffering capacity of the suspending solution. The work presented here illustrates ways in which the theory of linear system can be used to analyze the behavior of photosynthetic systems subject to time-varying light intensities. It is emphasized that the analysis of biological processes as linear systems is only a mathematical approximation. The value of the technique, as applied in this work, lies in its ability to suggest possible mathematical expressions to account for the observed time-dependent behavior, and also to relate to hypotheses about underlying mechanisms.
140
Conclusion Much of the research on photosynthesis addressed the fine details of pathways and of energy transfer. Although there are significant attempts to improve productivity of algal systems and to construct large, practical growth units, there are little data of use to engineers. In the work presented here, system engineering techniques have been applied to study the dynamic responses of photosynthesis to environmental changes. When light variations are not so extreme that nonlinearities cause distortions, the magnitude of oxygen response is sufficient for Bode diagram analysis over a range of frequencies of practical interest. Two time constants of 4 and 16 s have been determined by frequency response analysis with Bode diagrams that are in accord with two first-order systems in series. The proposed control model of algal photosynthesis allows fresh insights into photosynthesis and should be of value to engineers working with photosynthetic systems.
Acknowledgements Advice and encouragement from Dr. G.-Yull Rhee of the New York State Department of Health and from Dr. L.S. Clesceri were essential to this research. Financial support came from National Science Foundation Grant No. CPE82-17676.
Symbols
N-1 AAAAc c C(s) c, :o,l D Gi (S) hv I Is _I I(s) ki kr K
[NADP+]
Concentration of reduced electron carrier A Rate of change of [A-] Laplace transform of AADissolved oxygen concentration Deviation variable of C Laplace transform of _C Saturated dissolved oxygen concentration Steady-state of C Intracellular carbon dioxide concentration Diffusion coefficient Transfer functions One quantum Absorbed light intensity Steady-state of I Deviation variable of I Laplace transform of _I Kinetic constants Rate constant for photorespiration Mass transfer coefficient Concentration of nicotinamide-adenine dinucleotide phosphate
141
[NADPH]
iO,l
P E P(s) pss
i B R(s) Rc Rs s t X X0
WI KJ IX”1 x*w
Concentration of reduced NADP Intracellular oxygen concentration Photosynthetic oxygen production rate Deviation variable of P Laplace transform of _P Steady-state of P Intrinsic quantum yield [NADPH] Deviation variable of R Laplace transform of & Cellular respiration rate Steady-state of R A variable defined in the complex plane Independent time variable Depth of the measuring point from the surface Total thickness of film Concentration of an intermediate in PS II in normal state Total concentration of X and X* Concentration of an intermediate in PS II in excited state Laplace transform of [X*]
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