Introduction

Photosynthesis is achieved through interconnected biophysical and biochemical processes. The biophysical processes, which include light absorption, energy transfer to the water splitting reaction centers and the efficiency of the subsequent transfer of electrons through the electron transport chain, fundamentally determine the photosynthetic electron transport rate (ETR, e.g. White and Critchley 1999; Ralph and Gademann 2005; Abramavicius and Mukamel 2010). ETR generates energy (ATP) and reducing power (NADPH) required for photosynthetic carbon reduction and photorespiratory carbon oxidation (Cheng et al. 2001; Nelson and Yocum 2006); however, the ETR is also directly (electron exchange) or indirectly (through provision of ATP) tied to additional pathways; for example, nitrate reduction (Turpin et al. 1988; Cheng et al. 2001; Eichelmann et al. 2011), direct reduction of O2 in the Mehler reaction (Asada 1999, 2000; Badger et al. 2000; Ort and Baker 2002; Miyake 2010) and terminal oxidases (Streb et al. 2005; Bailey et al. 2008; Miyake 2010; Hemschemeier and Happe 2011). As such, biophysical processes driving ETR play a critical role in attempting to maintain optimum photosynthetic rates and ensuring an effective flow of ATP and NADPH to pathways required to fix inorganic carbon into organic skeletons for growth and/or maintenance.

Photosynthesis research has long focused on ETR through photosystem II (PSII) given its core role in water splitting (and hence photosynthesis) but also since PSII activity can be conveniently assayed via bio-optical techniques, e.g. chlorophyll fluorescence (e.g. Long and Bernacchi 2003; Buckley and Farquhar 2004; Papageorgiou and Govindjee 2004; Robakowski 2005; Baker 2008; Suggett et al. 2011; Pavlovič et al. 2011). Numerous studies have used these fluorescence techniques to determine ETR; however, in particular for optically thick organisms such as higher plants, most have been limited by an inability to evaluate the photosynthetic rates generated relative to the physical processes describing light absorption via the photosynthetic pigment matrix (e.g. Govindjee 1990); specifically, the light absorption cross-section of the pigment–reaction centre complex, average lifetime of pigment molecules in the excited state, degeneration of the energy level of photosynthetic pigment molecules in the ground state versus the excited state, and hence the overall transfer efficiency of excitons from light harvesting through to the core PSII reaction centres. Quantifying the nature and variability of the steps involved from light absorption to water splitting is still extremely limited but remains a key goal in order to improve models for predicting productivity from measurements of light absorption alone (e.g. Platt et al. 1995; Sathyendranath et al. 1995; Renk et al. 2000; Buckley and Farquhar 2004).

In the past 30–40 years, models of varying complexity have been constructed to describe the light response of photosynthesis; for example single exponential or tangent models (e.g. Webb et al. 1974; Jassby and Platt 1976), double exponential models (Platt et al. 1980), rectangular (e.g. Thornley 1998) and non-rectangular hyperbola (e.g. Farquhar et al. 1980) models (see also Eilers and Peters 1993). Common to all these models is how bulk properties of absorption may affect light-limited versus light-saturated photosynthesis; indeed such properties are already used in primary productivity models for optically thin microalgae (e.g. Platt et al. 1995; Renk et al. 2000). However, aside from mechanistic models describing the flow of electrons through PSII (Fasham and Platt 1983) and CO2 fixation processes via the Calvin Cycle (Sharkey et al. 2007), no previous light-response model has considered the core mechanisms determining light harvesting and subsequent productivity (ETR) via PSII. We therefore developed a mechanistic model for ETR based on the fundamental properties of light absorption and transfer of energy to the reaction centers via photosynthetic pigment molecules (see Ye 2012). Importantly, here we produce a novel approach to parameterize these light harvesting properties and hence for the first time are able to evaluate how light harvesting complexes may be modified between species and environments. We present the development of this model and its application to a range of organisms (vascular plants and microalgae) from previously published data of the light response of ETR for PSII.

Model development

Oxygenic photosynthesis via PSII is driven by light absorption from chromophoric pigment molecules, primarily chlorophyll a and phycobiliproteins, accessory chlorophylls and carotenoids (Jansson 1994; Valkunas et al. 1995; La Roche et al. 1996; Hu et al. 2002; MacIntyre et al. 2002; Richter et al. 2008). Absorbed energy is then transferred to the reaction centre where it drives primary charge separation (Oxborough 2004; Cogdell et al. 2006; Richter et al. 2008; Fassioli et al. 2009; Sener et al. 2011). Upon each transfer of an exciton between molecules, the electron in the receptor molecule raises to a higher energetic state whilst that of the donor returns to the ground state. Chromophoric pigments within the pigment bed are composed of both “photosynthetic” pigments and “photoprotective” pigments; the ultimate net transfer of energy to P680 (i.e. photosynthesis) versus emission as heat (photoprotection) is determined by the transfer efficiency of the individual pigment types but modified according to how they are inter-connected to each other within the pigment bed (e.g. Sener et al. 2011; see Suggett et al. 2004 and references within). Therefore, the core PSII chlorophyll a molecules (P680) will be excited to P680* when net energy transfer through this chromophoric pigment bed is sufficient to raise the P680 energetic state, P680* and subsequently donate an electron to pheophytin and in turn the various intermediary carriers in the electron transport chain according to their reduction potentials (Govindjee 1990; Baker 2008).

To simplify the question we address here, we will work under the fundamental premise that an exciton is initially generated within the associated pigment molecule matrix of PSII (and PSI) (Oxborough 2004); this exciton is then lost along one of three de-excitation pathways, photochemistry, heat dissipation and fluorescence (Govindjee 1990; Walters and Horton 1991; Oxborough 2004; Papageorgiou and Govindjee 2004; Baker 2008), operating in direct competition (Fig. 1). Thus the probability of photochemistry, non-radiation heat dissipation and fluorescence (respectively ξ 1, ξ 2 and ξ 3 (dimensionless)), ξ 1 + ξ 2 + ξ 3 = 1.

Fig. 1
figure 1

Simplified schematic of two-level model of a photosynthetic pigment molecule (Ye 2012). E i and E k denote the energy level of the photosynthetic pigment molecules in the ground state (i) and the excited state (k), respectively. n i and n k are the concentration (per unit volume), and g i and g k are degeneration of energy level, of photosynthetic pigment molecule for the i and k states, respectively

Full size image

Light absorption coefficient of photosynthetic pigment molecules

Equations theoretically describing the total light absorption coefficient of photosynthetic pigment molecules, the effective optical absorption cross-section of photosynthetic pigment molecules, the total numbers of photosynthetic pigment molecules in the excited states, photochemical quenching of chlorophyll fluorescence and non-photochemical quenching of chlorophyll fluorescence have been recently developed (Ye 2012); specifically, the total light absorption coefficient of pigment molecules based on uniform optical absorption in leaves (α t) can be defined as,

$$ \alpha_{t} = \frac{{\sigma_{\text{ik}} n_{0} }}{{1 + \frac{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}I}}\left[ {1 - \frac{{\left( {1 - {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}I} \right], $$
(1)

where σ ik is eigen-absorption cross-section from the ground state i to the excited state k in photosynthetic pigment molecules (σ ik expresses the ability to absorb light by photosynthetic pigment molecules), which importantly is dependent upon the light frequency (wavelength), but not light intensity, and n 0 is the total number of photosynthetic pigment molecules per unit volume. R 1 is the rate of decay from the excited state to the ground state to drive primary charge separation; similarly, R 2 is the rate of decay of pigment molecules from the excited to the ground state due to non-radiative heat dissipation. g i and g k are degeneration of the energy level of photosynthetic pigment molecules in the ground state i and the excited state k, respectively. \( \tau \) is the average lifetime of the photosynthetic pigment molecules in an excited state (it expresses the retention time of exciton in an excited state), and I is the light intensity, and finally ξ 1, ξ 2 and ξ 3 are the probability of photochemistry, non-radiation heat dissipation and fluorescence, respectively.

According to Eq. 1, α t increases with R 1, R 2, ξ 1, ξ 2 and ξ 3 increasing. Furthermore, since g i , g k , τ, R 1, R 2, n 0, ξ 1, ξ 2, ξ 3 and I are all positive, \( \alpha_{\text{t}} \) will nonlinearly decrease with increasing values for I. Thus, the light absorption coefficient of photosynthetic pigment molecules is not a constant according to Eq. 1. In the case of optical absorption via an alternative light frequency (wavelength), α t will be further modified to yield a total optical absorption coefficient of photosynthetic pigment molecules (α T) such that,

$$ \begin{aligned} \alpha_{\text{T}} = \sum\limits_{l} {\alpha_{{{\text{t}}(l)}} } = \sum\limits_{l} {\frac{{\sigma_{{{\text{ik}}(l)}} n_{0} }}{{1 + \frac{{\left( {1 + {{g_{i(l)} } \mathord{\left/ {\vphantom {{g_{i(l)} } {g_{k(l)} }}} \right. \kern-\nulldelimiterspace} {g_{k(l)} }}} \right)\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I}}{{\xi_{3(l)} + \left( {\xi_{1(l)} R_{1(l)} + \xi_{2(l)} R_{2(l)} } \right)\tau_{(l)} }}}}\left[ {1 - \frac{{\left( {1 - {{g_{i(l)} } \mathord{\left/ {\vphantom {{g_{i(l)} } {g_{k(l)} }}} \right. \kern-\nulldelimiterspace} {g_{k(l)} }}} \right)\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I}}{{\xi_{3(l)} + \left( {\xi_{1(l)} R_{1(l)} + \xi_{2(l)} R_{2(l)} } \right)\tau_{(l)} }}} \right]} . \\ \end{aligned} $$
(2)

To differentiate eigen-absorption cross-section of photosynthetic pigment molecules between the effective optical absorption cross-section of photosynthetic pigment molecules (termed \( \sigma^{\prime}_{\text{ik}} \)), \( {{\alpha_{\text{t}} } \mathord{\left/ {\vphantom {{\alpha_{\text{t}} } {n_{0} }}} \right. \kern-\nulldelimiterspace} {n_{0} }} \) is defined as \( \sigma^{\prime}_{\text{ik}} \), it expresses the effective optical absorption cross-section of photosynthetic pigment molecules when light intensity (I) > 0, and thus,

$$ \sigma^{\prime}_{\text{ik}} = \frac{{\sigma_{\text{ik}} }}{{1 + \frac{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau I}}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}}}\left[ {1 - \frac{{\left( {1 - {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau I}}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}} \right]. $$
(3)

As with α t, \( \sigma^{\prime}_{\text{ik}} \) increases with R 1, R 2, ξ 1, ξ 2 and ξ 3 increasing, and \( \sigma^{\prime}_{\text{ik}} \)will nonlinearly decrease with increasing values for I. The effective optical absorption cross-section of photosynthetic pigment molecules \( \sigma^{\prime}_{\text{ik}} \) = σ ik when I = 0. Again, for a case of optical absorption via a different light frequency (wavelength), the total effective optical absorption cross-section of photosynthetic pigment molecules (\( \sigma^{\prime}_{\text{T}} \)) will be,

$$ \begin{aligned} \sigma^{\prime}_{\text{T}} = \frac{{\sum\nolimits_{l} {\alpha_{{{\text{t}}(l)}} } }}{{n_{0} }} = \sum\limits_{l} {\frac{{\sigma_{{{\text{ik}}(l)}} }}{{1 + \frac{{\left( {1 + {{g_{i(l)} } \mathord{\left/ {\vphantom {{g_{i(l)} } {g_{k(l)} }}} \right. \kern-\nulldelimiterspace} {g_{k(l)} }}} \right)\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I}}{{\xi_{3(l)} + \left( {\xi_{1(l)} R_{1(l)} + \xi_{2(l)} R_{2(l)} } \right)\tau_{(l)} }}}}\left[ {1 - \frac{{\left( {1 - {{g_{i(l)} } \mathord{\left/ {\vphantom {{g_{i(l)} } {g_{k(l)} }}} \right. \kern-\nulldelimiterspace} {g_{k(l)} }}} \right)\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I}}{{\xi_{3(l)} + \left( {\xi_{1(l)} R_{1(l)} + \xi_{2(l)} R_{2(l)} } \right)\tau_{(l)} }}} \right]} . \\ \end{aligned} $$
(4)

It is also possible to express the total number of photosynthetic pigment molecules in the excited state as,

$$ N_{k} = \frac{{\sigma_{\text{ik}} \tau I}}{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau I + \left( {\xi_{3} + \xi_{1} R_{1} \tau + \xi_{2} R_{2} \tau } \right) + 1}}N_{0} . $$
(5)

In Eq. 5, N k increases with \( \tau \) but also with I until N k equals the maximum value N 0/(1 + g i /g k ). Once again, optical absorption via a different light frequency (wavelength) will modify the total number of photosynthetic pigment molecules in the excited state as,

$$ N^{\prime}_{k} = \sum\limits_{l} {\frac{{\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I}}{{\left[ {1 + {{g_{i(l)} } \mathord{\left/ {\vphantom {{g_{i(l)} } {g_{k(l)} }}} \right. \kern-\nulldelimiterspace} {g_{k(l)} }}} \right]\sigma_{{{\text{ik}}(l)}} \tau_{(l)} I + \left[ {\xi_{3(l)} + \xi_{1(l)} R_{1(l)} \tau_{(l)} + \xi_{2(l)} R_{2(l)} \tau_{(l)} } \right] + 1}}N_{0} } . $$
(6)

Using A ki = 1/τ (A ki is the Einstein’s coefficient for spontaneous emission, and τ is the inverse of the average lifetime of photosynthetic pigment molecules in the excited state k), R 1, R 2, ξ 1, ξ 2 and ξ 3, the photochemical quenching of chlorophyll fluorescence (q P) can be calculated by:

$$ q_{\text{P}} = \frac{{R_{1} \xi_{1} }}{{R_{1} \xi_{1} + R_{2} \xi_{2} + {{\xi_{3} } \mathord{\left/ {\vphantom {{\xi_{3} } \tau }} \right. \kern-\nulldelimiterspace} \tau }}}. $$
(7)

And the non-photochemical quenching of chlorophyll fluorescence (q N) can be calculated by:

$$ q_{\text{N}} = \frac{{R_{2} \xi_{2} + {{\xi_{3} } \mathord{\left/ {\vphantom {{\xi_{3} } \tau }} \right. \kern-\nulldelimiterspace} \tau }}}{{R_{1} \xi_{1} + R_{2} \xi_{2} + {{\xi_{3} } \mathord{\left/ {\vphantom {{\xi_{3} } \tau }} \right. \kern-\nulldelimiterspace} \tau }}}. $$
(8)

More detailed derivation of Eqs. 1–8 is described by Ye (2012).

Photosynthetic electron transport rate through PSII

For simplicity here we assume uniform light intensity through an optically dense layer (e.g. a leaf) and hence the photosynthetic pigment molecules absorb this light uniformly. Photons absorbed by the leaf can thus be described as \( \alpha^{\prime}\) IS (μmol photons s−1), where \( \alpha^{\prime}\) is the leaf absorptance (dimensionless), I is light intensity (μmol photons m−2 s−1), S is the measured leaf area (m2). Photons absorbed per unit leaf thickness will be α′ISα t (μmol photons m−1 s−1), where α t (m−1) is the total light absorption coefficient of the photosynthetic pigment molecules for uniform case of optical absorption, see above. Furthermore, photons absorbed by of the photosynthetic pigment molecules in leaf with thickness d are α′ISdα t (μmol photons s−1), where d is the leaf thickness (m), and photons absorbed by PSII are α′β′ISdα t (μmol photons s−1), β′ is the fraction of light absorbed by PSII (dimensionless). Therefore, photons absorbed by photosynthetic pigment molecules and transported to PSII per unit of time (termed N′) are given as,

$$ N^{\prime} = \alpha^{\prime}\beta^{\prime}ISd\alpha_{t} . $$
(9)

Assuming that the efficiency of exciton transfer via PSII reaction centres for charge separation of P680 is φ [μmol electrons (μmol photons)−1], then N′φ (μmol electrons s−1) will yield the number of electrons via charge separation of P680. However, since the current unit of ETR is μmol electrons m−2 s−1, using Eq. 9 to calculate ETR requires that Eq. 9 must first be divided by the measured leaf area S,

$$ {\text{ETR}} = \varphi N^{\prime}/S $$
(10)

where ETR is the photosynthetic electron transport rate via PSII (μmol electrons m−2 s−1).

Substituting Eq. 9 into 10 gives the ETR as,

$$ {\text{ETR}} = \alpha^{\prime}\beta^{\prime}\varphi Id\alpha_{\text{t}} = \alpha^{\prime}\beta^{\prime}\varphi IV\alpha_{\text{t}} /S, $$
(11)

where V (V = dS) is volume of the measured leaf. Substituting Eq. 1 into 11, further enables ETR to be calculated as,

$$ {\text{ETR}} = \frac{{\alpha^{\prime}\beta^{\prime}N_{0} \sigma_{\text{ik}} \varphi }}{S} \times \frac{{1 - \frac{{\left( {1 - {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}I}}{{1 + \frac{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }}I}}I, $$
(12)

where N 0 is total photosynthetic pigment molecules of the measured leaf. Equation 12 shows that ETR is dependent on α′, β′, N 0, σ ik, τ, φ, R 1, R 2, g i , g k , ξ 1, ξ 2, ξ 3 and I.

For any given species, σ ik, τ, ξ 1, ξ 2, ξ 3, g i , g k , R 1 and R 2 will inherently have specific values in Eq. 12 for any given and constant environmental conditions (e.g. temperature, CO2 concentration and relative humidity). Therefore, we can assume \( \alpha = \frac{{\alpha^{\prime}\beta^{\prime}N_{0} \sigma_{\text{ik}} \varphi }}{S} \) (m2 s μmol electrons [μmol photons]−1), \( \beta = \frac{{\left( {1 - {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }} \) (m2 s [μmol photons]−1), \( \gamma = \frac{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }} \) (m2 s [μmol photons]−1), so that Eq. 12 can be simplified as,

$$ {\text{ETR}} = \alpha \frac{1 - \beta I}{1 + \gamma I}I, $$
(13)

where α is the initial slope, β is the extent of dynamic down-regulation of PSII, and γ is defined as a saturation term of light response curve for photosynthetic electron transport rate (ETR-I). According to Eqs. 12 and 13, ETR increases with I before saturation irradiance but decreases thereafter via dynamic down-regulation of PSII, i.e. the typical pattern expected for ETR versus I response curves (White and Critchley 1999; Ralph and Gademann 2005)

The first derivative of Eq. 13 is

$$ \frac{{d{\text{ETR}}}}{dI} = \alpha \frac{{1 - 2\beta I - \beta \gamma I^{2} }}{{\left( {1 + \gamma I} \right)^{2} }}, $$
(14)

here, dETR/dI decreases as I increases. dETR/dI is initially high as I is increased from the minimum light intensity but goes to zero when I reaches the (saturation) light intensity (termed here as PARsat) corresponding to maximum ETR and finally is < 0 as I increases above PARsat. Where dETR/dI = 0, PARsat is calculated from Eq. 14 as,

$$ {\text{PAR}}_{\text{sat}} = \frac{{\sqrt {{{\left( {\beta + \gamma } \right)} \mathord{\left/ {\vphantom {{\left( {\beta + \gamma } \right)} \beta }} \right. \kern-\nulldelimiterspace} \beta }} - 1}}{\gamma }, $$
(15)

Thus PARsat corresponds with σ ik, τ, R 1, R 2, g i , g k , ξ 1, ξ 2 and ξ 3. Substituting Eq. 15 into 13, the maximum \( {\text{ETR}}\left( {{\text{ETR}}_{ \max } } \right) = \alpha \frac{{1 - \beta \cdot {\text{PAR}}_{\text{sat}} }}{{1 + \gamma \cdot {\text{PAR}}_{\text{sat}} }}{\text{PAR}}_{\text{sat}} , \) can be simplified as,

$$ {\text{ETR}}_{\max } = \alpha \left( {\frac{{\sqrt {\beta + \gamma } - \sqrt \beta }}{\gamma }} \right)^{2} , $$
(16)

Using \( \beta = \frac{{\left( {1 - {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }} \) and \( \gamma = \frac{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} \tau }}{{\xi_{3} + \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)\tau }} \), g i /g k is calculated by,

$$ \frac{{g_{\text{i}} }}{{g_{\text{k}} }} = \frac{\gamma - \beta }{\gamma + \beta } $$
(17)

Applying different values of β and γ to Eq. 13 will inherently affect the ETR–I curve (Fig. 2). Notably increasing values of both β and γ decreases ETRmax; however, decreasing values of β also increase the likelihood of inhibition. Overall, the effect of β values is greater than that of γ values on ETR.

Fig. 2
figure 2

Model responses of the photosynthetic electron transport rate (ETR, μmol electrons m−2 s−1) versus light intensity (I, μmol photons m−2 s−1) determined for different values of β and γ but where α = 0.2 μmol electrons (μmol photons)−1

Full size image

Thus Eqs. 12 and 13 (Fig. 2) closely reproduces the generally observed trend of the ETR–I curves (e.g. White and Critchley 1999; Ralph and Gademann 2005); specifically (i) an initial zone of low-intensity illumination in which ETR increases linearly with irradiance I, the maximum rate of ETR asymptotically at saturation light intensity, and (ii), a decrease in ETR with irradiance beyond the saturation light intensity via dynamic down-regulation of PSII when α and γ are kept constant in (Fig. 2, left panel). These components, respectively, describe light-dependent, light-saturated and high light-inhibited ETR that are commonly denoted by α, ETRmax and β in pre-existing models describing the ETR-I curves (Platt et al. 1980; Eilers and Peters 1993). Equations 12 and 13 can also well describe ETR–I curves that are characterised by only two phases, i.e. no dynamic PSII down-regulation under high light, when α and β are kept constant (Fig. 2, right panel).

σ ik can be estimated via \( \alpha = \frac{{\alpha^{'} \beta^{\prime}N_{0} \sigma_{\text{ik}} \varphi }}{S} \) since N 0 and S can be measured, and α′ can be measured using an integrating sphere (with a value typically returned of 0.84, Ehleringer 1981; Rascher et al. 2000; Lüttge et al. 2003; Evans 2009), and with β′ typically assumed to be 0.5 (Ehleringer and Pearcy 1983; Maxwell and Johnson 2000; Major and Dunton 2002; Evans 2009). Furthermore, α can be calculated through fitting ETR–I curves via Eq. 12 such that,

$$ \sigma_{\text{ik}} = \frac{S\alpha }{{\alpha^{\prime}\beta^{\prime}\varphi N_{0} }} $$
(18)

In addition, τ can be estimated by β or γ. Therefore, since β and γ can be calculated through fitting ETR–I curves via Eq. 13, we can also calculate τ as,

$$ \tau = \frac{{\frac{{\gamma \xi_{3} }}{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} }}}}{{1 - \frac{{\gamma \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)}}{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} }}}} $$
(19)

In theory the average lifetime can be calculated by Eq. 19. However, at present we are not able to measure directly R 1, R 2, ξ 1, ξ 2 and ξ 3. In order to estimate the magnitude of the average lifetime it is possible to calculate the minimum average lifetime (τ min) in an excited state. Since \( 0 < 1 - \frac{{\gamma \left( {\xi_{1} R_{1} + \xi_{2} R_{2} } \right)}}{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} }} < 1 \), Eq. 19 can be rearranged to estimate τ,

$$ \tau > \frac{{\gamma \xi_{3} }}{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} }} $$
(20)

For live plants, the maximum fluorescence yield in the photosynthetic apparatus is about 3–5 % (Krause and Weis 1991). By using the minimum value, and hence ξ 3 of 3 %, τ min is calculated as,

$$ \tau_{\min } > \frac{0.03\gamma }{{\left( {1 + {{g_{i} } \mathord{\left/ {\vphantom {{g_{i} } {g_{k} }}} \right. \kern-\nulldelimiterspace} {g_{k} }}} \right)\sigma_{\text{ik}} }} $$
(21)

where the units for γ in Eq. 21 are in [m2 s (photosynthetic pigment molecule)−1].

Examples of model application

Details of the growth conditions, species examined, and procedures for generating the ETR-I curves to be examined with our model have already been described (the conifers Abies alba Mill., Picea abies Karst., and Pinus mugo Turra, Robakowski 2005; the coccolithophore Emiliania huxleyi, Suggett et al. 2007). Representative ETR–I curves (fitting the model of Eq. 12) for the three conifer species are given in Fig. 3. For A. alba (Fig. 3a) and P. abies (Fig. 3b). ETR initially increased (almost linearly) with I towards saturation and subsequently, at the highest irradiances, exhibited a sharp decline. P. mugo exhibited little decline of ETR with increasing I beyond the saturation light intensity (Fig. 3c). In their natural geographic ranges, P. mugo grows in high altitudes (the upper limit of its range in Central Europe is 1,800 m a.s.l., but has been found as high as 2,400 m a.s.l) and thus is adapted to high irradiance at high elevations; however, A. alba is a low elevation mountain species (upper limit of ca. 1,100 m a.s.l in Polish mountains) and is the most shade-tolerant tree species in European forests except for Taxus baccata L; P. abies is most commonly found in a subalpine zone but at elevations less than those for P. mugo. Thus, as expected, the shape of ETR-I curves (Fig. 3) but also the estimates of PARsat obtained from Eq. 15 changed among the species in the order corresponding to their decreasing shade tolerance; i.e. values of PARsat where A. alba < P. abies < P. mugo.

Fig. 3
figure 3

Model fits to the light response of photosynthetic electron transport rate (ETR, μmol electrons m−2 s−1) for three conifer species A. alba (a), P. abies (b), and P. mugo (c) (original data taken from Robakowski 2005). Circle indicates measured points and solid line denotes fitted by Eq. 12 or 13

Full size image

Figure 4 shows further fits of our model (Eq. 12) to the microalgal ETR–I data of Suggett et al. (2007) for the coccolithophore Emiliania huxleyi strain B11, grown at three different light intensities (at I = 600, 300 and 150 μmol photons m−2 s−1). Here, the model fit well to the data and generally returned a pattern expected as cells acclimated increasing growth light intensities, notably an increase or ETRmax from ca. 45 to 75 μmol electrons m−2 s−1 from the lowest growth irradiances. Values of PARsat were also lowest (370 μmol photons m−2 s−1) at the lowest growth irradiance (Fig. 4c) but highest (633 μmol photons m−2 s−1) at the intermediate growth irradiance (Fig. 4b) likely reflecting an onset of down-regulation of ETR, as seen at where I is >PARsat for cells at the lowest growth irradiance.

Fig. 4
figure 4

Model fits to the light response of photosynthetic electron transport rate (ETR, μmol electrons m−2 s−1) for E. huxleyi grown under three different light intensities of 600 (a), 300 (b), and 150 (c) μmol photons m−2 s−1 (original data taken from Suggett et al. 2007). Circle indicates measured points and solid line denotes fitted by Eq. 12 or 13

Full size image

We further compared the ETR–I fit from our model with fits returned from an established ETR–I model. Most commonly, a double exponential decay function (Platt et al. 1980) and a single exponential function (Harrison and Platt 1986) have been used previously to describe (and parameterize) ETR–I curves in algae, coral and plants (e.g. Platt et al. 1980; Harrison and Platt 1986; Henley 1993; Rascher et al. 2000; Ralph and Gademann 2005; Karageorgou and Manetas 2006), for example,

$$ {\text{ETR}} = {\text{ETR}}_{\text{s}} \left( {1 - \exp \left( {{{ - \alpha I} \mathord{\left/ {\vphantom {{ - \alpha I} {{\text{ETR}}_{\text{s}} }}} \right. \kern-\nulldelimiterspace} {{\text{ETR}}_{\text{s}} }}} \right)} \right)\exp \left( {{{ - \beta I} \mathord{\left/ {\vphantom {{ - \beta I} {{\text{ETR}}_{\text{s}} }}} \right. \kern-\nulldelimiterspace} {{\text{ETR}}_{\text{s}} }}} \right) $$
(22)

where ETRs is a scaling factor defined as the maximum potential relative ETR, β (>0) is photoinhibition term (Harrison and Platt 1986) or dynamic down-regulation of PSII (Ralph and Gademann 2005) or characterizes the slope of the ETR–I where PSII declines (Henley 1993), α (>0) is the initial slope of the ETR–I curve before the onset of saturation, and I is light intensity. In the absence of the photoinhibition or the dynamic down-regulation of PSII (β = 0), the function becomes a single exponential function (Harrison and Platt 1986), and Eq. 22 can be simplified to,

$$ {\text{ETR}} = {\text{ETR}}_{\text{m}} \left( {1 - \exp \left( {{{ - \alpha I} \mathord{\left/ {\vphantom {{ - \alpha I} {{\text{ETR}}_{\text{m}} }}} \right. \kern-\nulldelimiterspace} {{\text{ETR}}_{\text{m}} }}} \right)} \right) $$
(23)

where ETRm is the photosynthetic capacity at saturation light intensity (Ralph and Gademann 2005). In the double exponential decay function (Platt et al. 1980; Ralph and Gademann 2005), the following parameters ETRmax and PARsat were calculated using the following equations, respectively, namely,

$$ {\text{PAR}}_{\text{sat}} = \frac{{{\text{ETR}}_{\text{s}} }}{\alpha }\ln \frac{\alpha + \beta }{\beta } $$
(24)
$$ {\text{ETR}}_{\max } = {\text{ETR}}_{\text{s}} \frac{\alpha }{\alpha + \beta }\left( {\frac{\beta }{\alpha + \beta }} \right)^{{{\beta \mathord{\left/ {\vphantom {\beta \alpha }} \right. \kern-\nulldelimiterspace} \alpha }}} $$
(25)

An example of estimated values of the parameters for our model (Eqs. 12, 15, 16) and a previously used model (Eqs. 22, 24, 25), i.e. including photoinhibition/dynamic down-regulation of PSII, fitted to the data in Figs. 3b and 4b are given in Table 1. The fitted results in Table 1 show that β < 0 in Fig. 4b so that PARsat and ETRmax cannot be calculated by Eqs. 24 and 25, respectively. To fit ETR–I curve for E. huxleyi at I = 300 μmol photons m−2 s−1, the single exponential function (i.e. Eq. 23) need to be used. However, Eq. 23 is not able to describe ETR–I curves in A. alba and P. abies, as well as E. huxleyi at I = 600 and 150 μmol photons m−2 s−1 since their ETR-I curves decline with I increasing when I is beyond PARsat.

Table 1 Comparison of results fitted by Eqs. 12 and 22 with measured data in P. abies and E. huxleyi (at I = 300 μmol photons m−2 s−1), respectively
Full size table

Discussion

We have theoretically investigated the relationship between ETR and I based on the underlying physical parameters governing photosynthetic pigment molecules. Fitting this model to previously collected data (of varying degrees of photosynthetic acclimation and PSII down-regulation) not only returned extremely good fits (R 2 > 0.99), but also returned values for ETRmax, α and PARsat that were consistent with those from more established models that only return knowledge of ‘bulk’ absorption properties. A number of studies have previously compared parameter returns from different ETR–I models (e.g. Jassby and Platt 1976; Frenette et al. 1993) and not surprisingly alternative models, e.g. single versus double exponential models, will inevitably return different fits (Frenette et al. 1993); the model (and in turn the accuracy of the fit) may inevitably depend on the quality of the data, e.g. whether sufficient PSII down-regulation preclude accurate use of a single exponential model. Therefore, a major advantage of our model, aside from the additional wealth of mechanistic information that can ultimately be returned, is that it can be employed widely across data sets with differing extent of PSII down-regulation.

For the model presented here, \( \sigma^{\prime}_{\text{ik}} \) decreases (whilst N k increases) with increasing I. More importantly, both σ ik and τ (and τ min) can be estimated by our model. For any given species we can determine the light absorption ability of photosynthetic pigment molecules by Eq. 18, and determine the minimum retention time of the exciton in an excited state by Eq. 21. In this case, more photosynthetic pigment molecules are occupied in excited states with longer values of τ min. Furthermore, numbers of the photosynthetic pigment molecules in an excited state increase with I. Such observations may help to investigate the underlying nature of plants’ photoprotection (e.g. Murchie and Niyogi 2011) under high light intensity since many photosynthetic pigment molecules occupied in an excited states will prevent the molecules from absorbing light energy further and could thus provide a novel means for additional photoprotective pathway(s) in plants beyond heat dissipation and chlorophyll fluorescence emission alone (Ye 2012). Here, transient bleaching of photosynthetic organisms cannot occur since photosynthetic pigment molecules in an excited state under high light would decay to the ground state (chloroplasts could move to cytoplasmic positions corresponding to anticlinal cell walls); however, this notion needs further investigation. In addition, the model outputs help to investigate dynamic down-regulation of PSII since \( \sigma^{\prime}_{\text{ik}} \) decreases with increasing I thereby reducing light absorption ability of the photosynthetic pigment molecules and ETR decreasing with increasing I. However, that said, it is still clear that an important element in more widespread use of this model will be the requirement to independently validate that parameterization of the properties generated by the model, e.g. PARsat and ETRmax (Eqs. 15 and 16), against more established approaches from other species and different environmental conditions that will modify the shape of the light response function (photoacclimation and stress). Beyond validating the light harvesting properties returned by the model, we also realise that the mechanistic model for ETR–I may be further improved since we presently do not know whether environmental factors, such as temperature and CO2 concentration will influence harvesting properties of photosynthetic pigment molecules and ETR.

In conclusion, the present proposed model provides a means to predict and simulate the ETR-I curves, and could become a key tool towards identifying novel mechanistic properties by which plants and algae modify their light harvesting properties underlying photoacclimation and stress. An important next step will be to further develop this current mechanistic model of ETR–I towards a dynamic model of the light response of photosynthesis (P n – I). In particular, the model will need to be parameterized for many more species and environmental conditions, including light so that biophysical and molecular mechanisms of photochemical adaptation to irradiance can be better understood.