Coupled carbon and water fluxes in CAM photosynthesis: modeling quantification of water use efficiency and productivity

Mark S. Bartlett; Giulia Vico; Amilcare Porporato

doi:10.1007/s11104-014-2064-2

Plant and Soil

October 2014, Volume 383, Issue 1–2, pp 111–138 | Cite as

Coupled carbon and water fluxes in CAM photosynthesis: modeling quantification of water use efficiency and productivity

Authors
Authors and affiliations

Mark S. Bartlett
Giulia Vico
Amilcare PorporatoEmail author

Regular Article

First Online: 10 June 2014

Received: 15 October 2013

Accepted: 12 February 2014

Abstract

Background and Aims

Due to their high water use efficiency, Crassulacean acid metabolism (CAM) plants are of environmental and economic importance in the arid and semiarid regions of the world. Moreover, many CAM plants (e.g., Agave tequilana) have attractive qualities for biofuel production such as a relatively low lignin content and high amount of soluble carbohydrates. However, the current estimates of CAM productivity are based on empirical stress indices that create large uncertainties. As a first step towards a more accurate quantification of CAM productivity, this paper introduces a new model that couples both soil and atmosphere conditions to CAM photosynthesis.

Methods

The new CAM model is based upon well established C₃ photosynthesis models coupled to a nonlinear circadian rhythm oscillator for the control of the photosynthesis carbon fluxes. The leaf-level dynamics are coupled to a simple, yet realistic description of the soil-plant-atmosphere continuum, including a plant water capacitance module.

Results

The resulting model reproduces the four phases of CAM photosynthes is and the evolution of their dynamics during a soil moisture drydown, as a function of soil type, plant features, and climatic conditions.

Conclusion

The results help quantify the impact of soil water availability on CAM carbon assimilation and transpiration flux.

Keywords

Crassulacean acid metabolism (CAM) Carbon assimilation Soil moisture Water stress Circadian rhythm oscillator Plant water capacitance

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Notes

Acknowledgments

This work was partially funded through the Agriculture and Food Research Initiative of the USDA National Institute of Food and Agriculture (2011-67003-30222); the National Science Foundation through grants CBET-1033467, EAR-1331846, and EAR-1316258; and by the U.S. Department of Energy (DOE) through the Office of Biological and Environmental Research (BER) Terrestrial Carbon Processes Program (DE-SC0006967). G. Vico acknowledges the support of “AgResource - Resource Allocation in Agriculture”, from the Faculty of Natural Resources and Agricultural Sciences, Swedish University of Agricultural Sciences. The comments and suggestions of the two anonymous reviewers are also gratefully acknowledged. The Wolfram Mathematica code for this CAM model is available upon request.

Appendix 1: Photosynthesis Model

Carbon assimilation model

The photosynthetic carbon demand, $A_{\phi , c_{i}, T_{l}}$, is a function of solar radiation, ϕ, the mesophyll CO₂ concentration, c _i (or the higher mesophyll CO₂ concentration, c _c) and the leaf temperature, T _l (Kelvin). Here it is modeled following Farquhar et al. (1980) and includes the corrections by Leuning (1995). Consequently, the net photosynthetic demand for CO₂ is given by

$$A_{\phi, c_{i}, T_{l}}=f(\phi, c_{i}, T_{l})=\text{Min}[A_{C}, A_{Q}],$$

(18)

where A _C is the Rubisco-limited photosynthesis, and A _Q is the electron transport- (or light-) limited photosynthesis. They are modeled as

$$A_{C} = V_{c, max}(T_{l})\frac{c_{i}-\Gamma^{*}}{c_{i}+K_{c}(1+o_{i}/K_{o})}$$

(19)

$$A_{Q} = \frac{J}{4}\frac{(c_{i}-\Gamma^{*})}{(c_{i}+2\Gamma^{*})},$$

(20)

where V _max(T _l) is the maximum carboxylation rate, K _c and K _o are the Michaelis-Menten coefficients for CO₂ and O₂, respectively, and Γ^∗ is the CO₂ compensation point. Photorespiration is included in the process since the coefficients K _c and K _o represent the competitive inhibition of CO₂ fixation by O₂. In A _Q, J is the electron transport rate and is equal to Min [J _max(T _l), Q] where Q is the amount of solar radiation ϕ (in W m⁻²) converted to μ CO₂ m⁻² s⁻¹. Q is given by

$$Q(\phi)= \frac{\phi \ \lambda}{N_{a} \ h \ c} \times 0.5 \times \kappa_{2},$$

(21)

where 50-percent of the incoming radiation is considered photosynthetically active radiation (PAR) (Jones 1992), λ is the average wavelength in micrometers for PAR (assumed to be 550 nm), h is Plank’s constant (J s), c is the speed of light (m /s), N _a is Avogadro’s constant (1 /mol), and κ ₂ is the quantum yield of electron transport in mol electrons mol⁻¹ photons. The value of κ ₂ is assumed to be 0.3 based upon a leaf absorbance of 0.8 and a photosynthetic quantum yield of 0.093 (Long et al. 1993). J _max is given by

$$J_{max}(T_{l})=J_{max0}\frac{\text{exp}\left[\frac{H_{vJ}}{R T_{0}}(1-\frac{T_{0}}{T_{l}})\right]}{1+\text{exp}\left(\frac{S_{vJ}T_{l}-H_{dJ}}{R T_{l}}\right)}.$$

(22)

Finally, the maximum carboxylation rate and the CO₂ compensation point are given by

$$V_{c, max}(T_{l})=V_{c, max0}\frac{\text{exp}\left[\frac{H_{vV}}{R T_{0}}(1-\frac{T_{0}}{T_{l}})\right]}{1+\text{exp}\left(\frac{S_{vV}T_{l}-H_{dV}}{R T_{l}}\right)}$$

(23)

$$\Gamma^{*}(T_{l})=\gamma_{0}\left[1+\gamma_{l}(T_{l}-T_{0})+\gamma_{2}(T_{l}-T_{0})^{2}\right].$$

(24)

A modified Arrhenius equation describes the temperature dependence of the Michelis-Menten constants

$$K_{x}(T_{l})=K_{x0} \ \text{exp}\left[\frac{H_{Kx}}{R \ T_{0}}(1-\frac{T_{0}}{T_{l}})\right],$$

(25)

where x = c or o. Parameter values can be found in Table 3.

Dark respiration CO₂ flux (R_dc and R_dv)

A modified Arrhenius equation also describes dark respiration as a function of the leaf temperature T _l (Kelvin) (Leuning 1995), i.e.,

$$R_{d}(T_{l})=R_{d0} \ \text{exp}\left[\frac{H_{Rd}}{R \ T_{0}}(1-\frac{T_{0}}{T_{l}})\right],$$

(26)

where the parameter values are found in Table 3. The flux of C from dark respiration R _d(T _l) is assumed to be redirected to either the Calvin cycle, R _dc, or to the the cell vacuole, R _dv. For simplicity, we also assume that, when solar radiation (ϕ) is non-zero, almost all dark respiration CO₂ is directed to the Calvincycle, i.e.,

$$R_{dc}=R_{d}(T_{l})(1-\text{exp}(-\phi)),$$

(27)

where the term (1−exp(−ϕ))≈1 when solar radiation is high. As shown in Eqs. 9 and 11, the flux of dark respiration CO₂, R _dc, reduces the net flux of C from either the stomatal cavity or the cell vacuole. When solar radiation is absent, i.e., ϕ = 0, we assume that all dark respiration CO₂ is converted to malic acid and transported to the cell vacuole, i.e.,

$$R_{dv}=R_{d}(T_{l})\text{exp}(-\phi),$$

(28)

where exp(−ϕ)≈1 when solar radiation ϕ≈0. This flux increases the concentration of malic acid in the cell vacuole, as accounted for by Eq. 2.

Effect of leaf water potential

Similarly to Daly et al. (2004), we assume that the net C assimilation is negatively impacted at leaf water potentials below ψ _l1 and linearly decreases to zero at $\psi _{l_{0}}$, (Table 1; Fig. 9b) i.e.,

$$f_{\psi_{l}}(\psi_{l})=\left\{ \begin{array}{l l} \mathrm{0} & \quad \quad \quad \ \ \psi_{l} < \psi_{l_{0}} \\ \frac{\psi_{l}-\psi_{l_{0}}}{\psi_{l_{1}}-\psi_{l_{0}}} & \quad \psi_{l_{0}} \leq \psi_{l} \leq \psi_{l_{1}} \ \\ 1 & \quad \psi_{l_{1}} < \psi_{l} \leq 0. \ \\ \end{array} \right.$$

(29)

Fig. 9
a Carbon function f _C(z, M); b leaf potential function $f_{\psi _{l}}(\psi _{l})$; c malic acid storage function f _M(z, M, T _l); d curve for plant potential, ψ _w, fitted to the pressure volume data of *Agave deserti* (Nobel and Jordan 1983) with curves for Ω_p and π_p also displayed; e curve for temperature dependence of nocturnal carbon uptake fitted to data of (Nobel and Hartsock 1978); where A_sv,max0 = 7.1μmol m⁻²s⁻¹ and f) values of the maximum malic acid storage concentration M _S(T _l) and the equilibrium concentration M _E(z, T _l) for different leaf temperatures. The circadian rhythm exists when the system maximum storage concentration M _S(T _l) exceeds the equilibrium concentration M _E(z, T _l)

Effect of temperature on nocturnal carbon uptake

The net carbon uptake of Phase I is attenuated by the leaf temperature (Nobel et al. 1998; Nobel and Hartsock 1978, 1984, 1981, Haag-Kerwer et al. 1992, Neales 1973, Nobel 1988, p. 96). Based on experimental data (Neales 1973; Nobel and Hartsock 1978; 1981; Nobel 1988; Yamori et al. 2014) we assume that this temperature dependence has a parabolic form centered on an optimum temperature, i.e.,

$$A_{sv, max}(Tl)=A_{sv, max0}\left[1-k(T_{l}-T_{opt})^{2}\right],$$

(30)

where T _l is the leaf temperature, T _opt is the optimal leaf temperature, A _{sv, max0} is the maximum rate of carbon fixation as malic acid, k is a scale parameter (see Table 1), and A _{sv, max}(Tl) ≥ 0. The parameters k and T _opt are estimated based on the data of Agave deserti (see Fig. 9e), which was measured under growth conditions similar to the laboratory conditions used to calibrate the model runs in this paper (i.e., Nobel and Valenzuela 1987; Lee et al. 1989).

Appendix 2: Circadian rhythm oscillator

The circadian rhythm of the system is represented by a nonlinear oscillator, here described by a second order differential equation for the circadian order variable z, i.e.,

$$\begin{array}{@{}rcl@{}} t_{r} M_{max}\frac{d^{2}}{dt^{2}}z&+&M_{E}^{\ \prime}(z, T_{l})\frac{d}{dt}z\notag\\ &+&(z, M, \phi, c_{i}, T_{l})=0, \end{array}$$

(31)

where $M_{E}^{\ \prime }=\frac {d}{dt}M_{E}$. The equilibrium malic acid function M _E(z, T _l) represents a nonlinear damping force, while g(z, M, ϕ, c _i, T _l) represents a restoring force for the circadian oscillator. This oscillator creates a limit cycle of the type often used to describe rhythms in biology (e.g., Brown et al. 2000; Rompala et al. 2007; Forest et al. 2007, Pavlidis 1973). For example, for g(z, M, ϕ, c _i, T _l) = z, t _r M _max=1/μ and $M_{E}(z, T_{l})=\frac {1}{3}z^{3}-z$, Eq. 31 is a Liénard system for the classic van der Pol oscillator (e.g., Strogatz 2001). Separating Eq. 31 into two first order differential equations (e.g., Strogatz 2001) provides the relaxation oscillation form of the system, Eqs. 2 and 8. For the circadian system (2 and 8), $g(z, M, \phi , c_{i}, T_{l})=-\frac {1}{L_{M}}(A_{sv}+ R_{dv}-A_{vc})$. Oscillations of the system represent the four phases of CAM (see Fig. 10a).

Fig. 10
a The circadian oscillator cycle of M and z and the four CAM Phases for the *Clusia minor* model run at T _l=293 K (see Table 2) with the equilibrium and maximum storage concentrations of malic acid displayed by M _E and M _S, respectively. The oscillation direction is shown by arrows. b The equilibrium values of malate normalized by the maximum, $\frac {M_{E}(z, T_{l})}{M_{max}}$, are displayed for various temperatures that vary between T _L and T _H; solid gray lines. Function f _O(z) for corresponding values of circadian order z; *dark solid line*

Equilibrium concentration of malic acid

The function M _E(z, T _l) represents the equilibrium concentration of vacuolar malic acid for the current global state of gene expression, enzyme activity, and/or the tonoplast state and is modeled with a cubic function of z, i.e.,

$$\begin{array}{@{}rcl@{}} M_{E}(z, T_{l})&\, =\, &M_{max}\left[\!\left(\frac{ T_{H}-T_{l}}{T_{H}-T_{L}}+1\right)c_{1}\left[\beta(z-\mu)\right]^{3}\right.\notag\\ &&\left.-\frac{ T_{H}-T_{l}}{T_{H}-T_{L}}\left[\beta(z-\mu)-c_{2}\right]\right], \end{array}$$

(32)

where T _l is the leaf temperature (Kelvin) and M _E(z, T _l) ≥ 0 (see Fig. 10b). The cubic form of the equation is archetypical of relaxation oscillations found in biological rhythms modeled with the van derPol or FitzHugh Nagumo systems (Brown et al. 2000; Rompala et al. 2007; FitzHugh 1961; Nagumo et al. 1962). The constant μ positions the curves around the circadian order value of 0.5, while the constant β sets the curve for M _E(z, T _L) (for the temperature T _L) equal to 0 at z equal to 0 (see Table 4). In addition, the constant c ₁ sets the maximum equilibrium concentration of malic acid at the low temperature T _L equivalent to M _max, i.e., max[M _E(z, T _L)] = M _max, while the constant c ₂ sets the minimum equilibrium concentration of malic acid at the low temperature T _L equivalent to 0.1 M _max, i.e., min[M _E(z, T _L)] = 0.1M _max (see Fig. 10b). Limiting the minimum equilibrium concentration to 0.1M _max allows the malic acid concentration M to decrease below equilibrium during Phase III and sustain the oscillatory rhythm of z per Eq. 8(see Fig. 10a).

Table 4

Circadian oscillator constants

Constant	Value^a
μ	0.5
β	2.764
c ₁	0.365
c ₂	0.55
c ₃	10

^a see Appendix 2

Entrainment with the local day/night cycle by temperature and light is typical of circadian rhythms (Nimmo 2000), and the temperature dependence of M _E(z, T _l) aids in the entrainment of the circadian rhythm. Here the temperature dependence is assumed to be linear in T _l, with T _H and T _L representing an average temperature range over which the circadian rhythm of CAM has been observed. Typically this range has been found to be between 10 and 29.5°C (283.15 -302.65 K), and consequently, the reference high temperature T _H is 302.65 K while the reference low temperature T _L is 283.15 K (Nobel and Hartsock 1978; Buchanan-Bollig 1984; Lüttge and Beck 1992; Haag-Kerwer et al. 1992; Wilkins 1992; Carter et al. 1996; Grams et al. 1997). The temperature dependence creates a family of cubic curves (see Fig. 10b) which mirror how the CAM system has been observed to achieve high malic acid levels at low temperatures and low malic acid concentrations at high temperatures (Anderson and Wilkins 1989; Haag-Kerwer et al. 1992; Grams et al. 1997; Wilkins 1992).

The temperatures T _H and T _L set the limits of the circadian oscillations. At low temperatures lower than T _L, the malic acid concentration of the system cannot exceed the maximum equilibrium concentration of malic acid, max [M _E(z, T _l)], because of the storage limitation M _S(T _l) (see Fig. 9f and Eq. 35), and so the circadian rhythm oscillator ceases. Non equilibrium values of the malic acid concentration in excess of max [M _E(z, T _l)] sustain the oscillatory rhythm of z per Eq. 8. For example in Fig. 10a, the malic acid concentration exceeds the maximum equilibrium concentration, max[M _E(z, T _l)], and proceeds towards a value of M _S(T _l). In addition, the rhythm ceases when temperatures higher than T _H set the equilibrium concentration of malic acid M _E(z, T _l) to zero (see Fig. 10b).

Order function

The order function, f _O(z), represents the activity of the enzyme PEPC, the vacuolar H⁺-ATPase and/or H⁺-PPase pump for malic acid, and the decarboxylation enzymes for malic acid. We assume that as z increases the CAM plant deactivates the H⁺-ATPase and/or H⁺-PPase pump (allowing malic acid release to the cytoplasm) and activates the enzymes for malic acid decarboxylation in the cytoplasm. The release and decarboxylation of malic acid occurs when f _O(z)≈0 (Fig. 10b). Conversely, as z decreases the CAM plant activates both the enzyme PEPC for malic acid creation and the H⁺-ATPase and/or H⁺-PPase pump for malic acid transport, and deactivates the enzymes for malic acid decarboxylation. This action occurs when f _O(z)≈1 (Fig. 10b). This functional dependence on z as a switch is here represented by

$$f_{O}(z)=\text{exp}\left[-\left(\frac{z}{\mu}\right)^{c_{3}}\right],$$

(33)

where μ is the position parameter from Eq. 32 and c ₃ controls the slope of f _O(z) and thus the sharpness of the division between low and high order values (see Table 4). As shown in Fig. 10, f _O(z) is a sigmoidal curve (complementary Weibull distribution), comprised between 0 and 1, of the type often used as a vulnerability function in plant physiology (Sperry et al. 2002; Vico and Porporato 2008).

Circadian control functions

The carbon function, f _C(z, M) (Fig. 9a) is a function of the order, z, and the malic acid concentration, M, and describes the circadian control that determines whether the flux of C for the Calvin cycle is a supply that originates from the malic acid stored in cell vacuole or directly from the atmospheric CO₂ via the stomatal cavity

$$f_{C}(z, M)=(1-f_{O}(z))\frac{M}{\alpha_{1}M_{max}+M},$$

(34)

where α ₁ is given in Table 1. The order function expression, 1−f _O(z), describes the relative rate of malic acid diffusion from the cell vacuole to the cytoplasm and the overall activation state of the decarboxylation enzymes. The supply of malic acid in the vacuole limits the process through the Michaelis-Menten term $\frac {M}{\alpha _{1}M_{max}+M}$. When f _C(z, M)≈0, malic acid neither diffuses from the cell vacuole nor is decarboxylated in the cytoplasm, and C flows from the stomatal cavity to the Calvin cycle, represented by the flux A _sc. Conversely, when f _C(z, M)≈1, A _sc ≈ 0 and C is exclusively supplied from the cell vacuole.

The malic acid storage function, f _M(z, M), is a function of the circadian order, z, and the malic acid concentration, M (Fig. 9c) and accounts for the circadian control that creates an influx of C to the cell vacuole

$$f_{M}(z, M, T_{l})= f_{O}(z) \frac{M_{S}(T_{l})-M}{\alpha_{2} M_{S}(T_{l})+ (M_{S}(T_{l})-M)}.$$

(35)

Here the function M _S(T _l) is the maximum storage concentration of malic acid and [M _S(T _l)−M]≥0. The order function f _O(z) represents the overall rate at which malic acid is stored in the cell vacuole based on the activity of the the H⁺-ATPas and/or H⁺-PPase pump and the PEPC enzyme. The limitation of the maximum vacuolar storage is estimated using the Michaelis-Menten term $\frac {M_{S}(T_{l})-M}{\alpha _{2} M_{S}(T_{l})+ (M_{S}(T_{l})-M)}$. When f _M(z, M)≈1 there is a flux of carbon to the cell vacuole. The concentration of malic acid that the system can store is given by

$$M_{S}(T_{l})=M_{max} \left[\frac{T_{H}-T_{l}}{T_{H}-T_{L}}(1-\alpha_{2})+\alpha_{2}\right],$$

(36)

where T _L and T _H are the temperature range values of the circadian rhythm (Table 1), T _l is the leaf temperature, M _max is the maximum malic acid concentration at T _L, and α ₂ is a parameter. When the value of α ₂ is small, the disparity between the maximum equilibrium value, max[M _E(z, T _l)], and the storage limitation M _S(T _l) is small, and a smaller half reaction constant of α ₂ M _S(T _l) is required to temper the malic acid flux to vacuole near the equilibrium concentration, M _E(z, T _l). Conversely when α ₂ is large, the disparity between max [M _E(z, T _l)] and M _S(T _l) is large, and a larger half reaction constant of α ₂ M _S(T _l) is required to temper the malic acid flux to the vacuole near the equilibrium concentration.

The control of the C fluxes of CAM photosynthesis is summarized in Table 5, which shows how the stomatal cavity fluxes, which are equal to A _n, and the Calvin cycle fluxes, which are equal to A _c, are determined by the functions f _M(z, M, T _l), f _C(z, M), and solar radiation ϕ. Fig. 10a provides an example of model run oscillation of z and M for Clusia minor, showing the the four CAM Phases, i.e., Phase 1, a buildup of malic acid; Phase II, a change of the circadian order z during both malic acid and Calvin cycle carbon fixation; Phase III, a release of malic acid; and Phase IV, a fixation of carbon in the Calvin cycle when the malic acid is depleted.

Table 5

Carbon fluxes as determined, f _M(z, M, T _l), f _C(z, M), and ϕ

Phase	Time	A _n	A _c	f _M(z, M, T _l)	f _C(z, M)	ϕ
I	night	=A _sv	=0	≈1	≈0	=0
II	morning	=A _sv+A _sc	=A _sc	1→0^a	0→1^a	>0
III	mid.day	=0	=A _vc	≈0	≈1	>0
IV	late afternoon	=A _sc	=A _sc	≈0	≈0	>0

^a Transitional phase where functions pass from 1 to 0 or 0 to 1, respectively

Appendix 3: SPAC model

Leaf specific humidity

Assuming equilibrium between the water vapor potential in the mesophyll and the leaf water potential, the specific humidity of the leaf mesophyll can be computed as (Jones 1992, p. 157)

$$q_{l}(T_{l}, \psi_{l})=q_{l, sat}(T_{l}) \ \text{exp}\left[\frac{V_{w} \ \psi_{l}}{R \ T_{l}}\right],$$

(37)

where V _w is the partial molal volume of water and T _l is the leaf temperature (Kelvin). The specific humidity at saturation q _{l, sat}(T _l) can be obtained from T _l as (Jones 1992)

$$q_{l, sat}(T_{l})=\frac{0.622}{p_{a}} \ a_{sat} \ \text{exp}\left[\frac{b_{sat}(T_{l}-273)}{c_{sat}+T_{l}-273}\right],$$

(38)

where p _a is the atmospheric pressure and a _sat, b _sat, c _sat are constants (Table 1).

Leaf temperature

The leaf temperature, T _l, can be obtained from a quasi-equilibrium plant energy balance, ϕ−H−λ _w ρ _w E = 0, where ϕ is the net flux of radiation per unit leaf area, λ _w is the latent heat of water, ρ _w is the density of water, and H is the sensible heat flux (H = c _p ρg _a(T _l−T _a), where c _p is the specific heat of air at constant pressure, and g _a is the atmospheric conductance). Substituting H into the leaf energy balance and rearranging leads to

$$T_{l}=T_{a}+\frac{\phi-\rho_{w} \ \lambda_{w} \ E}{c_{p} \ \rho \ g_{a}}.$$

(39)

At night, ϕ = 0 and the sensible heat flux equals the latent heat flux λ _w ρ _w E.

Conductances

The soil-root conductance is assumed to be proportional to the soil hydraulic conductivity, K, and the root area index, R _AI, divided by the root depth, Z _r, i.e.,

$$g_{sr}=\frac{K(s)\sqrt{R_{AI}s^{-a}}}{\pi g\rho_{w}Z_{r}},$$

(40)

where ρ _w is the density of water, and g is the gravitational constant (Daly et al. 2004; Katul et al. 2003). The term s ^−a (see Table 1 for a) attenuates the root area index so that CAM plants take advantage of soil moisture with root growth under moderate water stress (i.e., ψ _s < −1) (Nobel 1988; Larcher 2003), but disconnect from the soil (g _sr ≈ 0) due to soil and root shrinkage and reduced hydraulic conductivity of the soil during more severe water stress (i.e., ψ _s > −1) (Nobel 1988; Nobel and Cui 1992). The soil hydraulic conductivity is modeled as K(s) = K _s s ^2b+3 where the saturated hydraulic conductivity K _s and the constant b depend on the soil type (see Eq. 42 and Table 1). Finally, for relatively well watered conditions, g _p may be considered constant for analysis (Table 2); however, for extended droughts, which are not covered in this paper, cavitation may cause a reduction of g _p with ψ _l (Linton and Nobel 2001).

Also note that when the water supply from plant capacitance is negligible (g _w ≈ 0), the SPAC model (16) reduces to the often employed steady state model (e.g. Manzoni et al. 2013) where the flux of water to the leaf is proportional to the water potential gradient between the soil and leaves, i.e.,

$$E=q_{s}=g_{srp}(\psi_{s}-\psi_{l}),$$

(41)

where g _srp is given by (g _sr g _p L _AI)/(g _sr+g _p L _AI).

Water potentials

The soil water potential, ψ _s, is given by a Brooks-Corey type retention curve

$$\psi_{s}=\bar{\psi_{s}}s^{-b}$$

(42)

(Rodríguez-Iturbe and Porporato 2004), while the storage water potential, ψ _w, is the sum of the osmotic pressure and the turgor pressure (Ogburn and Edwards 2010).

The plant osmotic pressure is given by (e.g. Hem 1985, p. 29)

$$\Omega(\theta_{p})=\frac{R \ T_{l}}{V_{m}}\text{ln}\left[\frac{n_{w}(\theta_{p})}{n_{w}(\theta_{p})+n_{s}}\right],$$

(43)

where R is the ideal gas constant, T _l is the leaf temperature (Kelvin), V _m is the molar volume of liquid water (m³ mol⁻¹), 𝜃 _p is the relative water content of the plant, and n _s and n _w are the total moles of solute and water, respectively, on a total leaf area basis (mol m⁻²). The total moles of water is given by n _w(𝜃 _p) = 𝜃 _p D _max/V _m where V _m is the molar volume of water and D _max is the maximum water storage capacitance. The total moles of solute, n _s, determines the value of osmotic pressure when the plant is at full turgor, i.e., 𝜃 _p=1. Although the accumulation of malic acid creates fluctuations in the osmotic pressure on a daily basis (Nobel 1988), for simplicity, we assume that the number moles of solute present in the plant remains constant. At full turgor, the osmotic pressure balances the turgor pressure. The (positive) turgor pressure (in MPa) may be represented as a power law equation of the form (e.g. Ranney et al. 1990)

$$\Pi_{p}(\theta_{p})=(\theta_{p}-d_{1})^{d_{2}}.$$

(44)

Fig. 9d displays ψ _w, Ω_p(𝜃 _p), and π_p(𝜃 _p) versus cell relative water content, 𝜃 _p, for values of n _s, d ₁, and d ₂ (Table 2) that provide the best fit of ψ _w to the pressure volume data of Nobel and Jordan (1983) for Agave deserti. The relative water content is the ratio of the current plant water storage capacitance, D _w, and the maximum water storage capacitance at full turgor, D _max, i.e.,

$$\theta_{p}=\frac{D_{w}}{D_{max}},$$

(45)

where D _w and D _max are water depths per unit leaf area. Changes in the storage depth D _w are determined by the flux of storage water, q _w, as given by the rate equation

$$\frac{dD_{w}}{dt}=-\frac{q_{w}}{L_{AI}},$$

(46)

where L _AI normalizes q _w to a unit leaf area basis since it is for a unit ground area basis. The water storage flux, q _w, is given by the second term in Eq. 14, combined with Eq. 15 to eliminate the branch water potential, ψ _x,

$$q_{w}= g_{w} L_{AI}\left(\psi_{w}-E\frac{(1-f)}{g_{p} L_{AI}}-\psi_{l}\right).$$

(47)

Soil water balance terms

Here we discuss the terms of the soil water balance equation, i.e., Eq. 17. The soil water flux, q _s, (of Eq. 17) is given by the first part of Eq. 14. For simplicity the evaporation of soil water, E _s, (of Eq. 17) is assumed to have a linear dependence with the relative soil moisture, i.e.,

$$E_{s}(s)=\left\{ \begin{array}{l l} E_{smax}\frac{s-s_{h}}{1-s_{h}} & \quad s_{h} < s < 1\ \\ 0 & \quad \ 0 \leq s \leq s_{h}, \ \\ \end{array} \right.$$

(48)

where a maximum rate, E _smax (Table 1), linearly decreases to zero at the hygroscopic point, s _h (see Table 1). Because R = 0, runoff is zero, and L is assumed to be equal to the soil hydraulic conductivity (e.g., gravity drainage), i.e., L = K = K _s s ^2b+3 (see Table 1).

References

Anderson CM, Wilkins MB (1989) Period and phase control by temperature in the circadian rhythm of carbon dioxide fixation in illuminated leaves of Bryophyllum fedtschenkoi. Planta 177(4):456–469PubMedCrossRefGoogle Scholar
Aphalo P, Jarvis P (1991) Do stomata respond to relative humidity? Plant Cell Environ 14(1):127–132CrossRefGoogle Scholar
Ball J (1987) A model predicting stomatal conductance and its contribution to the control of photosynthesis under different environmental conditions. In: Prog. Photosynthesis Res. Proc. Int. Congress 7th, Providence. 10-15 Aug 1986. Vol4:221–224Google Scholar
Blasius B, Beck F, Lüttge U (1997) A model for photosynthetic oscillations in Crassulacean acid metabolism (CAM). J Theor Biol 184(3):345–351CrossRefGoogle Scholar
Blasius B, Beck F, Lüttge U (1998) Oscillatory model of Crassulacean acid metabolism: structural analysis and stability boundaries with a discrete hysteresis switch. Plant Cell Environ 21(8):775–784CrossRefGoogle Scholar
Blasius B, Neif R, Beck F, Lüttge U (1999) Oscillatory model of Crassulacean acid metabolism with a dynamic hysteresis switch. Proc Royal Soc Lond Ser B Biol Sci 266(1414):93–101CrossRefGoogle Scholar
Borland AM, Griffiths H (1996) Variations in the phases of crassulacean acid metabolism and regulation of carboxylation patterns determined by carbon-isotope-discrimination techniques. In: Crassulacean acid metabolism, Springer, pp 230–249Google Scholar
Borland A M, Hartwell J, Jenkins GI, Wilkins MB, Nimmo HG (1999) Metabolite control overrides circadian regulation of phosphoenolpyruvate carboxylase kinase and CO2 fixation in Crassulacean acid metabolism. Plant Physiol 121(3):889–896PubMedCrossRefPubMedCentralGoogle Scholar
Borland A M, Griffiths H, Hartwell J, Smith J A C (2009) Exploiting the potential of plants with Crassulacean acid metabolism for bioenergy production on marginal lands. J Exp Bot 60(10):2879–2896PubMedCrossRefGoogle Scholar
Brown E N, Choe Y, Luithardt H, Czeisler C A (2000) A statistical model of the human core-temperature circadian rhythm. Am J Physiol Endocrinol Metab 279(3):E669–E683PubMedGoogle Scholar
Buchanan-Bollig I C (1984) Circadian rhythms in Kalanchoë: effects of irradiance and temperature on gas exchange and carbon metabolism. Planta 160(3):264–271PubMedCrossRefGoogle Scholar
Calkin H W, Nobel PS (1986) Nonsteady-state analysis of water flow and capacitance for Agave deserti. Can J Bot 64(11):2556–2560CrossRefGoogle Scholar
Carlson T N, Lynn B (1991) The effects of plant water storage on transpiration and radiometric surface temperature. Agric Forest Meteorol 57(1):171–186CrossRefGoogle Scholar
Carter P J, Nimmo H, Fewson C, Wilkins M (1991) Circadian rhythms in the activity of a plant protein kinase. EMBO J 10(8):2063PubMedPubMedCentralGoogle Scholar
Carter PJ, Fewson CA, Nimmo HG, Wilkins MB (1996) Roles of circadian rhythms, light, and temperature in the regulation of phosphoenolpyruvate carboxylase in Crassulacean acid metabolism. In: Crassulacean Acid Metabolism, Springer, pp 46–52Google Scholar
Cockburn W, Ting I P, Sternberg L O (1979) Relationships between stomatal behavior and internal carbon dioxide concentration in crassulacean acid metabolism plants. Plant Physiol 63(6):1029–1032PubMedCrossRefPubMedCentralGoogle Scholar
Comins H, Farquhar G (1982) Stomatal regulation and water economy in Crassulacean acid metabolism plants: an optimization model. J Theor Biol 99(2):263–284CrossRefGoogle Scholar
Cowan I (1972) An electrical analogue of evaporation from, and flow of water in plants. Planta 106(3):221–226PubMedCrossRefGoogle Scholar
Cushman J, Bohnert H (1996) Transcriptional activation of CAM genes during development and environmental stress. In: Crassulacean Acid Metabolism, Springer, pp 135–158Google Scholar
Cushman J, Borland A (2002) Induction of Crassulacean acid metabolism by water limitation. Plant Cell Environ 25(2):295–310PubMedCrossRefGoogle Scholar
Cushman J C (2001) Crassulacean acid metabolism. a plastic photosynthetic adaptation to arid environments. Plant Physiol 127(4):1439–1448PubMedCrossRefPubMedCentralGoogle Scholar
Daly E, Porporato A, Rodriguez-Iturbe I (2004) Coupled dynamics of photosynthesis, transpiration, and soil water balance. Part I: Upscaling from hourly to daily level. J Hydrometeorol 5(3):546–558CrossRefGoogle Scholar
Dodd A N, Borland A M, Haslam R P, Griffiths H, Maxwell K (2002) Crassulacean acid metabolism: plastic, fantastic. J Exp Bot 53(369):569–580PubMedCrossRefGoogle Scholar
Escamilla-Treviño L L (2012) Potential of plants from the genus Agave as bioenergy crops. BioEnergy Res 5(1):1–9CrossRefGoogle Scholar
Farquhar G, von Caemmerer S, Berry J (1980) A biochemical model of photosynthetic CO2 assimilation in leaves of C3 species. Planta 149(1):78–90PubMedCrossRefGoogle Scholar
FitzHugh R (1961) Impulses and physiological states in theoretical models of nerve membrane. Biophys J 1(6):445–466PubMedCrossRefPubMedCentralGoogle Scholar
Flexas J, Ribas-Carbó M, Diaz-Espejo A, Galmes J, Medrano H (2008) Mesophyll conductance to CO2: current knowledge and future prospects. Plant Cell Environ 31(5):602–621PubMedCrossRefGoogle Scholar
Forest L, Glade N, Demongeot J (2007) Liénard systems and potential–Hamiltonian decomposition–Applications in biology. Comptes rendus biologies 330(2):97–106PubMedCrossRefGoogle Scholar
Friemert V, Heininger D, Kluge M, Ziegler H (1988) Temperature effects on malic-acid efflux from the vacuoles and on the carboxylation pathways in Crassulacean-acid-metabolism plants. Planta 174(4):453–461PubMedCrossRefGoogle Scholar
Grams T E E, Borland A M, Roberts A, Griffiths H, Beck F, Luttge U (1997) On the mechanism of reinitiation of endogenous Crassulacean acid metabolism rhythm by temperature changes. Plant physiol 113(4):1309–1317PubMedPubMedCentralGoogle Scholar
Griffiths H, helliker B, Roberts A, Haslam R P, Girnus J, Robe W E, Borland A M, Maxwell K (2002) Regulation of rubisco activity in Crassulacean acid metabolism plants: better late than never. Funct Plant Biol 29(6):689–696CrossRefGoogle Scholar
Haag-Kerwer A, Franco A C, Luttge U (1992) The effect of temperature and light on gas exchange and acid accumulation in the C3-CAM plant Clusia minor L. J Exp Bot 43(3):345–352CrossRefGoogle Scholar
Hartwell J, Smith L H, Wilkins MB, Jenkins GI, Nimmo HG (1996) Higher plant phosphoenolpyruvate carboxylase kinase is regulated at the level of translatable mRNA in response to light or a circadian rhythm. Plant J 10(6):1071–1078CrossRefGoogle Scholar
Hartwell J, Gill A, Nimmo GA, Wilkins MB, Jenkins GI, Nimmo HG (1999) Phosphoenolpyruvate carboxylase kinase is a novel protein kinase regulated at the level of expression. Plant J 20(3):333–342PubMedCrossRefGoogle Scholar
Hem JD (1985) Study and interpretation of the chemical characteristics of natural water, vol 2254, Department of the Interior, US Geological SurveyGoogle Scholar
Herrera A (2009) Crassulacean acid metabolism and fitness under water deficit stress: if not for carbon gain, what is facultative CAM good forAnn Bot 103(4):645–653PubMedCrossRefPubMedCentralGoogle Scholar
Holbrook NM, Putz F (1996) From epiphyte to tree: differences in leaf structure and leaf water relations associated with the transition in growth form in eight species of hemiepiphytes. Plant Cell Environ 19(6):631–642CrossRefGoogle Scholar
Jackson R, Mooney H, Schulze ED (1997) A global budget for fine root biomass, surface area, and nutrient contents. Proc Natl Acad Sci 94(14):7362–7366PubMedCrossRefPubMedCentralGoogle Scholar
Jones HG (1992) Plants and microclimate: a quantitative approach to environmental plant physiology. Cambridge University PressGoogle Scholar
Katerji N, Hallaire M, Menoux-Boyer Y, Durand B (1986) Modelling diurnal patterns of leaf water potential in field conditions. Ecol Model 33(2):185–203CrossRefGoogle Scholar
Kattge J, Knorr W (2007) Temperature acclimation in a biochemical model of photosynthesis: a reanalysis of data from 36 species. Plant Cell Environ 30(9):1176–1190PubMedCrossRefGoogle Scholar
Katul G, Leuning R, Oren R (2003) Relationship between plant hydraulic and biochemical properties derived from a steady-state coupled water and carbon transport model. Plant Cell Environ 26(3):339–350CrossRefGoogle Scholar
Kliemchen A, Schomburg M, Galla HJ, Lüttge U, Kluge M (1993) Phenotypic changes in the fluidity of the tonoplast membrane of Crassulacean-acid-metabolism plants in response to temperature and salinity stress. Planta 189(3):403–409PubMedCrossRefGoogle Scholar
Kluge M (2008) Ecophysiology: Migrations between different levels of scaling. Prog Bot 5–34Google Scholar
Kluge M, Kliemchen A, Galla HJ (1991) Temperature effects on crassulacean acid metabolism: EPR spectroscopic studies on the thermotropic phase behaviour of the tonoplast membranes of Kalanchoë daigremontiana. Botanica Acta 104(5):355–360CrossRefGoogle Scholar
Lal R (2004) Carbon sequestration in dryland ecosystems. Environ Manag 33(4):528–544CrossRefGoogle Scholar
Landsberg JJ, Sands P (2010) Physiological ecology of forest production: principles, processes and models, vol 4. Academic PressGoogle Scholar
Larcher W (2003) Physiological plant ecology: ecophysiology and stress physiology of functional groups. SpringerGoogle Scholar
Lee HS, Schmitt A K, Lüttge U (1989) The response of the C3-CAM tree, Clusia rosea, to light and water stress II. Internal CO2 concentration and water use efficiency. J Exp Bot 40(2):171–179CrossRefGoogle Scholar
Leuning R (1990) Modelling stomatal behaviour and and photosynthesis of Eucalyptus grandis. Funct Plant Bio 17(2):159–175Google Scholar
Leuning R (1995) A critical appraisal of a combined stomatal-photosynthesis model for C3 plants. Plant, Cell Environ 18(4):339–355CrossRefGoogle Scholar
Lhomme JP, Rocheteau A, Ourcival J, Rambal S (2001) Non-steady-state modelling of water transfer in a Mediterranean evergreen canopy. Agric Forest Meteorol 108(1):67–83CrossRefGoogle Scholar
Liguori G, Inglese G, Pernice F, Sibani R, Inglese P (2010) Co2 fluxes of opuntia ficus-indica mill. trees in relation to water status. In: VII International Congress on Cactus Pear Cochineal, 995, pp. 125–133Google Scholar
Linton MJ, Nobel PS (2001) Hydraulic conductivity, xylem cavitation, and water potential for succulent leaves of Agave deserti and Agave tequilana. Int J Plant Sci 162(4):747–754CrossRefGoogle Scholar
Lloyd J (1991) Modelling stomatal responses to environment in Macadamia integrifolia. Funct Plant Biol 18(6):649–660Google Scholar
Lohammar T, Larsson S, Linder S, Falk S (1980) FAST: simulation models of gaseous exchange in Scots pine. Ecol Bull 505–523Google Scholar
Long S, Postl W, Bolhar-Nordenkampf H (1993) Quantum yields for uptake of carbon dioxide in C3 vascular plants of contrasting habitats and taxonomic groupings. Planta 189(2):226–234CrossRefGoogle Scholar
Lüttge U (2000) The tonoplast functioning as the master switch for circadian regulation of Crassulacean acid metabolism. Planta 211(6):761–769PubMedCrossRefGoogle Scholar
Lüttge U (2002) CO2-concentrating: consequences in Crassulacean acid metabolism. J Exp Bot 53(378):2131–2142PubMedCrossRefGoogle Scholar
Lüttge U (2004) Ecophysiology of Crassulacean acid metabolism (CAM). Ann Bot 93(6):629PubMedCrossRefGoogle Scholar
Lüttge U, Beck F (1992) Endogenous rhythms and chaos in Crassulacean acid metabolism. Planta 188(1):28–38PubMedCrossRefGoogle Scholar
Lüttge U, Duarte HM (2007) Morphology, anatomy, life forms and hydraulic architecture. In: Clusia, Springer, pp 17–30Google Scholar
Manzoni S, Vico G, Porporato A, Katul G (2013) Biological constraints on water transport in the soil–plant–atmosphere system. Adv Water Resour 51:292–304CrossRefGoogle Scholar
Medlyn B, Dreyer E, Ellsworth D, Forstreuter M, Harley P, Kirschbaum M, Le Roux X, Montpied P, Strassemeyer J, Walcroft A, Wang K, Loustau D (2002) Temperature response of parameters of a biochemically based model of photosynthesis. II. A review of experimental data. Plant Cell Environ 25(9):1167–1179CrossRefGoogle Scholar
Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50(10): 2061–2070CrossRefGoogle Scholar
Neales T (1973) The effect of night temperature on CO2 assimilation, transpiration, and water use efficiency in Agave americana L. Aust J Biol Sci 26(4):705–714Google Scholar
Nimmo G, Nimmo H, Fewson C, Wilkins M (1984) Diurnal changes in the properties of phosphoenolpyruvate carboxylase in Bryophyllum leaves: a possible covalent modification. FEBS Lett 178(2):199–203CrossRefGoogle Scholar
Nimmo G, Wilkins M, Fewson C, Nimmo H (1987) Persistent circadian rhythms in the phosphorylation state of phosphoenolpyruvate carboxylase from Bryophyllum fedtschenkoi leaves and in its sensitivity to inhibition by malate. Planta 170(3):408–415PubMedCrossRefGoogle Scholar
Nimmo HG (2000) The regulation of phosphoenolpyruvate carboxylase in CAM plants. Trends Plant Sci 5(2):75–80PubMedCrossRefGoogle Scholar
Nobel P (1988) Environmental biology of agaves and cacti. Cambridge University Press, CambridgeGoogle Scholar
Nobel P, Castaneda M, North G, Pimienta-Barrios E, Ruiz A (1998) Temperature influences on leaf CO2 exchange, cell viability and cultivation range for Agave tequilana. J Arid Environ 39(1):1–9CrossRefGoogle Scholar
Nobel PS (1976) Water relations and photosynthesis of a desert CAM plant, Agave deserti. Plant Physiol 58(4):576–582PubMedCrossRefPubMedCentralGoogle Scholar
Nobel PS (1991) Achievable productivities of certain cam plants: basis for high values compared with C3 and C4 plants. New Phytol 119(2):183–205CrossRefGoogle Scholar
Nobel PS (1994) Remarkable agaves and cacti. Oxford University PressGoogle Scholar
Nobel PS (2009) Physicochemical and environmental plant physiology. Academic PressGoogle Scholar
Nobel PS, Cui M (1992) Hydraulic conductances of the soil, the root-soil air gap, and the root: changes for desert succulents in drying soil. J Exp Bot 43(3):319–326CrossRefGoogle Scholar
Nobel PS, Hartsock T L (1978) Resistance analysis of nocturnal carbon dioxide uptake by a Crassulacean acid metabolism succulent, Agave deserti. Plant Physiol 61(4):510–514PubMedCrossRefPubMedCentralGoogle Scholar
Nobel PS, Hartsock T L (1979) Environmental influences on open stomates of a Crassulacean acid metabolism plant, Agave deserti. Plant Physiol 63(1):63–66PubMedCrossRefPubMedCentralGoogle Scholar
Nobel PS, Hartsock T L (1981) Shifts in the optimal temperature for nocturnal CO2 uptake caused by changes in growth temperature for cacti and agaves. Physiol. Plant. 53(4):523–527CrossRefGoogle Scholar
Nobel PS, Hartsock T L (1984) Physiological responses of Opuntia ficus-indica to growth temperature. Physiol. Plant. 60(1):98–105CrossRefGoogle Scholar
Nobel PS, Jordan P W (1983) Transpiration stream of desert species: resistances and capacitances for a C3, a C4, and a CAM plant. J. Exp. Bot. 34(10):1379–1391CrossRefGoogle Scholar
Nobel PS, Valenzuela AG (1987) Environmental responses and productivity of the CAM plant, Agave tequilana. Agric Forest Meteorol 39(4):319–334CrossRefGoogle Scholar
Nungesser D, Kluge M, Tolle H, Oppelt W (1984) A dynamic computer model of the metabolic and regulatory processes in Crassulacean acid metabolism. Planta 162(3):204–214PubMedCrossRefGoogle Scholar
Ogburn R, Edwards EJ (2010) The ecological water-use strategies of succulent plants. Adv. Bot. Res 55:179–225CrossRefGoogle Scholar
Osmond C (1978) Crassulacean acid metabolism: a curiosity in context. Annu. Rev. Plant Physiol. 29(1):379–414CrossRefGoogle Scholar
Owen NA, Griffiths H (2013) A system dynamics model integrating physiology and biochemical regulation predicts extent of crassulacean acid metabolism (CAM) phases. New PhytologistGoogle Scholar
Pavlidis T (1973) Biological oscillators: their mathematical analysis. Academic pressGoogle Scholar
Ranney T, Whitlow T, Bassuk N (1990) Response of five temperate deciduous tree species to water stress. Tree Physiology 6(4):439–448PubMedCrossRefGoogle Scholar
Rodríguez-Iturbe I, Porporato A (2004) Ecohydrology of water-controlled ecosystems: soil moisture and plant dynamics. Cambridge University PressGoogle Scholar
Rompala K, Rand R, Howland H (2007) Dynamics of three coupled van der pol oscillators with application to circadian rhythms. Commun Nonlinear Sci Numer Simul 12(5):794–803CrossRefGoogle Scholar
Schulte P, Nobel P (1989) Responses of a CAM plant to drought and rainfall: capacitance and osmotic pressure influences on water movement. J Exp Bot 40:61–70CrossRefGoogle Scholar
Sheriff D (1984) Epidermal transpiration and stomatal responses to humidity: some hypotheses explored. Plant Cell Environ 7(9):669–677Google Scholar
Smirnoff N (1996) Regulation of Crassulacean acid metabolism by water status in the C3/CAM intermediate Sedum telephium. In: Crassulacean Acid Metabolism, Springer, pp 176–191Google Scholar
Smith J, Schulte P, Nobel P (1987) Water flow and water storage in Agave deserti: osmotic implications of crassulacean acid metabolism. Plant Cell Environ 10(8):639–648CrossRefGoogle Scholar
Sperry J, Hacke U, Oren R, Comstock J (2002) Water deficits and hydraulic limits to leaf water supply. Plant Cell Environ 25(2):251–263PubMedCrossRefGoogle Scholar
Strogatz S (2001) Nonlinear dynamics and chaos: with applications to physics, biology, chemistry and engineeringGoogle Scholar
Surendran Nair S, Kang S, Zhang X, Miguez F E, Izaurralde R C, Post W M, Dietze M C, Lynd L R, Wullschleger S D (2012) Bioenergy crop models: descriptions, data requirements, and future challenges. GCB Bioenergy 4(6):620–633CrossRefGoogle Scholar
Ting I, Patel A, Kaur S, Hann J, Walling L (1996) Ontogenetic development of Crassulacean acid metabolism as modified by water stress in peperomia. In: Crassulacean Acid Metabolism, Springer, pp 204–215Google Scholar
Vico G, Porporato A (2008) Modelling C3 and C4 photosynthesis under water-stressed conditions. Plant Soil 313(1–2):187–203CrossRefGoogle Scholar
Wilkins MB (1992) Tansley review No. 37 circadian rhythms: their origin and control. New Phytol 347–375Google Scholar
Yamori W, Hikosaka K, Way DA (2014) Temperature response of photosynthesis in C3, C4, and CAM plants: temperature acclimation and temperature adaptation. Photosynth Res 119(1–2):101–117PubMedCrossRefGoogle Scholar
Zañudo-Hernández J, González del Castillo Aranda E, Ramírez-Hernández BC, Pimienta-Barrios E, Castillo-Cruz I, Pimienta-Barrios E (2010) Ecophysiological responses of Opuntia to water stress under various semi-arid environments. J Prof Assoc Cactus Dev 12:20–36Google Scholar
Zotz G, Patiño S, Tyree MT (1997) Water relations and hydraulic architecture of woody hemiepiphytes. J Exp Bot 48(10):1825–1833CrossRefGoogle Scholar

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Authors and Affiliations

Mark S. Bartlett
- 1
Giulia Vico
- 2
Amilcare Porporato
- 1
Email author

1.Department of Civil and Environmental EngineeringDuke UniversityDurhamUSA
2.Department of Crop Production EcologySwedish University of Agricultural SciencesUppsalaSweden

Coupled carbon and water fluxes in CAM photosynthesis: modeling quantification of water use efficiency and productivity

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