Photosynthesis Research 12, 129-143 (1987) O Martinus Nijhoff Publishers, Dordrecht--Printed in the Netherlands
129
Regular paper
Generation of oscillatory behavior in the Laisk model of photosynthetic carbon assimilation P. H O R T O N l & H. N I C H O L S O N 2 Research Institute for Photosynthesis and Department of Biochemistry; 2Department of Control Engineering, University of Sheffield, Sheffield $10 2TN, UK Received 14 August 1986; accepted 17 November 1986 Key words: mathematical model, oscillations, photosynthesis, regulation of metabolism Abstract. The metabolic pathways in photosynthesis are modelled as an interconnected series of chemical reactions representing the electron transfer system, the carbon reduction cycle and starch and sucrose synthesis according to the model of Laisk and Walker [Proc R Soc Lond 22.7, 281-302 (1986)]. The model is formulated as a set of non=linear differential equations using mass-action kinetics, and stimulated for transient behaviour using an interactive simulation language. The model responses to switched light demonstrate the existence of oscillatory behaviour, similar to that found experimentally in 02 evolution and chlorophyll fluorescence, and explain known transient behaviour. The model is also used to investigate the source of oscillatory behaviour in the phosphate translocator, and other transient phenomena associated with the cyclic electron transfer system. Abbreviations: ~ PQ-plastoquinone;
PQH2-plastoquinol; ~PCr~-reduced plastocyanin; PCo~ oxidised plastocyanin; Pi-ortho (inorganic) phosphate in'chloroplasts; P~o-inorganic orthophosphate in cytosol; TP triose phosphate; Ru5P ribulose-5-phosphate; RuBP-ribulose bisphosphate; PGA-phosphoglyceric acid; HP-hexose phosphate; HPo-hexose phosphate-total sugar phosphate in cytoplasm; S starch; SU sucrose.
Introduction
The metabolic p a t h w a y s representing the reactions in photosynthesis were established in the 1950's. In m o r e recent years, a t t e m p t s have been m a d e to investigate the d y n a m i c behaviour o f the interconnected processes involved, using b o t h low-order mathematical models associated with the key biochemical reactions o f C O 2 assimilation with simplified light activation [2, 3, 7-9], and more complex models based on relatively large n u m b e r s o f k n o w n reactions and metabolites [6, 11]. Various models have also been developed to investigate conditions producing oscillatory responses [5, 10]. Competition between the A T P - c o n s u m i n g reactions and the phosphate translocator have b e e n suggested as possible mechanisms causing oscillations in photosynthesis, although precise details o f the feedback loops involved are still obscure. M o r e recently, a relatively large dynamic model exhibiting realistic transient responses and rapidly saturating steady-state CO2 and light curves, has been developed by Laisk and Wiilker [10]. Oscillatory behaviour is induced by incorporating a time delay in the p a t h w a y for sucrose synthesis, thereby affect-
130 ing the mechanism which determines the rate of phosphate turnover. It is known that sugar phosphate pools oscillate in counterphase with COz uptake, suggesting that inorganic phosphate might be involved in the oscillatory mechanism, and that an induced limitation might restrict photophosphorylation and thereby limit the rate of ATP production and C02 uptake [10]. The present paper uses the Laisk model in a slightly different format to highlight its structural properties, and particularly to investigate other sources and mechanisms of oscillatory behaviour and phenomena generated by changes in light intensity and kinetic parameters.
The mathematical model
The mathematical model developed to represent the interconnected processes in photosynthesis has been formulated using the following chemical reactions. The overall process is activated by light inputs LI, L2 to photosystems PSI, PSII in the electron transfer system, and by atmospheric CO2A entering the carbon reduction cycle in the chloroplasts. The process outputs are considered to be oxygen produced in the electron transport system, starch in the stroma and sucrose in the cytosol. The pathways for starch and sucrose synthesis are highly complex. A simplified representation of them are used in the present model with the activation of starch and sucrose phosphate synthase by organic phosphates and inactivation by orthophosphate in the chloroplast and cytosol regions respectively.
Electron transfer L2 + HzO + PQ
k,
02 + PQH2
(R1)
xADP + xP~ + PQH2 + PCox k2, xATP + PQ + P C ~
(R2)
L1 + PCred + NADP
(R3)
NADPH
k3 , NADPH + PCox
k, , NADP
(R4)
Calvin cycle RuBP + CO2
ks) PGA
PGA + ATP + NADPH
(R5) k6 k'6
TP + ADP + Pi + NADP
(R6)
131 TP
k7 ~ HP
TP + HP ~
(R7) k8
k_8
Ru5P + ATP
' Ru5P
(R8)
kg, RuBP + ADP
(R9)
Translocator TP + Pio ~
kl0 k-t0
~ HPo + Pi
(RI0)
Starch/sucrosesynthesis HP HPo
k.
S + Pi
(Rll)
kl2 SU -1- Pio
(R12)
The metabolic pathways representing the sequential 'light' reactions for electron transfer (R1-R4), including the photosystems I and II with donor and acceptor pools in reduced and oxidised states and the pseudocyclic flow of electrons 'out' of the pathways in R4, the dark reactions for RuBP carboxylation (R5), phosphorylation/reduction (R6), regeneration (R7-R9) and phosphate translocation (R10) in the Calvin cycle, and for starch and sucrose synthesis (R11, R12), are illustrated in block diagram form in Fig. 1. Oxidised acceptors (PQ, NADP) are reduced by electron transfer from reduced donors (H20, PCred), and photophosphorylation subject to photosynthetic control is coupled to electron transfer from acceptor PQH2 to donor PCo~. The dynamics of the light reactions are very rapid compared to those occurring in the Calvin cycle, and are regulated by the effects of pH, the ratio of the concentrations of ADP and ATP, and by photoactivated enzymes. In the Calvin cycle, CO2 combines with RuBP to give PGA, and the reversible reaction R6 links the electron transfer system and carbon reduction cycle via N A D P H and ATP produced by the 'light' reactions, and reduces PGA to TP. The multiplier 4/6 in the differential equation model represents the stoichiometry of electron and carbon flows (4 electrons per 6 atoms of carbon). The stoichiometric multipliers 9/15 (6/15) in the reversible reaction R8 accounts for 9(6) atoms of carbon removed from the trioses (hexoses) for 15 atoms of carbon used in Ru5P resynthesis. Triose phosphate acts as a central branching point for the fluxes of carbon compounds, diverting to the HP pool to regenerate RuBP or to the cytosolic pools via the phosphate translocator for sucrose synthesis.
132
®
Fig. 1. Diagramatic representation of the model of photosynthesis as formulated by Laisk and
Walker [10]. Each block represents an enzymicreaction as explained in the text. Starch synthesis in reaction R11 in the stroma acts as a regulating valve or buffer to balance stoichiometrically the flux of 2-carbon species to RuBP from the H P pool with the flux of 3-carbon species from TP. The phosphate translocator connects the stromal and cytosolic volumes, and effects the exchange of internal triose phosphate and inorganic phosphate (Pi) with external (Pio) and hexose phosphate (HPo) in the cytoplasm. The product activation of the enzyme phosphoribulokinase (with Ru5P) particularly, has been considered to be an important regulatory reaction which may produce instability and oscillatory behaviour [10]. Figure 1 highlights the complexity of the overall process, with many feedback connections and forward and reverse linkages producing action-reaction
133 phenomena between the individual coupled reactions. The Calvin cycle, particularly, is a closed chain of enzyme controlled reactions with a feedback connection formed by reactions R7-R9 providing the regeneration o f RuBP. Tight coupling and balance must exist between all the subsystems for efficient functioning of the whole system, and events in one can elicit immediate response in other subsystems (e.g., a restriction in the availability of phosphate in the cytoplasm has pronounced effects on the emission of fluorescence from the pigments in the chloroplast membrane [14, 15]). The model is represented by first-order nonlinear state differential equations and vector rate equations of the form ± =
F(v),
v =
f(x,u,y,k)
(1)
with initial conditions x(0), where xi is the concentration of the ith species, and Fi, f~ are polynomials in xj, Ul, Ym, and Vr is the rate of the rth reaction. The model contains 10 state variables, 12 reactions, 2 inputs (u =- L, CO2A) 3 output variables ( y - 02, S, SU), and 15 rate constants (kp, p = 1. . . . . 12). The velocity equations associated with the chemical reactions and the resulting state differential equations are as follows. v~
=
L2/[1 + L2/(k~PQ.H20)]
v2
=
k2PQH2PCo~ADP'Pi/[1 + Pi/(KMP)]
v3
=
L1/[1 + L1/(k3NADP-PC,ed)]
v4
=
k4NADPH/NADP
v5
=
ksRuBP"
CO2A
V6 = k 6 A T P - N A D P H , P G A - k _ 6 T P . A D P - N A D P . P i / [ 1 + Pi/(KMR)] v7
=
k7TP
v8
=
k s T P . t t P / [ K M F + HP] - k_sRu5P
v9 =
k9ATP'Ru5P
=
vsM =
k s R u B P M ' C O E g i f v 9 > vsM
vl0 =
kloTP'P~o/[1 + P~/(KITR)] - k_lonPo.Pi/[1 + P~o/(KITR)]
V~l =
knUP/[1 + P~/(KIS)]
v~2 =
k~EHPo'HPo(t - z)/[1 + Pio(t - Q/(KSUC)]
Symbols KMP, K M F , etc, represent Michaelis constants.
Conservation t e r m s
Total pools of phosphate in chloroplast and cytoplasm, of adenylates and electron transfer acceptors and d o n o r s - PiM
=
Pi + ATP + PGA/3 + TP/3 + HP/6 + Ru5P/5 ~ 2RuBP/5
134
PioM =
Pio + HPo/6
ADT
=
ADP + ATP
PQT
=
P Q + PQH2
PCT
=
PCox + PCred
NADPT
=
NADP + NADPH
Light i n p u t L
=
L1 + L2,
L1
=
yL
The model gives rise to the following model state equations. dPQH2/dt
=
dPCr~d/dt
=
Vl - v2 v2 -
v3
dNADPH/dt
=
dPGA/dt
6vf15 - v6
=
v3 -
dTP/dt
=
v6 -
dHP/dt
=
v7 - vtl -
v4 -
Vl0- v7-
dRu5P/dt
=
vs -- v9
dRuBP/dt
=
v9 -
4v6/6
9v8/15
6v8/15
v5
dATP/dt
=
xv2 -
2v6/6 - v9/5
dHPo/dt
=
vI0 - vlz.
The model is based on the principle o f mass action, and various reactions are considered to be saturating with respect to certain concentrations; thus velocity v2 is saturating with respect to Pi and rate control is determined by the deficiency of either A D P or Pi- Reaction R 7 was modelled linearly to avoid instability problems with a usual quadratic rate term (k7TP 2) [10], although no such difficulties were encountered with the present model. The phosphoribulokinase reaction R9 is assumed to be first-order with respect to R u 5 P and A T P , and the rate o f R u B P synthesis (v9) is limited to its m a x i m u m c o n s u m p t i o n rate (vsM) in reaction R5, to avoid the build-up o f a large pool o f free R u B P when its c o n s u m p t i o n rate is low. Sucrose synthase activity is considered to be subject to delayed allosteric feedback control with activation by HPo(t - z) and inactivation by o r t h o p h o s p h a t e Pio(t - z), where z represents a time delay o f the order o f 15 sec. N o m i n a l rate constants and initial concentrations were obtained f r o m the Laisk model [10] and by trial-and-error setting. SimiIar tuning to give reasonable 'order o f m a g n i t u d e ' steady-state values has been used previously [6]. Parameter estimation techniques can be used to obtain the best fit to experimental data, although practical difficulties can arise when fitting large numbers o f parameters [11]. The set o f constants used in the present model i's given in Table 1.
135 Table 1. Constants employed in the model simulations. For details see the text. Values are obtained from reference [10]. Constant
Value
Constant
Value
kt k2 k3 k4 k5
30 0.6 30 0.6 0.035
x y KMP KMR KMF
0,7 0.5 10 10 5
k6 k_ 6 k7 k8 k_ 8
0.3 0.04 3.5 10 0.1 6 0.6 0.01 0.035 0.01 0.5
KITR KSUC PiM PioM RuBPM ADT PQT PCT NADPT CO2A L
20 2 35 (1) 10 (1) 75 (2) 2 (I) 4 (3) 4 (3) 8 (1) 0-100 (4) 0-200 (5)
k9
kl0 k l0 kn kn kt3 ( 1 ) - - n m o l c m -2 cm -2 s - 1.
(2)--ngatCcm-:
( 3 ) - - n m o l e c m -z
( 4 ) - - n m o l c m -3
(5)--nmolquanta
Transient behaviour of the model was obtained by numerical integration using an interactive block-oriented simulation language (PSI) with a step length of 0.01 sec.
Model results
Three different oscillatory conditions have been investigated using the Laisk model [10]. Figure 2 shows the model responses using the published [10] parameters and initial conditions. Oscillations in the rate of photosynthesis (vi), ATP concentration, the level of stromal inorganic phosphate and in the redox state of plastoquinone are observed, exactly as described in reference 10. The rate of photosynthesis oscillates in phase with the level of ATP, consistent with the postulated limitation of ATP supply during oscillatory behaviour. It is notable also that a large initial peak in the level of PGA is associated with a severely inhibited rate of photosynthesis; again this is consistent with a restricted rate of A T P supply and has been demonstrated experimentally by direct assay of P G A in isolated protoplasts [13]. It is also of note that both N A D P H and PC are oxidised and PQ reduced at the minima of Vl; this is consistent with photosynthetic control via Pi availability. Laisk and Walker showed that the oscillatory behaviour is greatest at lower phosphate levels in the cytosol [10]. This phenomenon has been studied systematically and the results are shown in Figs 3-5. Sequestration of Pi by mannose feeding to leaves restricts the light saturated rate of photosynthesis without effect on the quantum yield [14, 15]. The light response curves of the model in normal and reduced cytoplasmic Pi are shown in Fig. 3; as expected the initial slope is unchanged, but the maximum
136 rate of O2 evolution is diminished by the decrease in PioM. It has been found that the light intensity and CO2 thresholds for oscillation are reduced upon depletion of cytoplasmic P~ [14, 15]. In the model, at a PioM of I0 nmol cm -2 oscillation appears when the light intensity exceeds 80 nmol quanta cm -2 s -l, and grows stronger up to approx 300nmol quanta cm-Zs -l (Fig. 4A). At lower P~oM, oscillation appears at intensities as low as 40nmol quanta and maximum strength is seen at 160nmol quanta c m - Z s -1 (Fig. 4B). As shown in Fig. 5, at 50 nmol quanta cm -2 s-1 strong oscillations were seen at low Pi and no oscillation at high Pi. In general, therefore, the model validates the hypothesis that oscillation in the rate of photosynthesis results from the control of sucrose synthesis and the resultant effects of Pj level on photosynthetic ATP production. However, it has recently been found that isolated chloroplasts can also reveal oscillatory behaviour if the supply of P~ is restricted [1] or the phosphate translocator inhibited [12], suggesting an oscillation 'generator' within the chloroplast reactions themselves. Decrease in kl0, the rate constant for the phosphate translocator, restricts the rate of photosynthesis, v~, to about 15% of its maximum value (Fig. 6).
- VI
Pi
ATP P6A P(/H 2
PCred
NADPH ~'
on
129
Regular paper
Generation of oscillatory behavior in the Laisk model of photosynthetic carbon assimilation P. H O R T O N l & H. N I C H O L S O N 2 Research Institute for Photosynthesis and Department of Biochemistry; 2Department of Control Engineering, University of Sheffield, Sheffield $10 2TN, UK Received 14 August 1986; accepted 17 November 1986 Key words: mathematical model, oscillations, photosynthesis, regulation of metabolism Abstract. The metabolic pathways in photosynthesis are modelled as an interconnected series of chemical reactions representing the electron transfer system, the carbon reduction cycle and starch and sucrose synthesis according to the model of Laisk and Walker [Proc R Soc Lond 22.7, 281-302 (1986)]. The model is formulated as a set of non=linear differential equations using mass-action kinetics, and stimulated for transient behaviour using an interactive simulation language. The model responses to switched light demonstrate the existence of oscillatory behaviour, similar to that found experimentally in 02 evolution and chlorophyll fluorescence, and explain known transient behaviour. The model is also used to investigate the source of oscillatory behaviour in the phosphate translocator, and other transient phenomena associated with the cyclic electron transfer system. Abbreviations: ~ PQ-plastoquinone;
PQH2-plastoquinol; ~PCr~-reduced plastocyanin; PCo~ oxidised plastocyanin; Pi-ortho (inorganic) phosphate in'chloroplasts; P~o-inorganic orthophosphate in cytosol; TP triose phosphate; Ru5P ribulose-5-phosphate; RuBP-ribulose bisphosphate; PGA-phosphoglyceric acid; HP-hexose phosphate; HPo-hexose phosphate-total sugar phosphate in cytoplasm; S starch; SU sucrose.
Introduction
The metabolic p a t h w a y s representing the reactions in photosynthesis were established in the 1950's. In m o r e recent years, a t t e m p t s have been m a d e to investigate the d y n a m i c behaviour o f the interconnected processes involved, using b o t h low-order mathematical models associated with the key biochemical reactions o f C O 2 assimilation with simplified light activation [2, 3, 7-9], and more complex models based on relatively large n u m b e r s o f k n o w n reactions and metabolites [6, 11]. Various models have also been developed to investigate conditions producing oscillatory responses [5, 10]. Competition between the A T P - c o n s u m i n g reactions and the phosphate translocator have b e e n suggested as possible mechanisms causing oscillations in photosynthesis, although precise details o f the feedback loops involved are still obscure. M o r e recently, a relatively large dynamic model exhibiting realistic transient responses and rapidly saturating steady-state CO2 and light curves, has been developed by Laisk and Wiilker [10]. Oscillatory behaviour is induced by incorporating a time delay in the p a t h w a y for sucrose synthesis, thereby affect-
130 ing the mechanism which determines the rate of phosphate turnover. It is known that sugar phosphate pools oscillate in counterphase with COz uptake, suggesting that inorganic phosphate might be involved in the oscillatory mechanism, and that an induced limitation might restrict photophosphorylation and thereby limit the rate of ATP production and C02 uptake [10]. The present paper uses the Laisk model in a slightly different format to highlight its structural properties, and particularly to investigate other sources and mechanisms of oscillatory behaviour and phenomena generated by changes in light intensity and kinetic parameters.
The mathematical model
The mathematical model developed to represent the interconnected processes in photosynthesis has been formulated using the following chemical reactions. The overall process is activated by light inputs LI, L2 to photosystems PSI, PSII in the electron transfer system, and by atmospheric CO2A entering the carbon reduction cycle in the chloroplasts. The process outputs are considered to be oxygen produced in the electron transport system, starch in the stroma and sucrose in the cytosol. The pathways for starch and sucrose synthesis are highly complex. A simplified representation of them are used in the present model with the activation of starch and sucrose phosphate synthase by organic phosphates and inactivation by orthophosphate in the chloroplast and cytosol regions respectively.
Electron transfer L2 + HzO + PQ
k,
02 + PQH2
(R1)
xADP + xP~ + PQH2 + PCox k2, xATP + PQ + P C ~
(R2)
L1 + PCred + NADP
(R3)
NADPH
k3 , NADPH + PCox
k, , NADP
(R4)
Calvin cycle RuBP + CO2
ks) PGA
PGA + ATP + NADPH
(R5) k6 k'6
TP + ADP + Pi + NADP
(R6)
131 TP
k7 ~ HP
TP + HP ~
(R7) k8
k_8
Ru5P + ATP
' Ru5P
(R8)
kg, RuBP + ADP
(R9)
Translocator TP + Pio ~
kl0 k-t0
~ HPo + Pi
(RI0)
Starch/sucrosesynthesis HP HPo
k.
S + Pi
(Rll)
kl2 SU -1- Pio
(R12)
The metabolic pathways representing the sequential 'light' reactions for electron transfer (R1-R4), including the photosystems I and II with donor and acceptor pools in reduced and oxidised states and the pseudocyclic flow of electrons 'out' of the pathways in R4, the dark reactions for RuBP carboxylation (R5), phosphorylation/reduction (R6), regeneration (R7-R9) and phosphate translocation (R10) in the Calvin cycle, and for starch and sucrose synthesis (R11, R12), are illustrated in block diagram form in Fig. 1. Oxidised acceptors (PQ, NADP) are reduced by electron transfer from reduced donors (H20, PCred), and photophosphorylation subject to photosynthetic control is coupled to electron transfer from acceptor PQH2 to donor PCo~. The dynamics of the light reactions are very rapid compared to those occurring in the Calvin cycle, and are regulated by the effects of pH, the ratio of the concentrations of ADP and ATP, and by photoactivated enzymes. In the Calvin cycle, CO2 combines with RuBP to give PGA, and the reversible reaction R6 links the electron transfer system and carbon reduction cycle via N A D P H and ATP produced by the 'light' reactions, and reduces PGA to TP. The multiplier 4/6 in the differential equation model represents the stoichiometry of electron and carbon flows (4 electrons per 6 atoms of carbon). The stoichiometric multipliers 9/15 (6/15) in the reversible reaction R8 accounts for 9(6) atoms of carbon removed from the trioses (hexoses) for 15 atoms of carbon used in Ru5P resynthesis. Triose phosphate acts as a central branching point for the fluxes of carbon compounds, diverting to the HP pool to regenerate RuBP or to the cytosolic pools via the phosphate translocator for sucrose synthesis.
132
®
Fig. 1. Diagramatic representation of the model of photosynthesis as formulated by Laisk and
Walker [10]. Each block represents an enzymicreaction as explained in the text. Starch synthesis in reaction R11 in the stroma acts as a regulating valve or buffer to balance stoichiometrically the flux of 2-carbon species to RuBP from the H P pool with the flux of 3-carbon species from TP. The phosphate translocator connects the stromal and cytosolic volumes, and effects the exchange of internal triose phosphate and inorganic phosphate (Pi) with external (Pio) and hexose phosphate (HPo) in the cytoplasm. The product activation of the enzyme phosphoribulokinase (with Ru5P) particularly, has been considered to be an important regulatory reaction which may produce instability and oscillatory behaviour [10]. Figure 1 highlights the complexity of the overall process, with many feedback connections and forward and reverse linkages producing action-reaction
133 phenomena between the individual coupled reactions. The Calvin cycle, particularly, is a closed chain of enzyme controlled reactions with a feedback connection formed by reactions R7-R9 providing the regeneration o f RuBP. Tight coupling and balance must exist between all the subsystems for efficient functioning of the whole system, and events in one can elicit immediate response in other subsystems (e.g., a restriction in the availability of phosphate in the cytoplasm has pronounced effects on the emission of fluorescence from the pigments in the chloroplast membrane [14, 15]). The model is represented by first-order nonlinear state differential equations and vector rate equations of the form ± =
F(v),
v =
f(x,u,y,k)
(1)
with initial conditions x(0), where xi is the concentration of the ith species, and Fi, f~ are polynomials in xj, Ul, Ym, and Vr is the rate of the rth reaction. The model contains 10 state variables, 12 reactions, 2 inputs (u =- L, CO2A) 3 output variables ( y - 02, S, SU), and 15 rate constants (kp, p = 1. . . . . 12). The velocity equations associated with the chemical reactions and the resulting state differential equations are as follows. v~
=
L2/[1 + L2/(k~PQ.H20)]
v2
=
k2PQH2PCo~ADP'Pi/[1 + Pi/(KMP)]
v3
=
L1/[1 + L1/(k3NADP-PC,ed)]
v4
=
k4NADPH/NADP
v5
=
ksRuBP"
CO2A
V6 = k 6 A T P - N A D P H , P G A - k _ 6 T P . A D P - N A D P . P i / [ 1 + Pi/(KMR)] v7
=
k7TP
v8
=
k s T P . t t P / [ K M F + HP] - k_sRu5P
v9 =
k9ATP'Ru5P
=
vsM =
k s R u B P M ' C O E g i f v 9 > vsM
vl0 =
kloTP'P~o/[1 + P~/(KITR)] - k_lonPo.Pi/[1 + P~o/(KITR)]
V~l =
knUP/[1 + P~/(KIS)]
v~2 =
k~EHPo'HPo(t - z)/[1 + Pio(t - Q/(KSUC)]
Symbols KMP, K M F , etc, represent Michaelis constants.
Conservation t e r m s
Total pools of phosphate in chloroplast and cytoplasm, of adenylates and electron transfer acceptors and d o n o r s - PiM
=
Pi + ATP + PGA/3 + TP/3 + HP/6 + Ru5P/5 ~ 2RuBP/5
134
PioM =
Pio + HPo/6
ADT
=
ADP + ATP
PQT
=
P Q + PQH2
PCT
=
PCox + PCred
NADPT
=
NADP + NADPH
Light i n p u t L
=
L1 + L2,
L1
=
yL
The model gives rise to the following model state equations. dPQH2/dt
=
dPCr~d/dt
=
Vl - v2 v2 -
v3
dNADPH/dt
=
dPGA/dt
6vf15 - v6
=
v3 -
dTP/dt
=
v6 -
dHP/dt
=
v7 - vtl -
v4 -
Vl0- v7-
dRu5P/dt
=
vs -- v9
dRuBP/dt
=
v9 -
4v6/6
9v8/15
6v8/15
v5
dATP/dt
=
xv2 -
2v6/6 - v9/5
dHPo/dt
=
vI0 - vlz.
The model is based on the principle o f mass action, and various reactions are considered to be saturating with respect to certain concentrations; thus velocity v2 is saturating with respect to Pi and rate control is determined by the deficiency of either A D P or Pi- Reaction R 7 was modelled linearly to avoid instability problems with a usual quadratic rate term (k7TP 2) [10], although no such difficulties were encountered with the present model. The phosphoribulokinase reaction R9 is assumed to be first-order with respect to R u 5 P and A T P , and the rate o f R u B P synthesis (v9) is limited to its m a x i m u m c o n s u m p t i o n rate (vsM) in reaction R5, to avoid the build-up o f a large pool o f free R u B P when its c o n s u m p t i o n rate is low. Sucrose synthase activity is considered to be subject to delayed allosteric feedback control with activation by HPo(t - z) and inactivation by o r t h o p h o s p h a t e Pio(t - z), where z represents a time delay o f the order o f 15 sec. N o m i n a l rate constants and initial concentrations were obtained f r o m the Laisk model [10] and by trial-and-error setting. SimiIar tuning to give reasonable 'order o f m a g n i t u d e ' steady-state values has been used previously [6]. Parameter estimation techniques can be used to obtain the best fit to experimental data, although practical difficulties can arise when fitting large numbers o f parameters [11]. The set o f constants used in the present model i's given in Table 1.
135 Table 1. Constants employed in the model simulations. For details see the text. Values are obtained from reference [10]. Constant
Value
Constant
Value
kt k2 k3 k4 k5
30 0.6 30 0.6 0.035
x y KMP KMR KMF
0,7 0.5 10 10 5
k6 k_ 6 k7 k8 k_ 8
0.3 0.04 3.5 10 0.1 6 0.6 0.01 0.035 0.01 0.5
KITR KSUC PiM PioM RuBPM ADT PQT PCT NADPT CO2A L
20 2 35 (1) 10 (1) 75 (2) 2 (I) 4 (3) 4 (3) 8 (1) 0-100 (4) 0-200 (5)
k9
kl0 k l0 kn kn kt3 ( 1 ) - - n m o l c m -2 cm -2 s - 1.
(2)--ngatCcm-:
( 3 ) - - n m o l e c m -z
( 4 ) - - n m o l c m -3
(5)--nmolquanta
Transient behaviour of the model was obtained by numerical integration using an interactive block-oriented simulation language (PSI) with a step length of 0.01 sec.
Model results
Three different oscillatory conditions have been investigated using the Laisk model [10]. Figure 2 shows the model responses using the published [10] parameters and initial conditions. Oscillations in the rate of photosynthesis (vi), ATP concentration, the level of stromal inorganic phosphate and in the redox state of plastoquinone are observed, exactly as described in reference 10. The rate of photosynthesis oscillates in phase with the level of ATP, consistent with the postulated limitation of ATP supply during oscillatory behaviour. It is notable also that a large initial peak in the level of PGA is associated with a severely inhibited rate of photosynthesis; again this is consistent with a restricted rate of A T P supply and has been demonstrated experimentally by direct assay of P G A in isolated protoplasts [13]. It is also of note that both N A D P H and PC are oxidised and PQ reduced at the minima of Vl; this is consistent with photosynthetic control via Pi availability. Laisk and Walker showed that the oscillatory behaviour is greatest at lower phosphate levels in the cytosol [10]. This phenomenon has been studied systematically and the results are shown in Figs 3-5. Sequestration of Pi by mannose feeding to leaves restricts the light saturated rate of photosynthesis without effect on the quantum yield [14, 15]. The light response curves of the model in normal and reduced cytoplasmic Pi are shown in Fig. 3; as expected the initial slope is unchanged, but the maximum
136 rate of O2 evolution is diminished by the decrease in PioM. It has been found that the light intensity and CO2 thresholds for oscillation are reduced upon depletion of cytoplasmic P~ [14, 15]. In the model, at a PioM of I0 nmol cm -2 oscillation appears when the light intensity exceeds 80 nmol quanta cm -2 s -l, and grows stronger up to approx 300nmol quanta cm-Zs -l (Fig. 4A). At lower P~oM, oscillation appears at intensities as low as 40nmol quanta and maximum strength is seen at 160nmol quanta c m - Z s -1 (Fig. 4B). As shown in Fig. 5, at 50 nmol quanta cm -2 s-1 strong oscillations were seen at low Pi and no oscillation at high Pi. In general, therefore, the model validates the hypothesis that oscillation in the rate of photosynthesis results from the control of sucrose synthesis and the resultant effects of Pj level on photosynthetic ATP production. However, it has recently been found that isolated chloroplasts can also reveal oscillatory behaviour if the supply of P~ is restricted [1] or the phosphate translocator inhibited [12], suggesting an oscillation 'generator' within the chloroplast reactions themselves. Decrease in kl0, the rate constant for the phosphate translocator, restricts the rate of photosynthesis, v~, to about 15% of its maximum value (Fig. 6).
- VI
Pi
ATP P6A P(/H 2
PCred
NADPH ~'
on
