Modeling of C3 photosynthesis

Introduction:

Carbon flux into the biosphere is mainly controlled by the global activity of Rubisco (ribulose-1,5-bisphosphate carboxylase/oxygenase) (Falkowski et al 2000, Cox et al 2000). The large majority of global CO₂ assimilation occurs in C3 plants in which Rubisco operates at relatively low carboxylation efficiency, in addition to oxygen inhibition and deficiency of CO₂ as substrate (Spreitzer and Salvucci 2002). Plant biologists have for decades attempted to model photosynthetic processes, in order to better understand the controlling mechanisms, and to predict the effects of greenhouse gas emissions, global warming, crop yield and biosphere productivity. More recently, modeling interest has focused on biome responses to global change (Baldocchi & Wilson 2001) and genetic manipulation for designing a better plant (Spreitzer and Salvucci 2002).

In the 1970’s two major schools of thought arose regarding modelling of photosynthesis at the biochemical level. Earlier models (van Bavel 1975, Ku & Edwards 1977, Hall 1979) argued that CO₂ assimilation in vivo had to reflect the simultaneous availability of the RuBP and CO₂ substrates, O₂ in its oxygenation role, and the kinetics of the Rubisco enzyme. Effects of the environment, such as light availability, sink strength, temperature, humidity and nutrient supply could be understood and modelled via the effects of combined CO₂ and RuBP substrate supply and Rubisco kinetics. Proponents of this, (the single enzyme school) argued that this approach was robust, because it followed biochemical theory and focussed the prediction of CO₂ assimilation upon the Rubisco reaction, with its inputs and outputs, rather than distributing the control of reaction over processes removed from carboxylation to varying degrees.

Later models emphasised the dichotomy between the capacity of Rubisco to consume RuBP, and the capacity of light harvesting, electron transport, and Calvin cycle to regenerate RuBP (Farquhar et al 1980, Sharkey 1985). In contrast to these dual system models, the single-enzyme model was hampered by greater complexity, and substantial uncertainty at the time.

Farquhar and co-workers based their two-system model on the assumptions that carboxylation reaction was assumed to reflect a) the minimum of either the Rubisco capacity to consume RuBP, (the steep part of the CO₂ response curve), or b), the regeneration rate of RuBP for its replacement, (the plateau of the CO₂ response curve), as determined by electron transport rate, Calvin cycle capacity, or triose-phosphate regeneration capacity (Sharkey 1985). The two-system model received initial experimental support shortly after development (von Caemmerer & Farquhar 1981, Sharkey 1985). It has since been used in a wide variety of applications, ranging from anti-sense manipulation at single enzyme levels (von Caemmerer et al 1994, Whitney et al 1999) to modelling global scale changes in carbon flux (Kaduk & Heimann 1997, Baldocchi & Wilson 2001) and global changes in ozone*, photosynthesis and biochemical cycle (Woodward et al 1994, 1995).

Despite its success, limitations persist in the predictive power of the two-system model. The model is considered to be Blackman (1905) type in nature, leading it to an abrupt change that requires empirical correction coefficients such as the convexity factor to produce a smooth transition that reflects a shift of control from Rubisco to RuBP regeneration pathway, limited by an empirical Jmax (Collatz et al 1990, de Pury & Farquhar 1997, von Caemmerer 2000).

The inability of the two-system model to cover different contingencies has led to attempts to improve the models (eg. Harley & Sharkey l99l), to reconsider the single enzyme approach (Farazdaghi & Edwards 1988a,b;1992), and other models (Laisk 1993). The single enzyme models, such as the model of Farazdaghi & Edwards (1989a, b) have been based on the controlling effect of substrates when their concentration is low, co-limitation of enzyme and substrates as the concentration of substrate increases, and enzyme limitation at saturated substrate levels. This is consistent with the widely accepted view that Rubisco is the “ultimate rate-limiting step” in carboxylation (Jensen 2000). The differences between the two schools and the scope of their effects can best be reflected by the following two quotations regarding the control over photosynthesis at ambient CO₂ (300mbar).

i- Eckardt et al (1997) write: “Rubisco activity is one of the principal factors limiting photosynthesis at saturating light and atmospheric CO₂ concentrations.”.

ii- Whitney et al (1999) write: “Since photosynthesis in control plants becomes limited by ribulose-P2 regeneration above 300 mbar CO₂ (Fig 6A), these plants are not suitable for measurement of wild-type Rubisco’s kinetic properties in vivo”.

The conclusion of Whitney et al (1999) is based on an important and basic assumption of the two-system theory of Farquhar et al (1980), that any deviation of the response of carboxylation to CO₂ from Michaelis-Menten curve should be due to a limitation in RuBP regeneration from Calvin cycle. This inevitable conclusion and its follow up recommendation can have significant impact on the direction of research on photosynthesis and genetic manipulation of C3 plants, and if incorrect, may have serious consequences.

The objective of this communication is to re-examine the two-substrate ordered reaction of Rubisco, and determine the hierarchy of the factors that control the rate-determining step (RDS). Also, to formulate the corresponding kinetic models for the single-enzyme two-substrate reaction of Rubisco that can be verified by current experimental observations.

The Single-Enzyme Two-Substrate Ordered Reaction Model:

Following Farazdaghi & Edwards (1988b), we consider that any change in the rate of RuBP regeneration in the Calvin cycle would alter the RuBP concentration that reacts with enzyme in a sequentially ordered two-substrate reaction. The inputs, outputs and feedbacks of Rubisco reaction can be experimentally evaluated, and the rate of reaction can be independently modelled through the enzyme and its substrates. The activity of Rubisco can be adjusted by the regulatory and feedback mechanisms (eg, Mg, CO₂, Rubisco Activase, temperature, pH, and so on), while the capacity of the system is controlled by the activated state and turnover rate of Rubisco that is influenced. In this way carboxylation rate can be fully explained by the kinetics of Rubisco enzyme.

Farazdaghi & Edwards (1988b) replaced RuBP with the equivalent radiation energy that was required for its regeneration in the Calvin cycle. They based derivation of their model on the initial slope or maximum efficiency of the carboxylation reaction (Ψ), which was more stable than Michaelis constant for Rubisco, and provided a simple linear function for CO₂ compensation concentration (Γ*) in relation to oxygen concentration as:

Γ* = 0.5(Vomax/Ko)*O/(Vcmax/Kc) = 0.5Ψ_o.O/ Ψ_c (1)

where, O is the partial pressure of oxygen at the site of reaction, Ψ_c and Ψ_o are the maximum carboxylation and oxygenation efficiencies respectively, and their ratio is consistent with the specificity constant of Jordan & Ogren (1984). The Km values for carboxylation (Kc) and oxygenation (Ko) are specific to the rectangular hyperbola and can be written as:

Km = Vmax/Ψ (2)

Therefore, the effects of competition between CO₂ and O₂ on the carboxylation and quantum efficiencies were described respectively as:

Ψa = Ψc/(1+O/Ko) (3)

Vc Ψ_c^.C C

Φ= Φm ————— = Φm —————— = Φm ——— (4)

Vc + Vo Ψ_c^.C + Ψ_o^.O C + 2 Γ*

Where Ψa is the maximum apparent carboxylation efficiency, Vc and Vo are the velocities of carboxylation and oxygenation, and Φ and Φm, the actual and the maximum quantum efficiencies for RuBP regeneration respectively. At CO₂ compensation point, where C = Γ* in absence “dark” or “day-respiration (Rd), the amount of energy used in oxygenation is twice that used in carboxylation.

The Theory: 1-Rubisco Reaction in a Sequential Order:

The following equation is an extension of Farazdaghi & Edwards (1988b, 1992) and describes both carboxylation and oxygenation via fully activated Rubisco for the two-substrate ordered reaction:

(5)

The k values represent the rate constants for the forward and reverse reactions. E is free enzyme, R is RuBP, C is CO₂, O is O₂; ER, ERC, and ERO are enzyme-substrate complexes with RuBP, RuBP plus CO₂ and RuBP plus O₂ respectively. ER* is the enediol intermediate resulting from the reversible reaction of ER « ER* (first step of the reaction). The reaction of ER* with CO_2, forming ERC, 3-keto-2-Carboxyarabinitol 2-phosphate, is irreversible (Andrews & Lorimer 1987 ; Chen & Spreitzer 1992). ERC undergoes a transformation, ERC « EP, leading to carbon-carbon cleavage and releases PGA in the second step of the reaction (Pierce et al 1986). EX is the enzyme-product complex for oxygenation, and P & Q are the oxygenation products. Because of the irreversibility of carboxylation and oxygenation of enol‑RuBP, k₆ and k₁₂ should be zero. Equation 6 provides a general form for the conservation of enzyme mass.

E_t = E + ER + ER* + ERC + ERO + EP + EX (6)

The components of equation 6 can be calculated in terms of EP for steady state conditions (Cleland 1991) as follows:

E_t/EP =(Kr/R)+(Kc/C).(1+O/K’o)+1+α₁ +α₂+Kr(k2k4/k5+k11.O/k5)/RC (7)

Where, Kr= (k2+k3)k9/k1k3; (8)

Kc= (k3+k4)k9/k3k5; (9)

K’o= k13.k15.(k3+k4)/[k3k11(k13+k14+k15)]. (10)

α1 = k9/k3, (11)

α2= Kp= (k9+k8)/k7. (12)

The enzyme and substrate components of equation 7 can be separated as:

[Et – (1+ α₁+ α₂ )EP]/EP = 1/Sf (13)

where, 1/Sf = (Kr/R)+(Kc/C).(1+O/K’o)+Kr(k2k4/k5+k11.O/k5)/RC (14)

With standard Michaelis-Menten procedures, in which both the transitional and steady state maximum velocities are the same, equation 13 also leads to a similar single step co-limiting model as that of Farazdaghi & Edwards (1988b). In such models the rates of steady state reactions for different substrate levels follow a rectangular hyperbolic function with an inhibition of the maximum velocity (uncompetitive inhibition) by the intermediate enzyme-substrate complexes. In such derivations, the kinetic equations for two-substrate reactions, of either random or sequentially ordered, are basically the same and the order of the reaction cannot be enforced (Johnson 1992). Experimentally, this is also inconsistent with the results reported for two-substrate ordered reactions in general (Johnson 1992, Moulis et al 1991, Jamin et al 1991), and for Rubisco specifically (Laisk 1985, Ruuska et al 1998, and Laisk and Oja 1998). Ruuska et al (1998) demonstrated that, the transitional rate of Rubisco carboxylation for wild-type tobacco (WT) followed a rectangular hyperbola with a maximum of Vcmax=72.5 mmolm^-2 s^-1. However, the steady state rate deviated significantly from the rectangular hyperbola, with a maximum of Vmax = 27 mmolm^-2 s^-1 that was less than 50% of Vcmax.

2- Kinetics of Limitations and Rate Determining Steps:

As the rate of each step in the reaction is proportional to the concentration of enzyme-substrate complex of that step, thus it can be limited either by the concentration of its substrate, or by the availability of enzyme for that step. The maximum velocity of the second step at RuBP saturation is dependent on the total capacity of Rubisco minus the concentration of enzyme that is held in the second step. Therefore, the maximal velocity of the reaction varies, and its lower limit is reached when both substrates are saturating. Thus, in substrate response curves for the two-step reaction of Rubisco, the following phases can be observed:

i- Transitional phase: When RuBP saturated enzyme reacts with CO₂, it will result a single-step transitional phase, for which EP reaches its maximum value that is equal to Et at CO₂ saturation, resulting an initial or transitional maximum velocity for the reaction (Vcmax).

ii- Substrate limitation phase: The initial single-step transitional phase would be followed by a two-step steady state, in which the rate of carboxylation will be limited by the enzyme-substrate complex of the slower step (the step with the limiting substrate). This will be considered as the Rate-Determining Step (RDS). Therefore, when RuBP is limiting, equation (14) will be reduced to:

1/Sf = Kr/R (15)

and when CO₂ is limiting:

1/Sf = (Kc/C).(1+O/K’o) (16)

This is based on the assumption that when free enzyme is available in the medium the reaction of the faster step will be completed sooner and cannot limit the overall rate of reaction.

iii- Co-Limitation Phase: When both substrates are available at relatively high concentrations, but below their respective saturation levels, the rate of carboxylation is proportional to the enzyme-substrate concentration of the limiting substrate (slower step), and its maximum is determined by the actual rate of the faster step. But, as free enzyme is reduced at higher substrate levels, the rate of the faster step becomes partly dependent on the release of enzyme that is controlled by the slower step. Thus as the substrate concentration of the slower step increases, the availability of enzyme for the faster step decreases and the two steps become co-limiting.

iv- Enzyme Limitation Phase: When the reaction has reached dual substrate saturation, the rate of reaction reaches its steady state maximum limit (Vmax). This is because no free enzyme is left in the medium, and the enzyme-substrate complexes of the two steps have cumulatively reached the limit of total enzyme. Therefore, the maximum concentration of EP, (EPmax), which produces the maximum steady state rate Vmax = k9Epmax, is smaller than the concentration of total enzyme that produces the transitional maximum,Vcmax = k9.Et, so that from equation (13) we get:

Et =(1+ α₁+ α₂ )EPmax. (17)

Or: Vcmax/Vmax = 1+ α1+ α2 (18)

3- Limitations of RuBP regeneration:

The continuity of the reaction for in-vivo steady state is dependent on the rate of RuBP regeneration. RuBP regeneration can be limited either by the supply of radiation energy, or other factors that are required in its regeneration pathway. In energy-limited RuBP regeneration, carboxylation responds positively to changes in energy supply. When carboxylation does not respond to any increase in radiation (energy-sufficient), any factor from a large group of parameters may be limiting the reaction. These include a limitation in CO₂ supply (limitation of PGA ), limited Rubisco capacity, or a limitation in the capacity of Calvin cycle enzymes and metabolites that influence the utilisation of energy for RuBP regeneration (eg. inorganic phosphate). When RuBP is Calvin cycle capacity limited, PGA is accumulated and causes product or feedback inhibition (Sun et al 1997). When both substrates are saturating, the total enzyme capacity, that is distributed in enzyme-substrate complexes of the first [α1.EP] and the second step [(1+α2)EP] would be co-limiting the reaction rate (co-limitation of the two steps). The magnitude of Vmax is found to be between 30% to 50% of Vcmax (Laisk 1985, Ruuska et al 1998 and Laisk & Oja 1998).

At sub-saturation, the substrate that is limiting determines the rate of reaction (RDS), and the enzyme component that is engaged with the limiting substrate (E.RDS), makes the co-limiting component of the enzyme for enzyme-substrate co-limitation. Then (Et-E.RDS) can be available in the form of enzyme-substrate complex of the faster step, and determines its potential rate which is the maximum rate of the slower step (often referred to as Kcat). Thus the apparent values of Kcat and Km vary for different reaction rates, within the limits of Vcmax and Vmax (Moulis et al. 1991, Jamin et al. 1991 Johnson 1992).

4- Effects of Limitations on Kinetic Equations:

When RuBP is limiting due to the limitation of energy, the limiting component of enzyme is α₁EP, that is engaged with RuBP in ER « ER*. α₁ is the ratio of the rate of product release in the second step to the rate of production of enol-RuBP in the first step. The slower the rate of enol-RuBP production (smaller k3), the larger is the value of α1 and its limitation or inhibition effect. This means that the maximum available enzyme from product release = Et-E.RDS = Et-α₁EP, and the fraction of total enzyme that is potentially available for maximum rate of the first step (Em1) is:

Em1/Et = (ET- α₁ EP)/Et (19)

When CO₂ is limiting the limitation is on the second step, and the maximum available enzyme for this step is given by the fraction of total enzyme (Em2/Et) that is available for producing ER*, i.e. total enzyme minus the component that is engaged in the slower step:

Em2/ET = [Et – (1+α₂ )EP]/Et (20)

Equations 19 and 20 are particularly significant because of the separation of the two steps by the irreversible link found by Andrews and Lorimer (1987).

When RuBP is limiting due to a deficiency in the Calvin cycle, it causes an accumulation of PGA which has inhibitory effect on enzyme (product inhibition) and produces a limitation in the second step. This case can also be represented by equation (20) through an equivalent increase in the value of α₂. Deficiencies in inorganic phosphate or its recycling rate observed by Sharkey (1985) or Zhang & Nobel (1996) can also be described by equation (20) in the same manner.

If the limitation in the second step is due to a limitation in recycling of inorganic phosphate, then it may also reduce the activity of Rubisco activase (Portis 1992, 1995) and may lead to an inhibition by RuBP substrate (Portis et al1995) or CA1P (Hammond et al 1998). Thus, with regards to RuBP, if there is a shift from the limitation in energy supply to a limitation in the Calvin cycle, it should be evident through a shift in their responses from equation 19 to equation 20. But, when limitation is in the second step no such shift can be observed.

Equation 13 was developed for full co-limitation of enzyme and substrates, without the consideration of RDS. In order to include RDS in this equation, equation 13 should be written for the limiting substrate with an adjustment in Kcat (or potential rate = Vm) for the fraction of total enzyme that can be available for that step. This is achieved by combining equation 13 for limitation of each substrate, with the fraction of enzyme used for its potential rate (equations 19 or 20) when the other substrate is not limiting, i.e.:

{Et – EP(1 + α₁ + α₂)}Et/{EP(Et- α1.EP)} = K’r/R (21)

{Et – EP(1 + α₁ + α₂ )}Et/{EP(Et- (1+α2).EP)} = (1+K’o/O)K’c/C (22)

{Et – EP(1+ α₁ + α₂ )}Et/{EP(Et-(1+ α2).EP)} = K’r/R (23)

Considering that the velocity of steady state reaction is determined from its last step, then for RuBP limited carboxylation, Vr= k9EP, and for CO₂ supply limited Vc = k9EP. As Vcmax = k9.Et, then, with simple substitution of Vr into equations (21) and (23), and Vc into equation (22), and rearrangements we get:

α1.Vr²- Vr [ (1+α1+α2).R.Vcmax/Kr + Vcmax] + Vcmax² .R/Kr = 0 (24)

(1+α2) Vc²- Vc [ (1+α1+α2).C.Vcmax/Kc + Vcmax] + Vcmax² .C/Kc = 0 (25)

(1+α2).Vr²- Vr [ (1+α1+α2).R.Vcmax/Kr + Vcmax] + Vcmax² .R/Kr = 0 (26)

Equations 24 to 26, can be written with respect to the maximum efficiency of the reaction according to equation (2) as follows:

α1.Vr²- Vr [ (1+α1+α2).ΨrR+ Vcmax] + Vcmax.Ψr.R = 0 (27)

(1+α2) Vc²- Vc [ (1+α1+α2).Ψc.C. + Vcmax] + Vcmax.Ψc.C = 0 (28)

(1+α2).Vr²- Vr [ (1+α1+α2).ΨrR+ Vcmax] + Vcmax.Ψr.R = 0 (29)

There is no basic difference between the two sets of equations (equations 24 to 26 and 27 to 29), the choice is rather based on clarity and ease of use, and the fact that Kc can not be measured accurately under steady state conditions (Laisk 1985, Ruuska et al 1998). In equations 27 to 29, the values of α1 and 1+α2 are related to the two separate steps of such reactions that can be applied to carboxylation. The findings of Andrews and Lorimer (1987) and Chen and Sprietzer (1992), of the irreversibility of the bindings of CO₂ with enolated RuBP provide experimental evidence for justification of such separation for Rubisco. However, there are additional complexities related to RuBP regeneration and oxygenation, but equations 27 to 29 provide logical bases for incorporation of such complexities based on the available information.

5-RuBP Regeneration and Radiation:

The amount of RuBP that is used in the reaction should be regenerated from the product of the reaction, PGA, in the triose- RuBP chain of the Calvin cycle, using the energy that is provided by NADPH and ATP. At CO₂ levels above compensation concentration (Γ), the PGA that is produced is not a limiting factor for the production of the consumed RuBP, thus, mole energy per mole RuBP can be substituted for RuBP with an appropriate change in the efficiency according to equation 4, therefore:

ΨrR = Φmax.Ei.I = Φm.I (30)

where, Φmax, is the maximum quantum efficiency for RuBP regeneration, the details of which are given by Farazdaghi & Edwards (1988b). Ei is the proportion of incident energy (I) that is absorbed, and Φm is the overall maximum quantum efficiency with respect to carboxylation, which is found to be constant among different C3 plants (Singsaas et al 2001). When CO₂ is not saturating, as the quantum efficiency for carboxylation is dependent on the strength of its competitor, i.e. oxygenation, therefore, the effect of competition for RuBP is reflected in the net quantum efficiency of carboxylase described by equation 4. When CO₂ is saturating, the response of carboxylation to radiation can set the Vmax of the CO₂ response at different light levels. By substituting equations (30) and (4) into equations (27) and (29) we get:

α1.Vi²-Vi[(1+α1+α2)Φ.I +Vcmax]+Φ.I.Vcmax = 0 (31)

(1+α2).Vi²-Vi[(1+α1+α2)Φ.I +Vcmax]+Φ.I.Vcmax =0 (32)

where Vi is the rate of carboxylation relative to I for a given CO₂ level.

6-Net Assimilation Rate: The rates of net CO₂ assimilation, with respect to light (Ai) or CO₂, (Ac) respectively are:

Ai = Vi – 0.5Vo – Rd (33)

Ac = Vc – 0.5Vo – Rd (34)

However, although the first step of oxygenation is the same as that of carboxylation, the remainder of the oxygenation pathway is more complex and its specific modelling is beyond the scope of this communication. But as light intensity does not change the competitive ratios of Vo/Vi, therefore models of net assimilation rates with respect to radiation are provided in equations 35 and 36.

α1.Ai²-Ai[(1+α1+α2)Φ.I +Vcmax-2α1.R] + Φ.(I-Ic).[Vcmax-R(1+α1+α2)] = 0 (35)

(1+α2)Ai²-Ai[(1+α1+α2)Φ.I+Vcmax-2(1+α2)R]+Φ.[I-Ic)][Vcmax-R(1+α1+α2)] =0 (36)

The compensation point for light, Ic, is approximated by:

(37)

Where I* and Id represent close approximations for the components of light energy used by oxygenation and day respiration respectively at light compensation point. Relationships similar to equation 37 can also be written with respect to CO₂ as follows:

(38)

where Γ* + Γd represent the components of CO₂ concentration related to photorespiration and day respiration respectively at CO₂ compensation point. The equation for net assimilation rate (39), can be written by considering equations 28 and 38, and an absence of photorespiration at CO₂ saturation.

(1+α2)Ac²-Ac[(1+α1+α2) Ψa.C+Vcmax-2(1+α2)R]+ Ψ.[C-Γ)][Vcmax-Rd(1+α1+α2)] =0 (39)

Equation 35 represents limitation in the first step of reaction, while equations 36 and 39 represent those in the second step. If both sides of the equations 35, 36 and 39 are divided by (1+α1+α2

Hadi Farazdaghi