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 *P*^{B} (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, *P*^{B} 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 *P*^{B} 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 *P*^{B} versus *E* relationship (cf., Talling 1957, Cullen et al. 2012) and thus apply to any determination of photosynthesis versus irradiance.

Symbol | Description | Units |
---|---|---|

a* (λ) | In vivo absorption spectrum normalized to chl a | m^{2} · mg chl a^{−1} |

Spectrally averaged chl a–specific absorption coefficient | m^{2} · mg chl a^{−1} | |

, | Normalized absorption spectrum for intercellular pigments dissolved in solution, its spectrally averaged value | m^{2} · mg chl a^{−1} |

A, B | Proportion of open and closed PSUs | n/a |

B _{eq} | Steady-state proportion of closed PSUs | n/a |

b, | Curvature parameter for the Bannister equation, its estimated value | n/a |

c_{i}, c | Intracellular pigment concentration of an algal cell, concentration in an algal culture | kg chl a · m^{−3} |

D | Diameter of an idealized spherical algal cell | μm |

E, E_{k}, E_{PAR} | Irradiance, saturation irradiance, photosynthetically available radiation | μmol photons · m^{−2} · s^{−1}, W · m^{−2} |

, | Estimated saturation irradiance influenced by pigment packaging, and self-shading in an algal culture | μmol photons · m^{−2} · s^{−1} |

E_{inc} (λ), E_{out} (λ) | Incident and transmitted spectral irradiance | μmol 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 PSUs | μmol O_{2} · mg chl a^{−1} |

P^{ B}, | Chl a–specific photosynthetic rate, maximum rate | μmol C · mg chl a^{−1} · s^{−1} or mg C · mg chl a^{−1} · h^{−1} |

PQ | Photosynthetic quotient | μmol C · μmol O_{2}^{−1} |

P | Curvature parameters for the Joliot equation | Transfer probability |

Q | Curvature parameters for the Honti equation | No. plastoquinone molecules |

T_{p} (λ), T_{p} (λ) | The theoretical and numerical particulate transmittance spectra, the proportion of the irradiance incident to a given volume that passes through it without being absorbed | Proportion per nm |

T _{PUR} | Percent irradiance transmitted | % |

α^{B} | Initial slope of the P^{B} versus E curve | μmol C · mg chl a^{−1} · m^{2} · μmol photons^{−1} |

σ_{PSU} | Optical absorption cross-section of a PSU | m^{2} · μmol O_{2}^{−1} |

σ_{PSII} | Functional absorption cross-section of a PSII | m^{2} · μmol photons^{−1} |

Τ | Turnover time for a PSU | s |

ϕ_{max} | Maximum quantum yield for O_{2} evolution | μmol O_{2} · μmol photons^{−1} |

In its simplest form, the observed *P*^{B} 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 H_{2}O to CO_{2}—at any one time. In this light-limited state, the chl *a*–specific photosynthetic rate *P*^{B} responds maximally to small increases in irradiance. When light is saturating, the ensemble of photosynthetic apparatuses functions at its maximum rate (). 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 *E*_{k} (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 α^{B} (μmol C · mg chl *a*^{−1} · m^{2} · μmol photons^{−1}), intersects the horizontal line at (Talling 1957). The curvature between the initial slope given by α^{B} and the ceiling imposed by 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.

The rate of photosynthesis can be reduced below 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 *P*^{B} 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 *P*^{B} versus *E* response examined here, but is left to future efforts.

The *P*^{B} 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* − *E*_{k}| (eq. 1). Unlike other *P*^{B} 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 *E*_{k}. 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.

No. | Equation | Authors | Form |
---|---|---|---|

(1) | Blackman 1905 | Bilinear | |

(2) | Baly 1935 | Rectangular Hyperbola | |

(3) | Smith 1936 | Modified Rectangular Hyperbola | |

(4) | Webb et al. 1974 | Exponential | |

(5) | Jassby and Platt 1976 | Hyperbolic Tangent | |

(6) | Bannister 1979 | Generalized 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 *P*^{B} 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 *P*^{B} 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 *P*^{B} 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, *E*_{k} and (where α^{B} = /*E*_{k}). While these parameters can be adjusted to fit the initial slope and the maximum photosynthetic rate of an observed *P*^{B} versus *E* curve, alone they are not sufficient to adjust the curvature between α^{B} and (Honti 2007), which is consequently an inherent property of a two-parameter *P*^{B} versus *E* equation. A set of observed *P*^{B} 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 *E*_{k} and (Frenette et al. 1993). The introduction of a third parameter into a *P*^{B} 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 *P*^{B} 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 *P*^{B} 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 *P*^{B} 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.