## Abstract

A computer model comprising light reactions, electron–proton transport, enzymatic reactions, and regulatory functions of C_{3} photosynthesis has been developed as a system of differential budget equations for intermediate compounds. The emphasis is on electron transport through PSII and PSI and on the modeling of Chl fluorescence and 810 nm absorptance signals. Non-photochemical quenching of PSII excitation is controlled by lumenal pH. Alternative electron transport is modeled as the Mehler type O_{2} reduction plus the malate-oxaloacetate shuttle based on the chloroplast malate dehydrogenase. Carbon reduction enzymes are redox-controlled by the ferredoxin–thioredoxin system, sucrose synthesis is controlled by the fructose 2,6-bisphosphate inhibition of cytosolic FBPase, and starch synthesis is controlled by ADP-glucose pyrophosphorylase. Photorespiratory glycolate pathway is included in an integrated way, sufficient to reproduce steady-state rates of photorespiration. Rate-equations are designed on principles of multisubstrate-multiproduct enzyme kinetics. The parameters of the model were adopted from literature or were estimated from fitting the photosynthetic rate and pool sizes to experimental data. The model provided good simulations for steady-state photosynthesis, Chl fluorescence, and 810 nm transmittance signals under varying light, CO_{2} and O_{2} concentrations, as well as for the transients of post-illumination CO_{2} uptake, Chl fluorescence induction and the 810 nm signal. The modeling shows that the present understanding of photosynthesis incorporated in the model is basically correct, but still insufficient to reproduce the dark-light induction of photosynthesis, the time kinetics of non-photochemical quenching, ‘photosynthetic control’ of plastoquinone oxidation, cyclic electron flow around PSI, oscillations in photosynthesis. The model may find application for predicting the results of gene transformations, the analysis of kinetic experimental data, the training of students.

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## Acknowledgements

This work was supported by Targeted Financing Theme 0182535s03 from Estonian Government and by Grants 6607 and 6611 from Estonian Science Foundation.

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## Appendices

### Appendix 1 Model constants and denotations

Acronyms and denotations used in Fig. 1 and in the text beyond model parameters and denotations: *A*, net CO_{2} assimilation rates; ADPGlu, ADP-glucose; *C*
_{a}, *C*
_{c}, CO_{2} concentration, ambient and at Rubisco sites, respectively; Cyt *f*, cytochrome *f*; DHAP, dihydroxyacetone phosphate; E4P, erythrose 4-phosphate; FBP, fructose 1,6-bisphosphate; F2,6BP, fructose 2,6-bisphosphate; Fd, ferredoxin; F6P, fructose 6-phosphate; GAP, glyceraldehyde phosphate; G6P, glucose 6-phosphate; G1P, glucose 1-phosphate; MDH, malate dehydrogenase; NPQ, non-photochemical quenching; OEC, oxygen evolving complex; OP, inorganic phosphate; OPOP, pyrophosphate; PFD, photon flux density and PAD, photon absorption density, mol quanta m^{−2} s^{−1}; PGA, 3-phosophoglycerate; PQ, plastoquinone; PS2, photosystem II; PC, plastocyanin; PGl, phosphoglycolate; P700, donor pigment of photosystem I; *R*
_{p}, rate of production of CO_{2} by photorespiration; R5P, ribose 5-phosphate; Ru5P, ribulose 5-phosphate; RuBP, ribulose 1,5-bisphosphate; SBP, seduheptulose 1,7-bisphosphate; S7P, seduheptulose phosphate; SP, sucrose phosphate; P5P, pentose 5-phosphates; T3P, triose 3-phosphates; UDPG, UDP-glucose.

Rubisco (ribulose 1,5-bisphosphate carboxylase-oxygenase) intermediate complexes: ER, with RuBP; EPP, with two PGA; EPG, with PGA and phosphoglycolate; EP, with one PGA; EOP, with phosphate; Ef, free enzyme.

Below the constants for individual reactions used in calculations are listed. References used as guides were given in the “Model” section. Some adjustments were made to fit the model responses to the kinetic experiments. Denotations here correspond to denotations in the program.

*Environmental and general constants*

Gas phase diffusion resistance (s mm^{−1}): *R*
_{gw} = 4.0E−1, varied during the light response curve;

Mesophyll diffusion resistance (s mm^{−1}): *R*
_{md} = 3.0E−2;

Leaf temperature: *T*
_{leaf} = 22 °C;

Stromal pH: pH_{s} = 8.04; Cytosolic pH: pH_{c} = 7.8;

Total chlorophyll content: Chl = 3.5E−4 mol m^{−2};

Distribution of Chl to PSII: *C*
_{chl2} = 0.48;

Distribution of Chl to PSI: *C*
_{chl1} = 0.48;

Thylakoid membrane volume, (l mol^{−1} Chl): *C*
_{vme} = 4.0;

Lumen volume, relative to membrane: *C*
_{lu} = 1.0;

Stroma volume, relative to membrane: *C*
_{st} = 8.0;

Cytosol volume, relative to membrane: *C*
_{cy} = 8.0;

*Standard midpoint redox potentials of electron carriers in Volts*

*E*
_{QA} = −0.05; *E*
_{PQ} = 0; *E*
_{Cytf} = 0.27; *E*
_{PC} = 0.29; *E*
_{P700} = 0.365;

*E*
_{Fd} = −0.35; *E*
_{NADPH} = −0.32; *E*
_{NADH} = −0.32 (difference *E*
_{Fd} − *E*
_{NADPH} was adjusted as required to fit the PSI acceptor side closure).

*Total pools, mol m*
^{−2}
*of leaf area*

PS1 density: PS1_{T} = 1.5E−6;

PSII density: PS2_{T} = 2.0E−6;

Plastoquinone: PQ_{T} = 13E−6;

Cytochrome *b*
_{6}
*f* Cytf_{T} = 1.5E−6;

Plastocyanin: PC_{T} = 4.5E−6;

NADP(H): NADP_{T} = 25E−6;

Adenylates: AD_{T} = 10E−6;

Orthophosphate in chloroplasts OP_{T} = 270E−6;

Orthophosphate in cytosol: OP_{Tc} = 400E−6;

(0) *Ribulose-bisphosphate carboxylase-oxygenase* (*Rubisco*)

Catalytic site turnover rate: *k*
_{cat} = 3 s^{−1}; *K*
_{m0}(CO_{2}) = 11.5E−6 M; CO_{2}/O_{2} specificity *K*
_{sp} = 92;

RuBP binding: *k*
_{1} = 2.0E+4; *k*
_{_1} = 9.0E−1; CO_{2} binding irreversible: *k*
_{_2} = 0; First PGA dissociation irreversible: *k*
_{_4} = 0; second PGA dissociation reversible: *k*
_{_5} = 7.0E+4;

O_{2} binding irreversible: *k*
_{_6} = 0; phosphoglycolate dissociation irreversible: *k*
_{_7} = 0.

Other Rubisco constants were computed as follows:

Total concentration of enzyme *E*
_{0T}: = *V*
_{m0}/*k*
_{cat}, where *V*
_{m0} was adjusted as measured;

*k*
_{4} = 2*V*
_{m0}/*E*
_{0T}; *k*
_{5} = *k*
_{4}; *k*
_{7} = 0.5*k*
_{4}; *k*
_{2} = *V*
_{m0}/*K*
_{m0}(CO_{2})/*E*
_{0T}; *k*
_{6} = *k*
_{2}/*K*
_{sp}.

*Reactions in chloroplast stroma* (*V*
_{m} in mol m^{−2} s^{−1}; *G*(= Δ*G*
^{′}_{0}
) in kcal mol^{−1}; *K*
_{m} in M). The *V*
_{m} values of the PGA reduction and RuBP regeneration chain were adjusted each to the metabolic control coefficient of less than 0.05 at the maximum CO_{2} and light-saturated rate.

(1) Complexed PGA-kinase-GAP-dehydrogenase (PGK-GAPDH): *V*
_{m1} = 2.0E-3; *G*
_{1} = 3;

(3) Aldolase (DHAP + GAP → FBP): *V*
_{m3} = 4.0E−4;

(4) Fructose-bisphosphate phosphatase: *V*
_{m4} = 5.0E−5;

(5 and 8) Transketolase, constants given for reverse reaction

*V*
_{m_5} = 3.0E−3; *V*
_{m_8} = 1.45E−3; *G*
_{5} = 1.4; *G*
_{8} = 0.1;

(6) Aldolase (DHAP + E4P → SBP): *V*
_{m6} = 2.0E−4;

(7) Seduheptulose bisphosphatase: *V*
_{m7} = 1.0E−4;

(9) Phosphoribulokinase: *V*
_{m9} = 1.0E−2.

Values for *K*
_{m} and other constants of the CRC, starch, and sucrose synthesis enzymes were the same as in (Laisk and Edwards 2000), except SPS was adjusted to *V*
_{m28} = 3.3E−6 in order to match the phosphate-limited, CO_{2} and light-saturated photosynthetic rate in the given leaf (strong metabolic control at CO_{2} and light saturation, shared with Cyt *b*
_{6}
*f*).

(12) ATP synthase: *V*
_{m12} = 3.0E−4 (corresponds to low metabolic control); *G*
_{12} = 7.3; *K*
_{m12ADP} = 3.0E−4; *K*
_{m12OP} = 3.0E−4; *K*
_{m12ATP} = 4.86E−1; *HPR* = 4.

*Electron/proton transport reactions*

(40) Rate constant for plastocyanin diffusion: *RC*
_{40} = 200 s^{−1} (low control);

(42) Cytochrome *b*
_{6}
*f*: *V*
_{m42} = 2.8E-4 mol leaf m^{−2} s^{−1} (relatively strong control shared with SPS at CO_{2} saturation); *K*
_{m42PQ_} = 2.0E−6 mol leaf m^{−2}; HER = 2.0;

(44) Proton leakage rate constant: *RC*
_{44} = 0;

(45) Electron transfer from PSII acceptor *Q*
_{A} to plastoquinone: *RC*
_{45} = 2.5E+9 (very fast);

(46) NDH-dependent cyclic e^{−} flow from NADPH to PQ:

*V*
_{m46} = 1.0E−5 mol m^{−2} s^{−1} (very slow); *K*
_{m46NADPH} = 1.0E−0006; *K*
_{m46PQ} = 1.0E−6; *K*
_{m46NADP} = 1.0E−6; *K*
_{m46PQ_} = 1.0E−6;

(47) Malate dehydrogenase-based NADPH-NAD shuttle: *V*
_{m47} = 30E−6 mol m^{−2} s^{−1} (being redox-activated shares the control of the alternative electron flow with the Mehler reaction at the atmospheric O_{2} concentration); *K*
_{m47NADPH} = 1.0E−6; *K*
_{m47NAD} = 1.0E−4; *K*
_{m47NADP} = 1.0E−6; *K*
_{m47NADH} = 1.0E−4 (here *K*
_{m} in mol m^{−2}).

(49) Mehler type O_{2} reduction rate constant: *RC*
_{49} = 4 s^{−1};

*Kinetics of non-photochemical excitation quenching*

(50) p*K* for *q*
_{E} site protonation: p*K*
_{50} = 5.65; rate constant for *q*
_{E} induction *RC*
_{50} = 3E−2 s^{−1}; *k*
_{f} = 0.149; *k*
_{N} = 0.603; *k*
_{I} = 0.015.

*Redox activation of CRC enzymes*

Redox potential of thioredoxin f: *E*
_{SHf} = −0.31 V;

*V*
_{m} of thioredoxin f reduction *V*
_{m52} = 7.2E−4;

Redox potential of carbon reduction enzyme SH groups: *E*
_{SHcrc} = −0.29 V;

*V*
_{m} of enzyme reduction from thioredoxin f: *V*
_{m54} = 3.6E−2.

*Redox activation control of the malate valve*

Redox potential of thioredoxin m: *E*
_{SHm} = −0.315 V;

*V*
_{m} of thioredoxin m reduction *V*
_{m53} = 3.0E−4;

Redox potential of MDH enzyme SH groups *E*
_{SHmdh} = −0.33 V (adjusted to fit the activation of the alternative electron transport);

*V*
_{m} of enzyme reduction from thioredoxin m: *V*
_{m55} = 3.0E−4 (also adjusted to fit the time kinetics of the alternative electron flow).

### Appendix 2 Budget equations

These are differential budget equations that describe the movement of metabolites through the chain of reactions, the rates of which are given by the concentration-dependent rate equations. One differential equation stands for every metabolite (except for those that are considered to be in equilibrium, where the whole group stands as one single pool). Since correct budgeting of carbon, adenylates and reducing equivalents is the primary prerequisite of such metabolic modeling, we list the complete set of equations here with comments (reaction numbering corresponds to Fig. 1).

*Electron–proton transport and ATP synthesis*

rates *V* are in mol e^{−} m^{−2} s^{−1}, *V*
_{46} is NDH;

electron budget for the redox-equilibrated Cytf–PC complex; *V*
_{61} is chlororespiratory terminal oxidase, a very slow rate (Eq. 36);

electron budget for the redox-equilibrated PC-P700 complex;

The proton budget is an example how compartment volumes are considered when substances are transported through the compartment boundaries.

the second term in parentheses considers electron consumption in the photorespiratory pathway;

*V*
_{19} is ATP consumption for starch synthesis; the last two terms consider ATP consumption in the photorespiratory pathway for glycerate phosphorylation and for ammonia re-assimilation.

*Carbon reduction cycle*

in the PGA budget the first two terms account for the production of PGA from the Rubisco reaction, the terms in the brackets account for the rebinding of PGA to the free and PGA-containing enzyme, *V*
_{O}/2 accounts for phosphorylated glycerate returning from glycolate pathway and *V*
_{PGAout} is the flux of PGA through the phosphate translocator.

*V*
_{3} and *V*
_{6} are aldolase, *V*
_{5} and *V*
_{8} are transketolase rates; the last term is phosphate translocator;

*V*
_{19} is the rate of starch synthesis pathway;

*Starch synthesis pathway*

*Reactions in cytosol, sucrose synthesis*

the last term considers PGA consumption for mitochondrial respiration;

*A*
_{rev} denotes the rate of ‘internal’ carbon feeding into the triosephosphate pool, probably due to glycolysis;

*dF*26*BP*
_{c} /*dt* = *V*
_{31} −*V*
_{32}; the budget of F2,6BP see Laisk et al. (1989);

*Non-photochemical excitation quenching*

*QH* is a quenching site that is immediately protonated, while *N*
_{q} follows the protonation state slowly. Equation 69 adjusts the time kinetics of the non-photochemical quenching.

*Ferredoxin-thioredoxin mechanism of enzyme activation*

In order to reproduce the relatively slow time kinetics of the process, enzyme activation was modeled with the help of two differential equations, the first for the reduction of a thioredoxin by ferredoxin and the second for the reduction of enzyme SS groups by the thioredoxin:

The driving force is the redox ratio of a compound, denoted *RFd*
^{−}/*RFd*, *RSH*
_{f}/*RSS*
_{f}, and *RSH*
_{crc}/*RSS*
_{crc}, where the subscript f stands for thioredoxin f and subscript crc for a regulated carbon reduction enzyme (redox potentials were assumed to be the same for all regulated CRC enzymes). Similar equations were used to model the activation state of malate dehydrogenase, activated by thioredoxin m (reaction 53) and MDH (reaction 55). By setting different *E*
_{m} for the sulfhydryl groups of the carbon reduction enzymes and MDH, as well as different *V*
_{m} values, different time kinetics of activation of the CRC enzymes and MDH were adjusted.

This system was computer-integrated using Euler’s method, programmed in Turbo Pascal 7.0. The integration step was made variable between 0 and 20 ms, calculated from the condition that during the step the most stiff (fastest-turnover) pool could change no more than by 0.1%. This guaranteed the stability of the system at practically all transients (although under stiff conditions the integration might virtually stop due to the very short step, run-time errors were avoided).

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Laisk, A., Eichelmann, H. & Oja, V. C_{3} photosynthesis *in silico*
.
*Photosynth Res* **90**, 45–66 (2006). https://doi.org/10.1007/s11120-006-9109-1

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DOI: https://doi.org/10.1007/s11120-006-9109-1