### Phytoplankton cultivation

The three strains of marine phytoplankton (*Isochrysis galbana*, *Dunaliella salina* and *Platymonas subcordiformis*) isolated from East China Seas were grown aseptically in f/2 medium. The four strains of freshwater phytoplankton (*Microcystis aeruginosa* FACHB-905, *Microcystis wesenbergii* FACHB-1112, *Scenedesmus obliquus* FACHB-116 and *Chlorococcum* sp. FACHB-1556) were purchased from the Freshwater Algae Culture Collection (FACHB-collection) of the Institute of Hydrobiology, Chinese Academy of Sciences (Wuhan, China) and cultivated in BG11 medium. The cultures were illuminated by cool white fluorescent bulbs (60 µmol photons m^{−2} s^{−1}) with a photoperiod of 12 h per day at 26 ± 1 °C.

### Measurement of photosynthetic oxygen evolution

After 7 to 10 days of incubation, the photosynthetic oxygen-evolving rate of microalgal cells reaching the exponential growth phase was determined using a bio-oxygen meter (Yaxin-1151, Beijing Yaxinliyi Science and Technology Co., Ltd., China). Eight mL cell suspensions of each strain were exposed to increasing orders of irradiance intensity (0, 25, 50, 100, 150, 200, 300, 400, 500, 600, 800, 1000, and 1200 µmol photons m^{−2} s^{−1}), given by a digital LED light source (YX-11LA, Beijing Yaxinliyi Science and Technology Co., Ltd., China), at 25 ± 1 °C. The meter took reads once every 3 s for 5 min in each irradiance measurement point, during which a linear relationship varying with time in oxygen concentration was obtained. Triplicate samples were prepared and measured for each test. The response of the photosynthetic oxygen-evolving rate to irradiance (*P*_{n}–*I*) was fitted with four *P*–*I* models [19,20,21,22, 29].

### Determination of chlorophyll *a* concentration and cell counts

The cells for photosynthetic oxygen-evolving measurement were collected by centrifugation (5600×*g*) for 10 min at 4 °C. Chlorophyll *a* (Chl *a*) was extracted from microalgal cells in 90% (v/v) acetone and left overnight at 4 °C in darkness. The extracts were then centrifuged at 3600×*g* for 10 min. The Chl *a* concentration was determined spectrophotometrically in the supernatant with a SP752 UV–Vis spectrophotometer (Spectrum Instruments, Shanghai, China) according to the method of Jeffrey and Humphrey [32]. One-mL cultures of each strain were taken and preserved in Lugol’s iodine solution for counting algal cells by a haemocytometer. Each test was conducted in triplicate.

### Model description

#### Model 1

The light dependence of the net photosynthetic rate (*P*_{n}) is expressed as [19]:

$$ P_{\text{n}} = P_{\text{nmax}} { \tanh }\left( {\frac{\alpha I}{{P_{\text{nmax}} }}} \right) - R_{\text{d}} , $$

(1)

where *P*_{n} (μmol O_{2} mg^{−1} Chl *a* h^{−1}) is the chlorophyll *a*-normalized net photosynthetic rate at irradiance *I*, *P*_{nmax} (μmol O_{2} mg^{−1} Chl *a* h^{−1}) is the light-saturated maximum rate of photosynthesis, *α* (μmol O_{2} mg^{−1} Chl *a* h^{−1}/μmol photons m^{−2} s^{−1}) is the light-limited initial slope of *P*_{n}–*I* curve, and *R*_{d} (μmol O_{2} mg^{−1} Chl *a* h^{−1}) is the dark respiration rate.

As Eq. (1) is an asymptotic function, the saturation irradiance cannot be directly calculated. Therefore, the saturation irradiance (*I*_{sat}, μmol photons m^{−2} s^{−1}) is obtained by drawing a line from tangent of the initial slope with the plateau of the *P*_{n}–*I* curve onto the *x*-coordinate [1]. *I*_{sat} is calculated by the following calculation formula:

$$ I_{\text{sat}} = \frac{{P_{\text{nmax}} - R_{\text{d}} }}{\alpha }. $$

(2)

But the analytic solution of the light compensation point (*I*_{c}, μmol photons m^{−2} s^{−1}) cannot be directly obtained by Eq. (1). In order to obtain *I*_{c}, Kok effect [33] must be ignored here, and then *I*_{c} can be calculated as [21]:

$$ I_{\text{c}} = \frac{{R_{\text{d}} }}{\alpha }. $$

(3)

The photosynthetic quantum efficiency (*P*_{n}′, μmol O_{2} μmol photons ^{−1}) is calculated as:

$$ P_{\text{n}}^{'} = \frac{\alpha }{{cosh^{2} \frac{\alpha I}{{P_{\text{nmax}} }}}}. $$

(4)

#### Model 2

The light dependence of *P*_{n} is expressed as [20, 21]:

$$ P_{\text{n}} = P_{\text{s}} \left[ {1 - { \exp }\left( {-\frac{\alpha I}{{P_{\text{s}} }}} \right)} \right]{ \exp }\left( {-\frac{\beta I}{{P_{\text{s}} }}} \right) - R_{\text{d}} , $$

(5)

where *P*_{n} is the chlorophyll *a*-normalized net photosynthetic rate at irradiance *I*; *α* is the light-limited initial slope of *P*_{n}–*I* curve; *β* is the dimensionless parameter reflecting the photoinhibition process; Without photoinhibition, *P*_{s} is the maximum photosynthetic output; *P*_{s} is the parameter reflecting the maximum, potential, light-saturated, rate of photosynthesis at *β* > 0; and *R*_{d} is the dark respiration rate.

The *I*_{sat} is calculated as:

$$ I_{\text{sat}} = \frac{{P_{\text{s}} }}{\alpha }\ln \frac{\alpha + \beta }{\beta }. $$

(6)

The *P*_{nmax} can be calculated as:

$$ P_{\text{nmax}} = P_{\text{s}} \left( {\frac{\alpha }{\alpha + \beta }} \right)\left( {\frac{\beta }{\alpha + \beta }} \right)^{{\frac{\beta }{\alpha }}} - R_{\text{d}} . $$

(7)

However, the analytic solution of *I*_{c} cannot be directly obtained by Eq. (5). To obtain *I*_{c}, the Kok effect must be ignored here, and then *I*_{c} can be calculated as:

$$ I_{\text{c}} = \frac{{R_{\text{d}} }}{\alpha }. $$

(8)

The photosynthetic quantum efficiency is calculated as:

$$ P_{\text{n}}^{'} = \exp \left( { - \frac{\beta I}{{P_{\text{s}} }}} \right)\left\{ {\alpha \exp \left( { - \frac{\alpha I}{{P_{\text{s}} }}} \right) - \beta \left[ {1 - \exp \left( { - \frac{\alpha I}{{P_{\text{s}} }}} \right)} \right]} \right\}. $$

(9)

#### Model 3

The light dependence of *P*_{n} is expressed as [22]:

$$ P_{\text{n}} = \frac{I}{{\alpha I^{2} + \beta I + \gamma }} - R_{\text{d}} . $$

(10)

Here, *P*_{n} is the chlorophyll *a*-normalized net photosynthetic rate at irradiance *I*; *α* and *β* are the fundamental parameters, nondimensional; and *R*_{d} is the dark respiration rate. The reciprocal of *γ* is the light-limited initial slope of *P*_{n}–*I* curve.

*I*_{sat} is calculated as:

$$ I_{\text{sat}} = \sqrt {\frac{\gamma }{\alpha }} . $$

(11)

*P*_{nmax} is given by:

$$ P_{\text{nmax}} = \frac{1}{{\beta + 2\sqrt {\alpha \gamma } }} - R_{\text{d}} . $$

(12)

When *P*_{n} = 0, *I*_{c} is given as follows:

$$ I_{\text{c}} = \frac{{1 - \beta R_{\text{d}} + \sqrt {\left( {1 - \beta R_{d} } \right)^{2} - 4\alpha \gamma R_{\text{d}} } }}{{2\alpha R_{\text{d}} }}. $$

(13)

The photosynthetic quantum efficiency is calculated as:

$$ P_{\text{n}}^{'} = \frac{{\gamma - \alpha I^{2} }}{{\left( {\gamma + \beta I + \alpha I^{2} } \right)^{2} }}. $$

(14)

#### Model 4

The light dependence of *P*_{n} is expressed as [29]:

$$ P_{\text{n}} = \alpha \frac{1 - \beta I}{1 + \gamma I}I - R_{\text{d}} . $$

(15)

Here *P*_{n} is the chlorophyll *a*-normalized net photosynthetic rate at irradiance *I*, *α* is the initial slope of the *P*_{n}–*I* response curve, *β* and *γ* are the nondimensional parameters reflecting photoinhibition and light saturation, respectively, and *R*_{d} is the dark respiration rate.

*I*_{sat} is calculated as:

$$ I_{\text{sat}} = \frac{{\sqrt {\frac{{\left( {\beta + \gamma } \right)}}{\beta }} - 1}}{\gamma }. $$

(16)

*P*_{nmax} is obtained by:

$$ P_{\text{nmax}} = \alpha \left( {\frac{{\sqrt {\beta + \gamma } - \sqrt \beta }}{\gamma }} \right)^{2} - R_{\text{d}} . $$

(17)

When *P*_{n} = 0, *I*_{c} is given as follows,

$$ I_{\text{c}} = \frac{{\alpha - \gamma R_{\text{d}} - \sqrt {\left( {\alpha - \gamma R_{\text{d}} } \right)^{2} - 4\alpha \beta R_{\text{d}} } }}{2\alpha \beta }. $$

(18)

The photosynthetic quantum efficiency is calculated as:

$$ P_{\text{n}}^{'} = \alpha \frac{{1 - 2\beta I - \beta \gamma I^{2} }}{{\left( {1 + \gamma I} \right)^{2} }}. $$

(19)

### Statistical analysis

*P*_{n}–*I* data were fitted using SPSS version 24.0 using nonlinear, least-squares fitting based on the Levenberg–Marquardt algorithm. Duncan’s post hoc tests (*p* < 0.05) were performed to establish differences among fitted results from model 1, model 2, model 3 and model 4. Data were reported as the means and standard errors in the calculations. Goodness of fit of the mathematical models to experimental data was assessed using the adjusted coefficient of determination (*R*^{2}). Akaike information criterion (AIC) is a standard to measure the best-fit of statistical models. When sample size (*n*) is small compared to the number of parameters (i.e., *n*/*k* < 40), the use of a second order, AIC_{c} (= AIC + 2*k*(*k *+ 1)/(*n *− *k *− 1)) is recommended [34]. In this paper, AIC_{c} of each model was calculated because of *n*/*k* = 1 for model 1, yet *n*/*k* = 0.75 for models 2, 3 and 4.