Curvature in models of the photosynthesis-irradiance response


  • Copyright changed July 3, 2014


An equation for the rate of photosynthesis as a function of irradiance introduced by T. T. Bannister included an empirical parameter b to account for observed variations in curvature between the initial slope and the maximum rate of photosynthesis. Yet researchers have generally favored equations with fixed curvature, possibly because b was viewed as having no physiological meaning. We developed an analytic photosynthesis-irradiance equation relating variations in curvature to changes in the degree of connectivity between photosystems, and also considered a recently published alternative, based on changes in the size of the plastoquinone pool. When fitted to a set of 185 observed photosynthesis-irradiance curves, it was found that the Bannister equation provided the best fit more frequently compared to either of the analytic equations. While Bannister's curvature parameter engendered negligible improvement in the statistical fit to the study data, we argued that the parameter is nevertheless quite useful because it allows for consistent estimates of initial slope and saturation irradiance for observations exhibiting a range of curvatures, which would otherwise have to be fitted to different fixed-curvature equations. Using theoretical models, we also found that intra- and intercellular self-shading can result in biased estimates of both curvature and the saturation irradiance parameter. We concluded that Bannister's is the best currently available equation accounting for variations in curvature precisely because it does not assign inappropriate physiological meaning to its curvature parameter, and we proposed that b should be thought of as the expression of the integration of all factors impacting curvature.


electron transport chain


photosynthetic quotient


photosynthetic unit


photosynthetically utilizable irradiance


Webb, Newton, and Starr

The aquatic food web is based on the conversion of inorganic carbon into organic compounds by photosynthetic organisms, including benthic microalgae, macroalgae, seagrasses, and phytoplankton. While the focus here is on phytoplankton, in part because of the relative simplicity of the optical fields that can be assumed for dilute samples of photosynthetic microbes, our findings are based on generally applicable principles and have some relevance to all photosynthetic systems.

Like all photosynthetic organisms, phytoplankton depends on light and nutrients for growth. Accurate estimation of the productive potential of an assemblage of phytoplankton is predicated on accurate quantification of the efficiency with which these resources are utilized, and can be crucial in the assessment of the structure and functioning of an aquatic ecosystem, estimation of the rate of carbon sequestration in climate models, and determination of the commercial feasibility of algal culture for the production of fuel, protein or other products. Estimation of productive potential requires, as an essential component, a model of the rate of photosynthesis, which is typically normalized to chl a concentration (chl a) as a proxy for biomass and represented by PB (μmol C · mg chl a−1 · s−1), as a function of irradiance, E (μmol photons · m−2 · s−1, see Table 1 for a list of symbols and units). Consequently, models of the P B versus E relationship have for many decades been centrally important to models of the rate of photosynthesis and the growth of phytoplankton (Ryther and Yentsch 1957, Saba et al. 2010). When their parameters are appropriately tuned to measurements, PB versus E equations can be used not only to predict the rate of photosynthesis at a given irradiance for a specified concentration of biomass but can also serve to diagnose the mean physiological status of the photosynthetic organisms. Specifically, parameters of a PB versus E equation can be related to the acclimation state of the phytoplankton and serve as indicators of the influences of environmental factors on primary production (Falkowski 1981, Cullen et al. 1992a,b, Henley 1993). Here, we describe an analytic framework in which photosynthetic parameters can be retrieved with improved consistency, and in which the range of physiological interpretations of the functional response is expanded. Photosynthesis is normalized to chl a to provide context (e.g., Jassby and Platt 1976), but our principal findings apply to the shape of the PB versus E relationship (cf., Talling 1957, Cullen et al. 2012) and thus apply to any determination of photosynthesis versus irradiance.

Table 1. Symbols and units.
a* (λ)In vivo absorption spectrum normalized to chl am2 · mg chl a−1
math formula Spectrally averaged chl a–specific absorption coefficientm2 · mg chl a−1
math formula, math formulaNormalized absorption spectrum for intercellular pigments dissolved in solution, its spectrally averaged valuem2 · mg chl a−1
A, BProportion of open and closed PSUsn/a
B eq Steady-state proportion of closed PSUsn/a
b,math formulaCurvature parameter for the Bannister equation, its estimated valuen/a
ci, cIntracellular pigment concentration of an algal cell, concentration in an algal culturekg chl a · m−3
D Diameter of an idealized spherical algal cellμm
E, Ek, EPARIrradiance, saturation irradiance, photosynthetically available radiationμmol photons · m−2 · s−1, W · m−2
math formula, math formulaEstimated saturation irradiance influenced by pigment packaging, and self-shading in an algal cultureμmol photons · m−2 · s−1
Einc (λ), Eout (λ)Incident and transmitted spectral irradianceμmol photons · m−2 · s−1 · nm−1
l (x, y, z)The path length from the illuminated surface of a theoretical spherical algal cell to the interior location (x, y, z)μm
N PSU Number of PSUsμmol O2 · mg chl a−1
P B, math formulaChl a–specific photosynthetic rate, maximum rateμmol C · mg chl a−1 · s−1 or mg C · mg chl a−1 · h−1
PQ Photosynthetic quotientμmol C · μmol O2−1
P Curvature parameters for the Joliot equationTransfer probability
Q Curvature parameters for the Honti equationNo. plastoquinone molecules
Tp (λ), Tp (λ)The theoretical and numerical particulate transmittance spectra, the proportion of the irradiance incident to a given volume that passes through it without being absorbedProportion per nm
T PUR Percent irradiance transmitted%
αBInitial slope of the PB versus E curveμmol C · mg chl a−1 · m2 · μmol photons−1
σPSUOptical absorption cross-section of a PSUm2 · μmol O2−1
σPSIIFunctional absorption cross-section of a PSIIm2 · μmol photons−1
ΤTurnover time for a PSUs
ϕmaxMaximum quantum yield for O2 evolutionμmol O2 · μmol photons−1

In its simplest form, the observed PB versus E response is a saturation curve, some examples of which are shown in Figure 1. At low irradiance, the rate of photon absorption is sufficient to recruit only a small proportion of the available photosynthetic apparatuses—the ensemble of components required for the transfer of electrons from H2O to CO2—at any one time. In this light-limited state, the chl a–specific photosynthetic rate PB responds maximally to small increases in irradiance. When light is saturating, the ensemble of photosynthetic apparatuses functions at its maximum rate (math formula). The transition from the light-limited to the light-saturated state occurs over an interval of intermediate irradiances defined in terms of the saturation irradiance Ek (μmol photons · m−2 · s−1), the point on the abscissa where the extension of the tangent line at E = 0, the slope of which is αB (μmol C · mg chl a−1 · m2 · μmol photons−1), intersects the horizontal line at math formula (Talling 1957). The curvature between the initial slope given by αB and the ceiling imposed by math formula that marks this transition is an expression of the rate of the reduction in the quantum yield of photosynthesis as progressively larger increments in irradiance are required to drive a unit increase in the photosynthetic rate.

Figure 1.

The Blackman equation (1) (dashed line) represents the photosynthetic response to irradiance when there are no inefficiencies in photon utilization. The Baly, WNS, and J&P equations (eqs. (2), (4), and (5) in Table 2) model inefficiencies by their inherent curvatures as quantified by the relative rate of photosynthesis at saturation irradiance, which are 0.50, 0.63, and 0.76 respectively.

The rate of photosynthesis can be reduced below math formula at high irradiance due to damage or destruction to components of the photosynthetic apparatus, a phenomenon known as photoinhibition (Long et al. 1994). Models of the PB versus E response that account for photoinhibition variously do (e.g., Neale and Richerson 1987) and do not (e.g., Platt et al. 1980, Cullen et al. 1992a,b) include a threshold level of irradiance below which photoinhibition does not occur. In the latter case, without an established framework for modeling curvature, it is unclear how inhibitory effects at low irradiance can be separated a priori from changes in curvature that are due to, as will be demonstrated herein, factors (including photoinhibition) that affect the efficiency of light utilization at multiple scales (i.e., at the level of the photosynthetic apparatus, the cell, and the algal culture). Photoinhibition is therefore not included in the models of the PB versus E response examined here, but is left to future efforts.

The PB versus E response has been modeled using a number of equations, exemplified by equations (1)–(6) in Table 2, the forms of some of which are shown in Figure 1. The simplest is the Blackman equation (Blackman 1905), which is bilinear in form and is normally written as a piecewise function of irradiance (e.g., see Jassby and Platt 1976), but here is written more sparingly by making use of the absolute value |E − Ek| (eq. 1). Unlike other PB versus E equations, all of which describe a smooth transition from the light-limited to the light-saturated state, the Blackman equation models this transition as a sudden shift that occurs at Ek. The second equation in Table 2, commonly called the Monod because of the appearance of an equation of the same rectangular hyperbolic form in a study on bacterial growth rates (Monod 1949), is analytically based on an assumption of first-order kinetics. As it was first used in the context of the photosynthesis-irradiance response by Baly (Baly 1935), equation (2) will be referred to as the Baly equation in this paper. A modified version of the Baly equation is the Smith equation (Smith 1936)(eq. 3, applied, for example, by Talling (1957) and Platt et al. (1988) to derive an expression for depth-integrated photosynthesis.

Table 2. A list of PB versus E equations prominent in the literature, recast in terms of the instantaneous rate normalized to chl a biomass, PB (cf. Jassby and Platt 1976), and saturation irradiance, Ek.
(1) math formula Blackman 1905Bilinear
(2) math formula Baly 1935Rectangular Hyperbola
(3) math formula Smith 1936Modified Rectangular Hyperbola
(4) math formula Webb et al. 1974Exponential
(5) math formula Jassby and Platt 1976Hyperbolic Tangent
(6) math formula Bannister 1979Generalized Rectangular Hyperbola

Commonly attributed to Webb et al. (1974) (e.g., Jassby and Platt 1976, Falkowski and Raven 2007), equation (4) has been rationalized using target theory by Dubinsky et al. (1986) and reviewed in the context of models of photosynthesis and growth by Cullen (1990). Although this exponential form has appeared in the literature before 1974 (e.g., Arnold 1932), equation (4) will be referred to as the Webb, Newton, and Starr (WNS) equation in deference to the authors of the 1974 paper (W. L. Webb, M. Newton, and D. Starr). The WNS equation is ubiquitous in the literature, probably because it is mathematically convenient and generally fits observations well. Nevertheless, Jassby and Platt (1976) found that their hyperbolic tangent equation (J&P, eq. 5) provided the best overall fit to close to 200 observed PB versus E curves in a comparison that included the Blackman, Baly, Smith, and WNS equations (Jassby and Platt 1976). The hyperbolic tangent equation is often used as the standard against which to compare the performance of novel PB versus E formulations (e.g., Honti 2007) for this reason.

Bannister (1979) developed a generalization of the Baly equation which contains an additional dimensionless parameter b to describe curvature, resulting in a family of equations of which the Baly equation (b = 1) and the Smith equation (b = 2) are members (eq. 6). The Bannister parameter b affords an extra degree of freedom (df) to equation (6), which consequently can often fit observations better than other PB versus E equations. Despite this, the Bannister equation seldom appears in contemporary studies, perhaps due to the difficulty in assigning a physical meaning to b.

With the exception of the Bannister, all of the equations in Table 2 contain two parameters, Ek and math formula (where αB = math formula/Ek). While these parameters can be adjusted to fit the initial slope and the maximum photosynthetic rate of an observed PB versus E curve, alone they are not sufficient to adjust the curvature between αB and math formula (Honti 2007), which is consequently an inherent property of a two-parameter PB versus E equation. A set of observed PB versus E curves often exhibits variations in curvature that may be due to taxonomic and/or acclimative differences between populations, but the practice of accounting for variations in curvature by using more than one equation to fit such data is problematic because inherent differences in curvature between functional forms can lead to inconsistencies in the estimates for Ek and math formula (Frenette et al. 1993). The introduction of a third parameter into a PB versus E equation to capture variations in curvature should in principle provide consistent parameter estimates. The utility, not to say necessity, of an additional parameter to account for changes in curvature, is therefore apparent. The extra parameter in the Bannister equation represents an empirical answer to this need, but, as mentioned, the Bannister equation has not been widely used, possibly due to its empirical origin. The purpose of our study was to explore analytic PB versus E equations that model variations in curvature via a physiologically meaningful parameter that would allow consistent parameter estimates and potentially provide the basis for enhanced diagnostics of the physiological conditions underlying the PB versus E relationship.

To the best of our knowledge, there is only one example in the literature of an analytic model developed specifically to account for variations in the curvature of the PB versus E relationship: Honti's Q-model (Honti 2007). Electrons generated by photon absorption in photosystem II (PSII) require transport to photosystem I (PSI) before they can be utilized in the reduction of carbon. In the Q-model (as in queue or quinone), the parameter q represents the number of intersystem electron transport molecules (i.e., the size of the plastoquinone pool) associated with a photosynthetic apparatus. Variations in curvature are accounted for by changes in q. Honti demonstrated that his model equation, which is not shown here because it has no closed form, fitted data better than the exponential and hyperbolic tangent equations (Honti 2007); the Bannister equation was not included in the analysis.

In this paper, we propose a second analytic model that accounts for variations in curvature. Our analysis begins with a theoretical comparison between the Baly and WNS equations to motivate the development of an analytic model of the photosynthesis-irradiance response that allows connections between light-harvesting antennae. The seminal work of Joliot and Joliot (1964) is used as a starting point to formulate the model, the solution of which we call the Joliot equation because of this association. Honti's (2007) Q-model is also described (without analytic detail). A comparison is then made between all equations (except the bilinear equation) fitted to the same observations as Jassby and Platt (1976). Although the Joliot equation is shown to have most frequently provided the best fit, with the Bannister a close second, the Bannister equation is shown to provide the better fit significantly more often when compared to the Joliot equation alone. By this result, combined with a consideration of other factors shown to impact curvature (specifically, inter- and intracellular self-shading), we suggest a meaningful interpretation of the Bannister parameter as a scalar quantity representing the integrated effect of all variables that impact curvature.

Materials and Methods

The photosynthetic unit

All of the models presented herein are based upon a convenient theoretical conceptualization of the photosynthetic apparatus commonly referred to as the photosynthetic unit (PSU), which we assume to be comprised of one PSII and an unspecified number of PSIs. The optical absorption cross-section of a PSU, σPSU (m2 · μmol O2−1) describes the absorption of light by both PSII and PSI (Mauzerall 1989 and references therein). In turn, the functional absorption cross-section of a PSII, σPSII (m2 · μmol photons−1), quantifies its capacity to utilize absorbed photons to promote charge separation (cf. Dubinsky et al. 1986). The maximum quantum yield for O2 evolution, ϕmax (μmol O2 · μmol photons−1), is the ratio ϕmax = σPSIIPSU.

We make the assumption that the energy of a single exciton generated by the absorption of one photon initiates the electron transport chain (ETC) of a PSU (i.e., is consumed by PSII) with probability ϕmax, and define the turnover time τ (s) to be the duration between the moment when an exciton is inducted into the ETC. and the moment when the PSU is ready to induct the next exciton, considered to occur upon the loss of one electron from the manganese cluster at PSII (i.e., four turnovers are required to produce one molecule of O2). A PSU is considered closed when it is not yet ready to utilize the next exciton, and is otherwise considered open. This definition of a PSU implies that the maximum rate of photosynthesis is directly proportional to the number of PSUs in the system, NPSU (μmol O2 · mg chl a−1), and inversely proportional to the turnover time, so that math formula = PQ·(NPSU/τ) (Falkowski and Raven 2007, p. 244). The proportionality constant PQ is the photosynthetic quotient (μmol C · μmol O2−1), used to relate oxygen evolved to carbon reduced (Williams and Robertson 1991).

Input irradiance

Irradiance absorbed depends on both the spectral absorption of the phytoplankton and the spectral quality of the incident irradiance. This is made explicit in the spectrally averaged chl a–specific absorption coefficient, math formula (m2 · mg chl a−1):

display math(7)

(Babin et al. 1996), where a*(λ) (m2 · mg chl a−1) is the in vivo absorption spectrum of the phytoplankton normalized to chl a at wavelength λ, Einc (λ) (μmol photons · m−2 · s−1 · nm−1) is the spectral irradiance assumed to be incident to all PSUs, and EPAR (PAR, μmol photons · m−2 · s−1) is the integral of Einc (λ) across wavelengths from 400 to 700 nm. The scalar irradiance absorbed by a PSU, Eabs (μmol photons · mg chl a−1 · s−1), is therefore math formula. This computation is not necessary for the analysis to follow, however, as the differential absorption of Einc (λ) across wavelengths is already accounted for by σPSU and σPSII. The irradiance E in the analytic PB versus E equations presented below is therefore the incident scalar irradiance EPAR. However, the spectrally averaged chl a–specific absorption coefficient computed using equation (7) is relevant in our discussion of inter- and intracellular self-shading.

Modeling the photosynthesis-irradiance response

Consistent with the generalized descriptions of the kinetics of photosynthesis (e.g., Joliot and Joliot 1964, Rehák et al. 2008), Figure 2 provides a common point of reference for modeling the photosynthesis-irradiance response via different approaches. PSUs are represented by two pools, one for the proportion of PSUs that are open (A) and another for the proportion that are closed (B), such that A + B = 1. The right-pointing arrow represents the instantaneous rate at which absorbed photons initiate the ETC. of open PSUs, causing the onset of primary charge separation at PSII. The left pointing arrows represents the instantaneous rate at which closed PSUs turn over to become open again, signifying their contribution toward the evolution of O2 via the oxidation of the manganese cluster at PSII. Individual PSUs transition from one pool to the other dynamically: the steady-state proportion of closed PSUs (Beq) is reached when the rates of open-to-closed and closed-to-open transitions are in balance. The point of balance as a function of irradiance can be determined using either a kinetic-type approach or target theory, both of which are described below in the context of the analytic development of the Baly and WNS equations, respectively. The steady-state rate of photosynthesis is directly proportional to the fraction of closed PSUs, and can be written as math formula for any theoretically derived or empirically determined function Beq (E). For simplicity, we write Beq in place of Beq (E) for the remainder of this paper.

Figure 2.

In this simple model of the photosynthetic irradiance response, open PSUs are closed at a rate that depends upon the rate at which absorbed irradiance is utilized by photosynthesis (ϕmaxσPSUE = σPSIIE), and the proportion of open PSUs (A). Closed PSUs open at a rate determined by the turnover time τ and the proportion of closed PSUs (B).

The Baly equation

Zonneveld (1997) and Han (2001) derived Baly's rectangular hyperbolic equation mechanistically using the kinetic approach to solve for the rate of photosynthesis in continuous light. Equating the instantaneous rates of open-to-closed and closed-to-open transitions in Figure 2maxσPSUE(1 − B) = B/τ), and solving for the steady-state proportion of closed PSUs yields the equation:

display math(8)

By multiplying both sides by math formula and substituting Ek for (σPSIIτ)−1 (Falkowski and Raven 2007, p. 246), equation (8) can be shown to be identical to the rectangular hyperbolic Baly equation (2).

The WNS equation

The exponential form of the WNS equation (4) can be rationalized using target theory, which quantifies the probability that a photon will strike a target of specified size at least once during a specified time interval (Ley and Mauzerall 1982). Returning to Figure 2, we consider a population of PSUs placed in the dark long enough that they are all in the open state (A = 1). When subjected to a fixed level of irradiance for a duration Δt, the probability that a PSU is hit by at least one photon and becomes closed can be modeled using the exponential probability distribution:

display math(9)

The integrand of equation (9) is the exponential probability distribution for a process—in this case the absorption of quanta by PSUs—occurring at an average rate of ϕmaxσPSUE (s−1). If Δt is very short so that no closed PSU has time to turn over, then the expected proportion of closed PSUs upon the cessation of irradiance will be equal to the probability on the right hand side of equation (9). With E and Δt known, equation (9) can be used to empirically determine a value for σPSII (Ley and Mauzerall 1982).

Although equation (9) is strictly valid only in the context described above, it has been applied heuristically to the case of continuous irradiance (Dubinsky et al. 1986). The probability that a PSU is hit by at least one photon during the time interval [0, τ] can be computed by substituting τ in place of Δt in equation (9). If it is assumed that any PSU that is hit remains closed, then at the end of [0, τ] the steady-state proportion of closed PSUs will be approximately:

display math(10)

Multiplying both sides of equation (10) by math formula and substituting 1/Ek for σPSIIτ yields the WNS equation (4).

Comparison of the Baly and WNS equations

A point that has been recognized (e.g., Dubinsky et al. 1986) but not well represented in the literature is that kinetic theory and target theory are not different models of the photosynthesis-irradiance response, but are mathematical approaches used to determine the proportion of closed PSUs under different light regimes. The above rationalization of the WNS equation is not consistent with the case of continuous irradiance because it only considers open-to-closed transitions. In flash experiments where one can assume that all PSUs are initially open and that the flash is brief compared to τ, closed-to-open transitions are very unlikely. Under continuous irradiance, some PSUs are certain to be closed at the beginning of the time interval, and closed-to-open transitions are likely to occur on an interval as long as [0, τ]. By taking closed-to-open transitions into account, it can be shown that the exact solution of target theory applied to the case of continuous irradiance is in fact the Baly equation (demonstrated in Appendix S1 in the Supporting Information). Given this result, it would seem that the exact Baly equation should fit observations better than the approximate WNS equation, but just the opposite is the case (Jassby and Platt 1976, Chalker 1981). This suggests the existence of at least one important process not accounted for by the model in Figure 2 that is implicit in the rationalization of the WNS equation.

As shown in Figure 1, the WNS equation predicts a higher rate of photosynthesis, or equivalently a greater quantum yield of photosynthesis, than the Baly equation at levels of irradiance corresponding to the transition between αB and math formula. Explicit in Figure 2 is the assumption that only photons directly absorbed by the antenna of an open PSU are utilized by photosynthesis, the so-called “puddle” model of antenna organization (Lavorel and Joliot 1972). This excludes the possibility of the transfer of excitons between PSUs characterized by connectivity. Connectivity enhances the quantum yield of photosynthesis by allowing excitons in the antennae of closed PSUs to be transferred to the antennae of open PSUs where they can be utilized (Joliot and Joliot 1964, Lavorel and Joliot 1972, Krause and Weis 1991). We hypothesized that the WNS equation tends to fit observations better than the Baly because it implicitly allows excitons to be used more efficiently in a way analogous to connective transfer, which has been observed in certain species of phytoplankton (e.g., Kolber et al. 1998) and may be ubiquitous. If this hypothesis can be supported, connective transfer should be included in a model of the photosynthesis-irradiance response.

The Joliot equation

Connectivity first appeared in a model of photosynthesis in a study by Joliot and Joliot (1964). In their model, excitons in the antennae of closed PSUs have a probability p of being transferred to another PSU, open or closed, and a probability 1 − p of being quenched by thermal dissipation or fluorescence. By a succession of transfers, an exciton either eventually arrives at an open PSU, where it is utilized with probability ϕmax = σPSIIPSU, or is quenched by a nonphotosynthetic process. The probability that a photon is utilized by photosynthesis, they demonstrated, is given by the infinite sum:

display math(11)

Each term ϕmaxA(pB)n in equation (11) gives the probability that the photon is first absorbed by a closed PSU (probability B), and that the resulting exciton visits a sequence of n – 1 closed PSUs (probability (pB)n−1) before finally being transferred to, and utilized by, an open PSU (probability ϕmaxpA). Note that the first term ϕmaxA in the sequence is the probability that a photon is first absorbed and utilized by an open PSU. The rate of photosynthesis (expressed as μmol C · mg chl a−1 · s−1) is the product of the rate at which an ensemble of PSUs absorb photons (NPSUσPSUE), the probability of photon utilization (eq. (11)), and the photosynthetic quotient:

display math

Making the substitution

display math

yields the equation reported in Joliot and Joliot (1964):

display math

To go one step further, PB/math formula can be substituted for Beq in the above equation, so that solving for PB yields what will be referred to as the Joliot equation:

display math(12)

Validation of equation (12) as a formulation comparable to equations (1)–(6) requires confirmation that its derivative at E = 0 is math formula/Ek, and that it approaches math formula as E becomes large. These requirements were easily verified.

It can be demonstrated that equation (12) is the Baly when p is zero in limit, and the Blackman when p = 1. The Joliot equation and its connectivity parameter therefore express a continuum in the degree of connectivity between the Baly (no connectivity) and the Blackman (absolute connectivity, in which no PSU is left open as long as there are excess excitons to close it), and changes in the connectivity parameter modulate curvature between αB and math formula. Equation (12) has been presented in the botanical literature (e.g., Thornley 1998), but there it was characterized as being empirical in origin, with p described as a curvature parameter and assigned no physiological meaning.

The Q-model

The ETC is initiated at PSII when a permanently bound plastoquinone molecule QA is reduced to math formula in the process of primary charge separation. Each PSU, assumed to have one QA molecule, is closed upon the QAmath formula reaction and must await the availability of a temporarily bound plastoquinone molecule QB to oxidize math formula before reopening. After it has collected two electrons and two protons in sequence (QBmath formulaQH2), the twice-reduced QB molecule detaches from PSII, and by diffusion transports the electron pair to cytochrome b6f. The model in Figure 2 is consistent with the assumption that the rate of oxidation of math formula by QB is the slowest step in the ETC, and therefore, by Blackman's law of limiting factors (Blackman 1905), determines the rate of photosynthesis. Honti's (2007) Q-model relegates the bottleneck to the next step down-stream, where electrons are transported from PSII to cytochrome b6f.

As depicted in Figure 2, the rate of photosynthesis is a function of the proportion of closed PSUs (i.e., PB = math formula Beq). In the Q-model, the rate of photosynthesis is proportional to the rate at which electrons are delivered to cytochrome b6f by q molecules of plastoquinone associated with each PSU. Hence, PBαReq/τQ, where τQ (s) is the average turnover time required for the QBmath formulaQH2QB transformation (including the time required for the newly oxidized QB molecule to diffuse from cytochrome b6f back to PSII) and Req is the steady-state proportion of QH2 molecules in the plastoquinone pool. In a sense, the average proportions of QB and QH2 molecules in the plastoquinone pool at steady state take the place of the average proportions of open and closed PSUs in Figure 2. The dynamic responses of Req and Beq to changes in irradiance are quite different, however in the latter case, all of the PSUs are exposed to irradiance simultaneously, while in the former case plastoquinone molecules must queue up to receive electrons from math formula. Honti (2007) accounted for the effect of queuing by using a model for stochastic parallel processes, which captured the dynamics of the oxidative state of the plastoquinone pool in response to changes in irradiance. By this model, the average size q of the plastoquinone pool modulates changes in curvature between αB and math formula.

As stated in the introduction, there is no closed form for the Q-model. In order to fit observations to the model, we used a MATLAB® (Mathworks Corporate Headquarters, Apple Hill, MA, USA) function kindly provided by Dr. Mark Honti. The interested reader will find details in Honti (2007) and in the electronic supplementary material for that paper.


Curvature between αB and math formula

As illustrated in Figure 1, the Blackman, Baly, WNS, and J&P equations each have a characteristic curvature between αB and math formula. All plots were generated using the same values for math formula and Ek, so differences are due only to the inherent curvature of each equation. The bilinear equation (dashed line) represents the maximum potential rate of photosynthesis over the interval [0, Ek] as every absorbed photon is assumed to result in the recruitment of resources (PSUs in the model of Fig. 2, plastoquinone molecules in the Q-model) that are not yet engaged in photosynthesis. Only when all resources are engaged does the rate of photosynthesis abruptly level off at math formula. The other plots in Figure 1 follow the Blackman closely at very low irradiance, but fall short in sequence as irradiance increases, depending on the inherent curvature of each.

This descriptive comparison of curvature can be expressed quantitatively in numerous ways, for example by computing the ratio of the rate of photosynthesis at saturation irradiance to the maximum rate of photosynthesis, PB (Ek)/math formula (cf. the deficit of photosynthesis in Honti (2007)). The computed values were PB (Ek)/math formula = 0.50, 0.63, 0.76, and 1 for the Baly, WNS, J&P, and Blackman equations, respectively. The same computation was applied to the Bannister, Joliot, and Honti equations over a range of values for their curvature parameters. The results are shown in Figure 3, where it can be seen that the Bannister, Joliot, and Honti equations each model changes in curvature differently. These differences impact how well each equation fits observed PB versus E.

Figure 3.

The curvature of the (a) Bannister, (b) Joliot, and (c) Honti equations measured by PB (Ek)/math formula responds differently to changes in their respective parameters, b, p, and q. These differences impact how well each equation fits observed PB versus E data.

Equations fitted to the Jassby and Platt data

Jassby and Platt (1976) fitted a set of PB versus E equations, including the Baly, WNS, and J&P equations, to observations collected at three locations near Halifax, Nova Scotia, consisting of a total of 185 (not 188 as has been reported elsewhere) experiments each with 12–20 paired observations of PB (mg C · mg chl a−1 · h−1) and E (W · m−2; for data and a description of methods see Irwin et al. 1975). We performed a similar analysis on the same data set to compare the WNS, J&P, Bannister, Joliot, and Honti equations. To allow for cases with a nonzero intercept, we introduced an intercept parameter P0 to each PB versus E equation, as Jassby and Platt did in their analysis. Model fits were performed in MATLAB® (Mathworks) using the nonlinear regression procedure lsqcurvefit, which obtains parameter estimates by minimizing the sum of squared errors via the trust-region-reflective algorithm (Coleman and Li 1996). For each observed PB versus E curve, the equation that provided the best fit was determined by comparing the sum of squared residual errors (SSE). In each case, the equation with the minimum SSE was deemed to provide the best fit.

To confirm that our methods were consistent with those used previously, we compared the Baly, WNS, and J&P equations (as in Jassby and Platt 1976), and the WNS, J&P, and Honti equations (as in Honti 2007). The hyperbolic tangent equation fitted observations best in 51% of the cases when compared to the Baly and WNS equations, consistent with the finding by Jassby and Platt (Table 2 in Jassby and Platt 1976). The Honti equation fitted best in 68% of the cases when compared to the WNS and J&P equations, similar to Honti's own results that used a different data set and for which his equation provided the best fit in 57% of the cases (Table 1 in Honti 2007).

Using the Jassby & Platt data, we next made a comparison of all five equations, the results of which are shown in Table 3. The Joliot provided the best fit for 37% of the cases, the Bannister for 29%, and the Honti for only 9% of the cases. The apparent superiority of the Joliot and the Bannister must be taken cautiously; however, as the median coefficient of determination (R2), a measure of the proportion of the variance in the data captured by a given equation, was found to be very high and the same to two decimal places for all five equations. The values for R2 that appear in the fourth column of Table 3 suggest that modulations in the curvature between αB and math formula captured by the curvature parameters in the Bannister, Joliot, and Honti equations accounted for only a median value of between 0.20 and 0.40 percent of the total variance in the data. For example, the difference in the median R2 for WNS and the Bannister equation is 0.9838 − 0.9801 ≅ 0.004 or 0.40%. Such minor changes, in combination with the reduction in the df attendant with the inclusion of an additional parameter, indicate that the curvature parameter in each equation does not significantly (in the statistical sense) improve the equation's fit to the data compared to the equations with fixed curvature. From a statistical point of view, and for the sake of parsimony, we would therefore reject the Bannister, Joliot, and Honti equations in favor of the equations with one less parameter.

Table 3. The Joliot and Bannister equations provided the best fits most frequently, but corresponded to a negligible improvement in the quality of fit as measured by the median coefficient of determination (R2).
 Best fit frequency (%)Median R2
WNS21 (11)0.9801
J&P26 (14)0.9807
Bannister53 (29)0.9838
Joliot69 (37)0.9831
Honti16 (9)0.9828
Total185 (100) 

The minor improvement in the quality of fits engendered by the inclusion of a curvature parameter may be at least partly due to the physiological state of the algal samples from which measurements were taken. Greater curvature is indicative of greater efficiency in the utilization of photons at intermediate irradiances, and the majority of the data exhibited what we interpreted to be large enhancements in efficiency. For example, Jassby and Platt found that the Blackman equation (representing maximum efficiency) fitted best in about 14% of the 185 cases whereas the Baly equation (representing minimum or baseline efficiency) did so in only 5% of the cases, suggesting a bias in the data favoring higher efficiency. The utility of the Bannister and Joliot equations, we suggest, is that they can model the photosynthesis-irradiance response for phytoplankton physiologically adapted to utilize photons over the full range of efficiencies. We maintain that, even if a curvature parameter is not statistically significant, its inclusion in a photosynthesis-irradiance model is desirable because it offers consistent estimates of Ek and math formula when PB versus E curves exhibit a wide range of curvature, thereby addressing a pervasive problem associated with using different two-parameter models, each with its own fixed curvature, for different circumstances (Frenette et al. 1993).

In a direct comparison of the Bannister and Joliot equations made by fitting the data to these two equations only, the Bannister provided better fits in 58% of the cases, which is significantly greater than the outcome expected if both equations had fit equally well (i.e., 58% is significantly greater than 50%, P = 0.015, n = 185). This suggests that the Bannister might be the better choice of an equation that can account for variations in the curvature between αB and math formula. This statement is further supported by an examination of the impact of nonphysiological factors on curvature given in the following two sections.

Examples of the Baly, WNS, J&P, and Bannister equations fitted to data are shown in Figure 4. Differences in curvature are clearly evident in Figure 4a, where the Bannister equation provided the best fit to the data (SSE = 0.27), followed by the WNS (SSE = 0.32), the Baly (SSE = 0.40), and the J&P (SSE = 0.50). The curvature estimated by the Bannister equation (math formula) was 1.5, which is larger than the fixed curvature of the Baly (b = 1) and smaller than the curvatures of both the WNS equation (which corresponds closely to the Bannister with b = 2.2) and the J&P equation (b = 3.1). The estimated saturation irradiance was 7.9, 15, 21, and 15 W · m−2 for the Baly, WNS, J&P, and Bannister equations, respectively. In Figure 4b, the curvature estimated by the Bannister equation was math formula = 2.6. Here, the J&P equation provided the best fit (SSE = 0.17), followed by the Bannister (SSE = 0.21), WNS (SSE = 0.22), and Baly (SSE = 1.20) equations. The estimates for saturation irradiance were 11, 16, 21, and 22 W · m−2 for the Baly, WNS, J&P, and Bannister equations, respectively.

Figure 4.

The variable-curvature Bannister model (heavy line) is compared to fixed-curvature models in two examples from the experimental data. (a) The Bannister provided the best fit due to the curvature, math formula = 1.5, exhibited by the data, which is greater than that of the Baly (b = 1) and less than that of the WNS (b ≈ 2.2) and J&P (b ≈ 3.1) equations. (b) The curvature exhibited by the data is quite large, math formula = 2.6. Consequently, the Baly equation fits the data poorly. In this case, the J&P equation fitted the data slightly better than the Bannister.

In both illustrated examples, Ek estimated by the Bannister equation was in closest agreement with that estimated by the fixed-curvature equation with the smallest SSE. This relationship is consistent, as illustrated in Figure 5. The abscissa is Ek estimated by the Bannister equation and the ordinate is that provided by the fixed-curvature equation that resulted in the smallest SSE. The line is one-to-one. The Bannister equation gives estimates for Ek that are consistent with estimates obtained by fitting the data to the most appropriate fixed-curvature equation (the correlation coefficient is 0.98), although there is evidence of a slight bias at higher values (the slope of the line of best fit is 0.88).

Figure 5.

A comparison of the saturation irradiances estimated using the Bannister equation to that estimated using the fixed-curvature equation that provided the best fit to the data demonstrates that using the Bannister equation is consistent with using the most appropriate fixed-curvature equation.

Apparent changes in efficiency (1)—pigment packaging

The parameters p and q describe the influence of connectivity and the plastoquinone pool on curvature in observed PB versus E using a modeling framework that assumes uniform illumination of all PSUs, but this assumption is not always valid. Intracellular self-shading (a.k.a. pigment packaging), especially in large algal cells with high pigment concentration, has a “flattening” effect on the phytoplankton absorption spectrum (Duysens 1956), and results in a gradient in the spectrally integrated scalar irradiance within the cells. Because irradiance incident to internally shaded PSUs will be overestimated if this effect is not taken into account, we hypothesized that pigment packaging would lead to an apparent reduction in efficiency with an attendant reduction in the curvature parameter.

This hypothesis was tested using a numerical simulation of the irradiance field inside a spherical algal cell with diameter d (μm) and intracellular pigment concentration ci (kg chl a · m−3), based on the assumption, following Duysens (1956), that incident spectral irradiance Einc (λ) emanating from one direction is attenuated as described by the Beer-Lambert law, with no diffraction or scattering as it passes through the cell. The spectrally integrated photosynthetically utilizable irradiance (PUR) EPUR (after Markager and Vincent 2001, cf. Morel 1978) at any location (x, y, z) within the cell is given by:

display math(13)

The absorption spectrum for intracellular pigments in solution math formula (m2 · mg chl a−1), provides the weightings for EPUR when normalized by its spectrally averaged value math formula (m2 · mg chl a−1). The path length math formula is the distance from the illuminated surface of the sphere to the interior location (x, y, z) over which light is attenuated (see Fig. 6). By using EPUR in the simulation, we explicitly took into account changes in the amplitude and shape of the irradiance spectrum at varying locations within the cell. Note that chl-specific absorption coefficients a*(λ) (m2 · mg chl a−1) for whole cells would be used instead of math formula to calculate EPUR for a suspension of phytoplankton (Markager and Vincent 2001).

Figure 6.

The diagram on the left demonstrates how the irradiance within a theoretically homogeneous spherical cell was computed. The irradiance incident to the surface of the sphere is Einc (λ), and the path length over which irradiance is attenuated from the surface to the interior point is l (x, y, z). The diagram on the right depicts the numerical calculation of math formula, the particulate transmittance spectrum.

Duysens (1956) did not show the analog of equation (13), but only the particulate transmittance spectrum Tp(λ), the ratio of the incident irradiance integrated over the illuminated hemisphere to the transmitted irradiance integrated over the distal hemisphere of the spherical cell (represented by the two shaded disks in Fig. 6), given by the theoretical expression:

display math(14)

A numerical estimate of the particulate transmittance spectrum:

display math(15)

was computed for comparison with equation (14). Eout (λ, x, y) in equation (15) is the integrand of equation (13) at the point (x, y, z) with z constrained to be on the surface of the distal hemisphere of the cell: it quantifies the irradiance at wavelength λ transmitted through the cell. The double integral ∬Eout (λ, x, y) dxdy is the summation of all ∬Eout (λ, x, y) over the distal hemisphere of the cell. Likewise, ∬Einc (λ, x, y) dxdy is the sum of the irradiance over the illuminated hemisphere of the cell (see Fig. 6). The caret over Tp indicates that the value is a numerical estimate subjected to discretization error. The chl a–specific absorption spectrum for Prochlorococcus, a picoplankton with cell diameter of less than 1 μm, was used to approximate math formula (Ciotti et al. 2002) as the effects of pigment packaging are minimal due to its very small size and intracellular pigment concentration. The discrepancy between the simulated math formula of equation (15) and the theoretical value given by equation (14) for cell diameters of 4 and 20 μm was less than 5% at each wavelength, and this was determined to be entirely due to the discretization, demonstrating that our numerical model of spectral irradiance within a cell is consistent with the original Duysens (1956) theoretical model.

The chl a–specific absorption spectrum can be obtained from the transmittance spectrum using the following theoretical relationship (Duysens 1956):

display math(16)

The effect of pigment packaging on the chl-specific absorption spectrum of a hypothetical spherical algal cell as determined by equations (15) and (16) is demonstrated in Figure 7. The upper-most spectrum is our approximation of phytoplankton absorption unaffected by pigment packaging, from Ciotti et al. (2002). The middle spectrum resulted from the numerical simulation with ci = 2.86 kg chl a · m−3, a representative value reported in Morel and Bricaud (1981), and d = 4 μm, similar to that of Coccolithus (Emiliania) huxleyi (ibid.). The bottom-most spectrum resulted by using the same value for ci, but with d = 20 μm (e.g., Platymonas suecica, ibid.). Duysens's theoretical spectra computed using equations (14) and (16) are shown as dashed lines for comparison. Differences in the chl a–specific absorption spectra shown in Figure 7, due only to differences in cell diameter, are consistent with observations of the effects of pigment packaging on the absorption spectra of phytoplankton (Berner et al. 1989, Ciotti et al. 2002 and references therein).

Figure 7.

The chl a–specific absorption coefficient a* (λ) was determined from equation (16) using both Duysens’ (1956) theoretical approach and our numerical approach to calculate the particulate transmittance. The chl a–specific absorption coefficient for Prochlorococcus was used to approximate math formula (top-most spectrum). The solid a* (λ) spectra resulted using equation (15) to estimate math formula for homogeneous spherical cells of diameters d = 4 μm and d = 20 μm assuming an intracellular chl concentration of ci = 2.86 kg chl a · m−3. The dashed a* (λ) spectra were obtained using equation (14) from Duysens (1956) to compute Tp (λ). The numerical simulation of the light field within a cell is strongly consistent with the Duysens’ theoretical approach.

The impact of pigment packaging on saturation irradiance and curvature was determined by numerically integrating the rate of photosynthesis throughout the volume of a spherical cell using the equation:

display math(17)

The function f is a specified PB versus E equation. This formulation, which makes the assumption that PSUs are uniformly distributed throughout the cell's volume, was applied using the Joliot equation with p = 0.50, math formula = 0.1 μmol C · mg chl a−1 · s−1, Ek = 200 μmol photons · m−2 · s−1, ci = 2.86 kg chl a · m−3 and di = 4 and 20 μm. To simulate Einc(λ), an area-normalized solar spectrum obtained from the American Society for Testing and Materials (ASTM G-173) was scaled to give EPAR values between 0 and 2,000 μmol photons · m−2 · s−1. Each simulated PB versus E curve, where math formula, was then fitted to the Joliot equation. The resulting estimate of saturation irradiance, different from the inherent Ek at the level of the PSU, was labeled math formula to represent the influenced of pigment packaging.

Results are shown in Figure 8, where the thin black line shows the PB versus E response without intracellular self-shading (i.e., the Joliot equation with p = 0.50). With d = 4 μm (thicker black curve), the simulated data fitted to the Joliot equation yielded math formula = 250 μmol photons · m−2 · s−1 and the estimated connectivity parameter math formula = 0.49. The minor change in the curvature parameter reflects the small cell diameter for which the pigment packaging effect is only slight. Setting d to 20 μm (gray curve) yielded larger changes in parameter values, with math formula = 453 μmol photons · m−2 · s−1 and math formula = 0.29. Pigment packaging resulted in the overestimation of the saturation irradiance and the underestimation of the curvature parameter in both cases due to the assumption that shaded PSUs are exposed to the full intensity of the incident irradiance math formula. This demonstrates that intracellular self-shading can potentially lead to erroneous estimations of physiological parameters at the level of the PSU if not accounted for.

Figure 8.

Effects of pigment packaging on curvature of the PB versus E relationship. A simulation of pigment packaging using the Joliot equation with p = 0.50 and Ek = 200 μmol photons · m−2 · s−1 was used to produce normalized PB versus E curves for cell diameters d = 4 μm (black) and 20 μm (gray). Estimates of math formula and math formula, the parameters that would be retrieved by analyzing results for cells subject to pigment packaging, were obtained by fitting the simulated data to the Joliot equation; they increasingly depart from their modeled values as cell diameter increases. The thinner black line is the PB versus E curve with no pigment packaging.

Apparent changes in efficiency (2)—self-shading

Intercellular self-shading in a dense algal culture has an effect similar to pigment packaging, but the effects depend on the geometry of the experimental system. To provide an example, we simulated a flat-plate photobioreactor with two thicknesses zmax of 0.02 and 0.05 m, illuminated from above. The path length in equation (13) was thus replaced with depth. Equation (17) was then used to determine the integrated rate of photosynthesis using the Baly equation (i.e., the Bannister equation with b = 1) for f with math formula = 0.1 μmol C · s−1, and Ek = 200 μmol photons · m−2 · s−1 and with a culture concentration of c = 1 kg chl a · m−3. Results are shown in Figure 9, where the thin black line shows the PB versus E response without intercellular self-shading (i.e., the Baly equation). Parameters were estimated by fitting the simulated PB versus E curves to the Bannister equation. The parameters estimated for zmax = 0.02 m (thicker black curve) were math formula (the apparent saturation irradiance for a dense algal culture) = 321 μmol photons · m−2 · s−1 and math formula = 0.98, indicating that the culture depth was sufficient to cause an apparent change in saturation irradiance but only a very minor change in curvature. Setting zmax to 0.05 m (gray curve) resulted in math formula = 513 μmol photons · m−2 · s−1 and math formula = 0.90. The curvature in this case was PB (Ek)/math formula = 0.24, less than half of the baseline curvature PB (Ek)/math formula = 0.50 for the Baly equation. Only the Bannister equation can capture curvature less than 0.50 insofar as a negative value for p in the Joliot equation that would be required to produce the same reduction in curvature is inconsistent with its physiological meaning. This adds weight to our suggestion that the Bannister equation is the better choice.

Figure 9.

Effects of optical thickness of the culture vessel on the shape of measured PB versus E curves. A simulation of a flat-plate photobioreactor using the Baly equation (i.e., the Bannister equation with b = 1) with Ek = 200 μmol photons · m−2 · s−1 was used to produce normalized PB versus E curves for culture depths zmax = 0.02 m (black) and 0.05 m (gray) with a culture concentration c of 1 kg chl a · m−3. The thinner black line is the PB versus E curve with no intercellular self-shading. Estimates of math formula and math formula obtained by fitting the simulated data to the Bannister equation increasingly depart from their modeled values as culture depth increases. Changes in saturation irradiance and curvature in both this example and Figure 8 are suggestive of differences in physiology at the level of photosystems, but they are entirely due to self-shading.

Adjusting for self-shading

The departures in the estimates of the parameters for curvature and saturation irradiance from their corresponding modeled values were solely due to overestimation of the amount of irradiance incident to self-shaded algal cells. In the context of a flat-plate photobioreactor, the proportion of irradiance transmitted through an algal culture (TPUR,%), defined as the ratio of PUR emerging from the bottom of the culture to that incident at its top surface can be used as a proxy for the extent of self-shading in a dense culture. The relationships between TPUR and estimates of math formula and p were simulated using the Bannister equation with b = 1, Ek = 200 μmol photons · m−2 · s−1, and at a fixed concentration of c = 1 kg chl a · m−3 over a range of maximum depths. The results shown in Figure 10 demonstrate how Ek and b are overestimated and underestimated, respectively, when TPUR is low. The discord between the true parameter value and its estimate in both cases is consistent with a decrease in the efficiency at which irradiance is utilized. It is evident therefore that estimates of model parameters intended to quantify factors that impact efficiency at the level of the PSU can be confounded by inefficiencies in light utilization at the scale of the entire culture.

Figure 10.

Intercellular self-shading in dense cultures can be quantified by the ratio of transmitted to incident PUR, TPUR (%), on the abscissa. The ordinate on the left is the ratio of the estimated saturation irradiance math formula for the algal culture to the modeled value at the level of the cell Ek; on the right it is the estimate of the curvature parameter. The curves demonstrate that by not accounting for the steep gradient in the irradiance field when transmittance is low, PB versus E curves fitted to the Bannister equation overestimate saturation irradiance, and underestimate curvature. The plots were obtained by fitting to the Bannister equation PB versus E curves simulated using the Baly equation in the context of a flat-plate photobioreactor of variable maximum depth zmax.


The objective of this paper was to formulate an equation containing a physiologically meaningful parameter that captures the variability in the curvature between αB and math formula that has been observed in some sets of PB versus E data. Our results indicate that both the degree of connectivity between PSUs (p) and the size of the plastoquinone pool (q) capture some of this variability. This is consistent with ample evidence in the fluorescence literature of variability in the rate of exciton transfer from closed to open PSUs and in the size of the plastoquinone pool (e.g., Krause and Weis 1991), so there is no question that p and q at least approximate the effect of these processes on the rate of photosynthesis. However, other factors that can impact curvature in experimentally determined PB versus E, such as inter- and intracellular self-shading, are not accounted for by the Joliot and Honti equations and weaken the case for either analytic approach.

Honti described his Q-model as being “phenomenological” because “most of the known complexity of photosynthesis was omitted” (Honti 2007). By this definition, all of the analytic models presented in this paper are phenomenological. In the face of unaccounted for and potentially confounding factors that affect curvature, it is uncertain whether estimates of p and q can be interpreted as accurate measures of connectivity and plastoquinone pool size, respectively. Changes in the photosynthetic apparatus with irradiance over the minutes or hours used to measure PB versus E likely mean that observations at different levels of irradiance correspond to different physiological states (Marra 1978, Berner et al. 1989), possibly affecting curvature and further obfuscating the interpretations of p and q. Specifically, any irradiance-dependent process that draws photons or electrons away from the photosynthetic or carbon-reduction pathways will reduce efficiency and impact curvature. These include changes in the number or effectiveness of functional PSU's (e.g., by photoinhibition; Rehák et al. 2008, Han et al. 2000), nonphotochemical quenching via the induction of the xanthophyll cycle (Polimene et al. 2012), Mehler reactions and photorespiration (Bender et al. 1999), and the diversion of electrons to alternate sinks such as nitrate reduction (Lomas and Gilbert 1999). All of these processes may occur over the range of irradiances used to produce a PB versus E curve. The difficulty in separating this multiplicity of potential sources of variation supports an empirical approach to modeling curvature.

We suggest that Bannister's is the most appropriate of existing PB versus E equations with a curvature parameter not only because it most frequently provided the best fit to the data used in this study but also because its curvature parameter does not have specific, and potentially inappropriate physiological meaning. The Bannister parameter may be interpreted as representing the integration of all factors that affect curvature, including but not limited to connectivity, plastoquinone pool size, competition for excitons and electrons by nonphotochemical pathways, and self-shading. This interpretation is consistent with the fact that, unlike the Joliot and Honti equations, the Bannister equation allows for a reduction in efficiency below the baseline predicted by the Baly equation.

Even though variations in b cannot be related directly to any one optical or physiological property of phytoplankton, we speculate that Bannister's parameter may be useful in exploring fundamental relationships between biooptical properties of phytoplankton and the photosynthesis-irradiance response. For example, b may have the power to discriminate between broad classes of community structure via the relationship between curvature and the pigment packaging effect quantified by the product of cell diameter and intracellular pigment concentration dci (Morel and Bricaud 1981, Ciotti et al. 2002). It might also be correlated with aspects of the natural growth environment, such as variations in efficiency as a function of the mixing rate through the irradiance gradient of the photic zone (e.g., Cullen and Lewis 1988, Macintyre and Geider 1996) or as a function of trophic status. These hypotheses are purely speculative however, since the Jassby & Platt data do not include environmental parameters with which to test them. The salient idea is that, because it is independent (in the mathematical sense) of Ek and math formula, Bannister's parameter provides an additional dimension of variability with which to explore the response of natural aggregations of phytoplankton to environmental changes that has been hitherto unexamined.

Our study of the effects of self-shading within or between algal cells has implications for the estimation of the parameters of photosynthesis or specific growth rate in large-scale biogeochemical ocean general circulation models, or BOGCMs (Friedrichs et al. 2009). Typically, models based on experimental observations provide the foundations for the functions that describe PB versus E and variations in the chl content of phytoplankton in BOGCMs. One approach is to apply a dynamic regulatory model (e.g., Geider et al. 1998) in which photosynthetic parameters are related mechanistically to acclimating chemical composition (chl content and nutrient quota); the other is to derive photosynthetic parameters using a simple empirical approach and to back-calculate the ratio of chl to carbon (e.g., Flynn 2003). Regardless of the approach taken, estimation of model parameters is centrally important for model development and validation. It was shown that self-shading has an impact not only on the efficiency of light utilization and therefore curvature but also on estimates of Ek (see Figs. 8 and 9). This is, in the absence of photoinhibition, concomitant with variations in estimates of the initial slope αB. Together, these variations suggest the necessity of confirming that estimates of Ek and αB used in the formulation of BOGCMs were made on optically thin samples (i.e., TPUR approaching 1.0) or that they are somehow corrected for the biases illustrated in Figure 10.

A plot such as Figure 10 might provide an approximate means of adjusting experimental determinations of model parameters for the effects of intercellular self-shading. At 20% transmittance of EPUR for example, the apparent saturation irradiance math formula is about 2.2 times the actual Ek, and math formula is about 0.93 of its true value. Estimates of the true parameter values might be acquired by adjusting accordingly. It may therefore be possible to obtain parameter estimates that more accurately reflect productive potential even when intercellular self-shading alters the shape of an observed PB versus E curve.

A similar approach might be taken to adjust for intracellular self-shading, which in principle leads to a bias in the estimation of photosynthetic efficiency at the scale of the PSU, and could be a factor in the development and interpretations of mechanistic models that make assumptions about the constancy of αB (e.g., Geider et al. 1998). Our results suggest that the variation in photosynthetic parameters αB or Ek with environmental conditions at the level of the cell will not be the same as at the level of the PSU if pigment packaging changes significantly.

More practically, perhaps, our quantitative description of the effects of culture density on the determination of photosynthetic parameters (Fig. 10) demonstrates clearly that experimental determinations of saturation irradiance can be strongly biased when conducted on optically thick cultures, and that resulting estimates are uncertain unless the transmission of PUR is determined. This result is directly relevant to the development of strategies to optimize the optical properties of microalgae for enhanced biofuels production (Mitra and Melis 2008). Even if the chl concentrations of experimental cultures are matched (ibid.), expected changes in absorption normalized to chl will lead to differences in the transmission of PUR and thus biases in the estimation of the saturation irradiance, an index of productive potential. Robust comparisons between PB versus E curves for experimental cultures require either optically thin suspensions, or quantitative consideration of PUR transmission.

Although it was not considered in our analysis, a complete model of the photosynthesis-irradiance response must take photoinhibition into account. To estimate parameters for PB versus E data exhibiting photoinhibition, the simpler choice would be to use a threshold model such as that in Neale and Richerson (1987). This approach introduces a factor containing two additional parameters, the threshold level of irradiance ET below which inhibition is absent, and an exponential slope β that quantifies the reduction in the maximum potential rate as a function of irradiance above ET. The product of this factor with the Bannister equation would provide a complete model. However, this approach would likely be insufficient for cases in which inhibition manifests well before the maximum rate is reached (e.g., Platt et al. 1980). In such a case, the rate at which the PB versus E curve levels off would be influenced by the decline in efficiency modeled by Bannister's b as well as by the influence of photoinhibition (cf. Frenette et al. 1993), complicating the interpretation of parameters in ways that could be addressed in future studies.

Fasham and Platt developed a four-parameter physiological model with inhibition based on the kinetics of electron flow through PSII to produce a PB versus E equation that includes a parameter which they call as χ (Fasham and Platt 1983). It was shown that χ modulates curvature when the inhibition parameter is set to zero (see panel (a) of their Fig. 1). Although the Fasham & Platt equation is germane to our analysis, it was not included because we became aware of its existence only during the review. Nevertheless, our demonstration that processes occurring at multiple scales—at the level of the PSU, the cell, and the culture—can potentially impact curvature makes suspect the interpretation of any curvature parameter that is based on specific physiological processes. We therefore place the Fasham & Platt equation in the same class as the Joliot and Honti equations with little risk, although further work should be conducted to verify this placement.


The search for an equation containing a parameter to account for variations in the curvature between αB and math formula in observed PB versus E data was motivated by a need for consistency in the estimates of the conventional parameters Ek and math formula. Although several PB versus E equations with fixed curvature fit observations well, each describes curvature in a different way, so that estimates of Ek and math formula are not necessarily comparable (cf. Frenette et al. 1993). A single model with variable curvature could solve this problem, and ideally such a model would have a mechanistic foundation.

The mechanistic rationalization of the exponential WNS equation (Webb et al. 1974) was based on target theory, which describes the proportion of dark-acclimated PSUs that become closed during a single brief saturating flash. But because this rationalization does not consider the turnover of closed PSUs, it overestimates the rate of photosynthesis compared to the Monod-type rectangular hyperbolic equation introduced by Baly (1935). This explains why the WNS equation exhibits greater curvature than the Baly equation. The fact that most PB versus E measurement are more consistent with the WNS equation suggests that the existence of an efficiency-enhancing process, not included in the mechanistic model from which the Baly was derived, that is explicit in the WNS equation. Connectivity transfer between PSUs is one such process.

Two models were presented that describe curvature in the PB versus E relationship mechanistically: Honti's (2007) Q-model and one derived from relationships presented by Joliot and Joliot (1964). Along with Bannister's (1974) empirical model, each describes curvature effectively, but in a different way. A comparison of the Baly, WNS, J&P, Bannister, Joliot, and Honti equations fitted to a set of PB versus E curves demonstrated that the Joliot equation provided the best fit most frequently (37% of the cases) compared to the Bannister equation (29% of the cases), with each of the remaining equations accounting for 14% or less of the cases. However, when the Joliot and Bannister equations were compared directly, the Bannister equation provided the better fit in 58% of the cases.

Two well-recognized phenomena, intracellular self-shading (Duysens 1956) associated with large pigment-rich cells, and intercellular self-shading associated with optically thick suspensions, strongly influence the curvature of observed photosynthesis versus irradiance as compared to PB versus E at the level of the PSU. Estimates of curvature and saturation irradiance are thus biased and physiological interpretations of curvature are compromised unless accounted for. Although this problem can be resolved by explicitly modeling for self-shading, the possibility of other factors that impact curvature still exists. The likely differences between the physiological states of PSUs at different levels of irradiance that can develop during PB versus E measurements, for example, may affect curvature and remain a factor that is difficult to account for.

We conclude that the Bannister equation best satisfies the need for a PB versus E equation that captures variations in curvature between αB and math formula, and that its parameter b can be interpreted as quantifying the integration of all factors that impact curvature.

The authors thank Dr. Mark Honti for providing code for his Q-model and the two anonymous reviewers for pointing to lacunae in the manuscript, the filling of which has sharpened our arguments. This work was partially supported by NSERC grants to JJC.