Modelling C₃ and C₄ photosynthesis under water-stressed conditions

Abstract

Despite the observed impact of water stress on photosynthesis, some of the most used models of CO₂ assimilation in C₃ and C₄ functional types do not directly account for it. We discuss an extension of these models, which explicitly includes the metabolic and diffusive limitations due to water stress on photosynthesis. Functional relationships describing the photosynthetic processes and CO₂ diffusion inside leaves are modified to account for leaf water status on the basis of experimental results available in the literature. Extensive comparison with data shows that the model is suitable to describe the reduction in CO₂ assimilation rate with decreasing leaf water potentials in various species. A simultaneous analysis of photosynthesis, transpiration and soil moisture dynamics is then carried out to explore the actual impact of drought on different photosynthesis processes and on the overall plant activity. The model well reproduces measured CO₂ assimilation rate as a function of soil moisture and could be useful to formulate hypotheses for detailed experiments as well as to simulate in detail transpiration and photosynthesis dynamics under water stress.

Appendices

Appendix 1: CO₂ assimilation model

The assimilation models we use follow with some modifications the parameterizations by Berry and Farquhar (1978) and Farquhar et al. (1980) and their subsequent evolutions (e.g., Chen et al. 1994; Collatz et al. 1991; Collatz et al. 1992; von Caemmerer 2000). The CO₂ assimilation rate is described as the minimum of two potential capacities (Eq. 1): Rubisco activity (A _C ), and photosynthetic electron transport, driven by available solar radiation (A _J ). The finite rate of export or utilization of photosynthesis products may result in a further limitation to photosynthesis (Harley and Sharkey 1991): while some models take into account this third limitation by means of dependences on the maximum activity of Rubisco, V _{c, max} (Collatz et al. 1991; Foley et al. 1996), it is not considered here (as, e.g., in Farquhar et al. 1980; Leuning 1995; von Caemmerer 2000).

To introduce a gradual, more realistic transition from one limitation to the other, the minimum law in Eq. 1 is often substituted by the lower root of the following quadratic relation (e.g., Collatz et al. 1991, 1992; Foley et al. 1996)

$$\kappa _A A_{gross}^2 - \left( {A_C + A_J } \right)A_{\operatorname{gross} } + A_C A_J = 0,$$

(5)

where κ _A is the curvature factor.

CO₂ limitation

Following Farquhar et al. (1980) and von Caemmerer (2000), the Rubisco limited rate of photosynthesis is given by

$$A_C = V_{c,\max } \frac{{c - \Gamma ^ * }}{{c + K_C \left( {1 + \frac{o}{{K_o }}} \right)}},$$

(6)

where c and o are the CO₂ and O₂ concentrations at the site of photosynthesis (the mesophyll cell in C₃ species, thus the symbols c and o in Eq. 6 will have subscript m; the bundle sheath in C₄ species, thus c and o in Eq. 6 will have subscript bs), V _{c ,max} is the maximum catalytic activity of Rubisco at current leaf temperature and water status, Γ* is the equilibrium CO₂ compensation point for gross photosynthesis, K _C and K _O are the coefficients for CO₂ and O₂ of the Michelis–Menten kinetics, accounting for the competitive inhibition by O₂. Some experimental results show a dependence of V _{c ,max} on leaf water potential, as discussed in the text (see also Fig. 1a). Moreover, it is well known that the maximum carboxilation rate under well-watered conditions (V _{c ,max,ww}), and the parameters of the Michelis–Menten kinetics (K _C and K _O ) depend on leaf temperature, T _L ; we adopt the formulation of Leuning (1995) to express such dependences.

In C₃ leaves, photosynthesis takes place in the mesophyll cell, and the CO₂ concentration there, c _m , is to be determined through the mesophyll conductance (see text for details). In C₄ leaves, bundle sheath cells are functionally similar to mesophyll cells in C₃ plants; the CO₂ concentration there, c _bs , represents the equilibrium between the influx, V _P , driven by the C₄-cycle, and the sinks, represented by CO₂ assimilation, A, bundle sheath leakage, L _bs , and mitochondrial respiration, R _m , i.e. $V_P = A + L_{bs} + R_m$. The C₄-cycle is assumed to be controlled solely by PEP carboxylation rate (Berry and Farquhar 1978), and modelled with a Michelis–Menten type dependence upon CO₂ concentration in the mesophyll and an upper-bound PEP regeneration rate, V _Pr , (Chen et al. 1994; von Caemmerer 2000), i.e.

$$V_P = \min \left( {\frac{{c_m V_{P,\max } }}{{c_m + K_P }},V_{Pr} } \right).$$

(7)

As discussed in the text, experimental results show that V _{P ,max} may be reduced by plant water stress, along with PEP carboxylation rate, hence we assume that $V_{P,\max } \left( {\psi _L } \right) = V_{P,\max ,\operatorname{ww} } \eta _P \left( {\psi _L } \right)$, where V _{P ,max,ww} stands for V _{P ,max} rate under well-watered conditions (Fig. 1b). The diffusion flux of CO₂ between the bundle sheath and the mesophyll, L _bs , is driven by the difference in CO₂ concentration and the conductance between them, i.e. $L_{bs} = g_{bs} \left( {c_{bs} - c_m } \right)$ (Chen et al. 1994; von Caemmerer 2000). As in C₃ leaves, the CO₂ concentration in the mesophyll cell, c _m , is a function of mesophyll and stomatal conductances.

For the sake of simplicity, the oxygen concentration at the photosynthetic site (o _m and o _bs for C₃ and C₄ functional types respectively) is kept constant.

Light limitation

Despite a high potential rate of RuBp (ribulose-1,5-biphosphate) carboxylation/oxygenation by Rubisco, photosynthesis can be limited by RuBp regeneration rate. The latter is driven by available energy in the form of ATP and NAPDH, supplied by the electron transport rate, J. Following Leuning (1995), the rate of RuBP-limited CO₂ assimilation is described by

$$A_J = \frac{J}{4}\frac{{c - \Gamma ^ * }}{{c + 2\Gamma ^ * }},$$

(8)

where c is the CO₂ concentration at the photosynthetic site (c _m for C₃ plants, c _bs for C₄ species). The dependence of the electron transport rate J on absorbed photosynthetically active radiation (PAR) is given by the lower root of the equation (Farquhar and Wong 1984)

$$\kappa _\phi J^2 - \left( {J_\phi + J_{\max } } \right)J + J_\phi J_{\max } = 0,$$

(9)

where $\kappa _\phi$ is the curvature factor. Eq. 9 represents the continuous equivalent of the minimum between the limitation to the electron transport rate exerted by adsorbed PAR, $J_\phi$, and its maximum potential rate, J _max, which is assumed to depend only on leaf temperature (as in Leuning 1995). The PAR-dependent electron transport rate, $J_\phi$, is here determined after Genty et al. (1989) as

$$J_{\phi } = \frac{1}{2}\phi _{{PSII}} \phi _{{sw}} ,$$

(10)

where φ _sw is the PAR absorbed by the leaf, and φ _PSII is the quantum yield of electron flow through the photosystem II (PSII), which is assumed to be reduced with decreasing leaf water potential (Fig. 2). Theoretical considerations and experimental measurements (Krall and Edwards 1992; Loreto et al. 1994; Sharkey et al. 1988) show that Eq. 10 correctly estimates the electron transport rate necessary to support the photosynthetic process under relatively low light intensities, while overestimating it under high light intensity. Such possible overestimate is avoided here by using Eq. 9, which limits the calculated electron transport rate, J, to its maximum potential value, J _max.

Drawing on the similarities between the C₃-cycle in C₄ plants and the photosynthesis in C₃ plants, the same model is adopted for C₄ functional type as well. In fact, if the dependence of PEP regeneration on solar radiation is neglected, light availability does not impact the CO₂ concentration mechanism, while it plays a role in the C₃-cycle. Eq. 8 retains the main features of the formulation proposed by Chen et al. (1994), since it includes the dependence on CO₂ availability, and a saturating dependence on adsorbed photo-irradiance, which is upper-bounded by the potential rate of electron transport (Eqs. 9 and 10). This represents an improvement with respect to the models used by Collatz et al. (1992) and Foley et al. (1996), and has the advantage of allowing the inclusion of the impact of plant water status on light-limited assimilation rate.

Appendix 2: Soil water balance and transpiration flux in the plant

According to the cohesion theory, water moves through the SPAC along a path of decreasing water potential, from the soil (at ψ _S ), up to the leaves (at ψ _L ), and to the surrounding atmosphere (at ψ _a ). The water flux inside the plant, E, is assumed to be proportional to the water potential gradient through the soil–root–plant system,

$$E = g_{srp} \left( {\psi _S - \psi _L } \right),$$

(11)

where soil–root–plant conductance, g _srp , is the series of the soil-root and plant conductance (g _sr and g _p ). The first one is here described through a simple cylindrical model (Katul et al. 2003), including soil hydraulic conductivity, K _s (s), active soil depth, Z _r , and root area index, R _AI, as

$$g_{sr} = \frac{{K_s \left( s \right)\sqrt {R_{\operatorname{AI} } } }}{{\pi g\rho _w Z_r }}.$$

(12)

The hydraulic conductivity is reduced with soil moisture s(t) as $K_s \left( s \right) = K_{sat} s\left( t \right)^{2b + 3}$, where b is the exponent of the retention curve, $\psi _S = \bar \psi _S s^{ - b}$, linking s(t) to the soil water potential, ψ _S (t) (e.g., Laio et al. 2001; Rodriguez-Iturbe and Porporato 2004). To account for root response to declining soil moisture availability (Larcher 1995), root area index is corrected with a simple multiplicative term that attenuates the reduction in g _sr due to the reduced hydraulic conductivity during the dry-down, as $R_{{AI}} = R_{{AI,ww}} s^{{ - \omega }}$, where $R_{AI,ww}$ is the root area index under well watered conditions, and ω is a parameter greater than one accounting for root response (Daly et al. 2004a). The reduction of g _p with declining leaf water potential may be described by the following vulnerability curve (Lambers et al. 1998; Sperry et al. 2002)

$$g_p = g_{p,\max } \exp \left[ { - \left( {\frac{{ - \psi _L }}{{d_c }}} \right)^{b_c } } \right],$$

(13)

where g _{p ,max} is the xylem conductance when no cavitation occurs, while the parameters b _c and d _c account for the resistance to cavitation of the plant xylem. C₄ plants from arid and semi-arid ecosystems have lower hydraulic conductivity and less vulnerable vessels (i.e., a safer xylem) than most of C₃ species. On the contrary, C₄ plants adapted to resource-rich areas have parameters similar to those of C₃ plants, but a greater leaf area index, L _AI , which results in a more advantageous water use efficiency (Kocacinar and Sage 2003). Neglecting possible capacitances inside the plant, the transpiration rate at the leaf is equal to the flux in Eq. 11. The lower and upper boundary conditions to the SPAC are set by soil water potential and atmospheric humidity respectively.

The temporal evolution of the soil water potential, ψ _S , is determined by the soil water balance. In the absence of rain (i.e., during a dry-down), the evolution of the vertically-averaged relative soil moisture, s(t), may be expressed by the water balance over the root depth Z _r (e.g., Laio et al. 2001; Rodriguez-Iturbe and Porporato 2004)

$$nZ_r \frac{{ds\left( t \right)}}{{dt}} = - E - L,$$

(14)

where n is the soil porosity, E is the plant transpiration (Eq. 11), and L accounts for other losses, such as evaporation from soil and deep infiltration. For simplicity, L is assumed to follow the behaviour of the hydraulic conductivity K _s (s), plus a constant, E _v , accounting for soil direct evaporation. In some experiments (like those in Fig. 7), both losses are completely prevented (i.e., L = 0).

The upper boundary condition can be set by using the big-leaf scheme and linking the transpiration E to the specific humidity difference between the stomatal air and the atmospheric bulk air through the leaf conductance,

$$E = g_{sba} \left( {q_L - q_a } \right),$$

(15)

where g _sba is the series of stomatal, leaf boundary layer and atmospheric conductance for water vapour per unit ground area (g _s , g _b and g _a respectively, which relate to the corresponding conductances to CO₂ as specified in Table 1). Assuming equilibrium between the leaf water potential and the potential of water vapour in the stomatal cavities, one can write

$$q_L \cong \frac{{0.622}}{{P_a }}e_{sat} \left( {T_L } \right)\exp \left[ {\frac{{V_w \psi _L }}{{RT_L }}} \right],$$

(16)

where $e_{sat} \left( {T_L } \right) \cong a_{sat} \exp \left[ {b_{sat} \left( {T_L - 273} \right)\left( {c_{sat} + T_L - 273} \right)^{ - 1} } \right]$ is the saturated water vapour pressure at leaf temperature T _L (with a _sat  = 613.75 Pa, b _sat  = 17.502, c _sat  = 240.97 K; e.g., Jones 1992, p. 110), R is the gas constant, V _w is the water partial molal volume and ρ _a is the air density. Leaf temperature T _L may be obtained from the leaf energy balance, i.e. $H = c_p \rho _a g_a \left( {T_L - T_a } \right)$. The sensible heat flux is given by $H = \phi _{net} \left( t \right) - \lambda _w \rho _w E$, where φ _net (t) is the net flux of radiation per unit leaf area. In natural environments net solar radiation would be a function of latitude, day of the year, time of the day and atmospheric transmissivity. However, under artificial conditions, incident radiation is generally kept constant during the whole light period, and net radiation may be easily estimated from growing conditions, once the absorption coefficient is known (for our simulation we assumed an absorption coefficient of 0.85; Jones 1992).