All batch experiments were conducted in 125-mL serum bottles (Wheaton Science Products, USA) made of clear borosilicate glass and with a liquid volume of 100 mL. The experimental set up is shown schematically in Fig. S1 of Supporting Information. The mouth was sealed with a snap-on rubber stopper and an aluminum cap after filling the bottle with medium. Gas-flow controllers, connected to an air pump (KNF Neuberger, USA) with an air-filter (0.2 mm PTFE; Whatmann, USA) inlet, regulated the supply rate of CO₂ by maintaining an aeration rate of 5 L_air/L_reactor/min through a tube inserted into the bottle through the rubber cap. The rapid gas bubbling make the liquid contents completely mixed, and off gas was vented out through a syringe needle while preventing mass transfer limitation of C_i.

The batch experiments were performed in a photo-incubation chamber (TC30; Conviron Inc., USA) maintained at 30°C. Light panels inside the chamber were equipped with white-fluorescent lamps and supplied photosynthetically active radiation (PAR)[15] with a controllable LI in the range of 3 – 20 W/m² (14 – 92 mmol photons/s/m²) at the outside of the bottle. For attaining LI levels up to 48 W/m² (221 mmol photons/s/m²), additional lamps were 92 installed. The PAR LI was measured with a PAR sensor (LI-190 Quantum sensor; LI-COR Biosciences, USA) connected to a digital multi-meter; thus, all the LI values reported in this study are on a PAR basis. The photon flux was converted to energy flux using 4.6 mmol photons/s/m² per W/m² [15].

PCC6803 was cultivated in a 5-L mother culture grown in a transparent 10-L reservoir bottle (KIMAX, Germany) to which we sparged filtered air (2 L_air/L_liquid/min). The bottle was continuously illuminated using fluorescent lamps (20 W/m² on the exterior) in the same photo-incubator chamber. Non-limiting N_i, P_i, and other nutrients were supplied using standard BG-11 medium to which we added supplemental P_i using a semi-batch mode of operation (hydraulic retention time ≅ 10 d) [16]. Thus, inocula taken from the mother culture had been experiencing feast-and-famine nutrient conditions. Inoculum cultures were kept at a 730-nm optical density (OD₇₃₀) of 2.3, which is defined as -log₁₀ (I/I₀), where I is the intensity of light at a 730-nm wavelength that has passed through a sample and I₀ is the intensity of the light before it enters the sample.

For the batch experiments, the standard BG-11 growth medium [17] was modified to achieve limitation independently by each nutrient. For all experiments, we removed the usual amount of Na₂CO₃ from standard BG-11 solutions in order to have independent control of C_i; we call this BG-11C, which contained 1.5 g NaNO₃, 40 mg K₂HPO₄·3H₂O, 75 mg MgSO₄·7H₂O, 36 mg CaCl₂·2H₂O, 6 mg citric acid, 6 mg ferric ammonium citrate, 1 mg EDTA disodium salt, and 1 ml mixed trace metal solution. Each liter of trace metal solution contained 2.9 g H₃BO₃, 1.8 g MnCl₂·4H₂O, 0.22 g ZnSO₄·7H₂O, 0.39 g NaMoO₄·2H₂O, 79 mg CuSO₄·5H₂O, and 49 mg Co(NO₃)₂·6H₂O in deionized water (conductivity = 18.2 MΩ ·cm) produced by Purelab Ultra (ELGA lab water, USA). Further modifications were made to BG-11C for each batch experiments, as described below, and all media were autoclaved before use.

Table 1 summarizes the four sets of batch experiments to estimate each kinetic parameter. We first used batch experiment 1 to obtain the parameters for LI-controlled kinetics first. It had ample N_i, P_i, and C_i at all times so that only LI was limiting. Using optimal-LI conditions revealed in experiment 1, we then conducted batch experiments 2, 3, and 4 to determine parameters K_S,Ni, K_S,Pi, and K_S,Ci, respectively. The rightmost column in Table 1 indicates the range we tested for each independent variable. In all cases, the stoichiometric consumption of the limiting nutrient was small compared to its starting concentration.

Each series of batch experiment was begun by filling all the test bottles with 97 mL of modified medium according to Table 1 and adjusting the pH to 8±0.2 using a phosphate buffer (H₂PO₄⁻ and HPO₄²⁻ at a 7:43 mole ratio) and 0.2 M hydrochloric acid or 0.2 M sodium hydroxide. The pH was monitored using a pH meter (Accumet AB15; Fisher scientific, USA) with a glass pH electrode. We put the reactors inside the incubator overnight to establish the same temperature. The next day, we began aeration for the reactors of experiments 1 to 3 using gas-flow-rate controllers (5 L_air/L_liquid/min) and inoculated 3 mL of mother culture with disposable syringes and needles. The C_i-supply rate (mg C/L_reactor/min) was based on the CO₂-aeration rate (L_air/L_R/min), the CO₂ content in the air (~380 ppm, v/v), the ideal-gas assumption (1 mol = 22.4 L) under standard temperature and pressure, and the molar mass of C (12 g/mol). For estimating C_i kinetics in experiment 4, we added different amounts of NaHCO₃ to meet the C_i goal of each test, instead of supplying C_i by aeration; this approach allowed us to maintain a constant total-C content (= residual C_i + C in biomass) in each bottle. Although experiment 4 was not aerated, we inoculated it in the usual way and shook the bottles by hand to mix their contents at the start of the experiment and before each sampling. In addition, any produced gas was vented using a disposable syringe needle to relieve pressure build up. Since PCC6803 remained in a stable suspension for up to two days, the liquid contents for experiment 4 experienced no loss of biomass concentration due to sedimentation between samplings. The pH of BG-11C was 8±0.2, which indicates that dominant C_i species was HCO₃⁻[18].

The initial OD₇₃₀ ranged from 0.08 – 0.10, which corresponded to 21 – 27 mg/L as dry weight (DW). This low initial biomass concentration and excess concentrations of nutrients guaranteed minimal LI attenuation at the start of the experiment 1. Furthermore, we took samples during the initial 4 hours, when OD₇₃₀ remained less than 0.3 (Fig. S2 in supporting information). Based on our previous research [16], the specific growth rate adjusts to the new experimental conditions for LI, C_i, N_i, and P_i within an hour; thus, the OD₇₃₀ was utilized results from 1 to 4 hours.

We performed quadruplicate experiments in four reactors for each experimental condition. Thus, we present the average specific growth rate with a standard deviation from the 4 reactors. Variability was small among the tests and may have originated from small differences in acclimation to the new conditions, inoculum size, carryover of nutrients with the inoculums, and sampling timing.

We also carried out an upper control experiment with all nutrients present and a series of lower control experiments with one nutrient totally absent (Fig. S2 of the Supporting Information). The upper control results indicate that the maximum μ was > 1.6/d. For the lower controls, the inocula were washed three times in 0.9% NaCl solution to remove residual nutrient from the inoculum’s growth medium before conducting each experiment. The lower control results document that eliminating any one nutrient suppressed biomass growth so that the biomass increase over 4 h was a maximum of 8.0 mg/L or 0.03 OD₇₃₀. Because the change in biomass was so small in the lower controls, we were able to use directly the biomass concentrations in non-control experiments (Table 1).

The growth kinetics of PCC6803 were monitored by analyzing 2-mL liquid samples taken at 0, 1, and 4 hours with a disposable syringe using a needle inserted through the rubber cap. OD₇₃₀ was directly measured with a UV-visible spectrophotometer (Cary 50 Bio; Varian Inc., USA) at a wavelength of 730 nm and was converted to DW using calibration curve that we determined for PCC6803, shown in Fig. S3 of Supporting Information. For the calibration, DW was determined using total suspended solids (Method 2540D in Standard Methods) [19].

The specific growth rate was computed for synthesis based on biomass concentrations in samples taken at the beginning and end of a time interval:

where μ is the observed net specific growth rate (/d), μ_syn is the synthesis specific growth rate (/d), b is the endogenous decay rate (/d), and X₁ and X₄ are the biomass concentrations (expressed by mg DW/L) at time 1 and 4 hours (t1 and t4), respectively. Independently, b was estimated by the decrease of biomass concentration under complete darkness for one day; the value was 0.056/d, as presented in Table S1 of Supporting Information.

To obtain the best estimates of the kinetic parameters for LI, N_i, P_i and C_i from each set of experiments, we conducted non-linear regression using computer software (Sigmaplot10; Systat Software Inc., USA) that related the observed μ_syn = m + b to the spatially averaged light intensity (LI_SA) and the concentration of the limiting nutrient at a fixed LI_SA. For estimating parameters associated with N_i, P_i, and C_i kinetics in experiments 2 – 4, we employed the best-fit constants for LI revealed previously in experiment 1. This strategy made it possible for us to use the experimental data for each nutrient to estimate only one corresponding parameter (K_S).

Average LI integrated over the entire PBR volume representing spatially averaged value, LI_SA, was computed in W/m² as PAR based on these assumptions: a transparent cylindrical shape, uniform illumination from all sides, no light absorption by the biomass-free medium, no top and bottom effects, and light attenuation according to the Beer-Lambert law [12]. Supporting Information (Fig. S4) provides details of the geometry involved in the calculation. The equation for LI_SA is:

where LI₀ = LI at the light-entry surface (W/m²), ɛ = the Beer-Lambert constant (m³/g/m), X = biomass concentration (g/m³), and d = depth of light path from the light-entry surface (m). Suh and Lee[11] verified that e for Synechococcus sp. PCC6301 is 0.255 m³/g/m. Our preliminary study with PCC6803 found a similar value from light-attenuation experiments conducted within our ranges of biomass concentration using PCC6803. Therefore, we utilized ɛ= 0.255 m³/g/m to calculate LI_SA, and the LI_SA was used in Eq. (1).

Table S2 shows the calculated LI_SA values using the measured biomass concentrations and LI₀ at the start of each batch experiment. With LI₀ outside the batch reactor varying from 2.6 to 48 W/m², LI_SA ranged from 2.3 to 43 W/m². The attenuation was ~13% of LI₀ with the biomass concentrations of ~ 27 mg DW/L.

Fig. 1 shows the observed μ_syn values based on biomass measurements taken at 1 and 4 hours for each LI_SA. These results demonstrate the expected trends for LI self-inhibition kinetics: increasing μ_syn for LI_SA up to about 8 W/m², relatively steady μ_syn from about 10 to 25 W/m², and decreasing μ_syn for LI_SA greater than ~30 W/m². The highest observed μ_syn was approximately 1.5/d for LI_SA of ~30 W/m².

Non-linear regression yielded μ_max of 2.8/d, K_L = 11 W/m², and K_L,I = 39 W/m². The value of μ_max was significantly larger than 1.5/d because of LI inhibition, which is quantified by the relatively low value of K_L,I. Based on the LI kinetic model, PCC6803 achieve a benchmark μ_syn ≥ 1/d when LI_SA is in the range of 7 to 62 W/m².

Using the results for LI kinetics (experiment 1, which had sufficient N_i, P_i, and C_i), we could obtain parameter estimates for the growth kinetics of PCC6803 for each nutrient limitation (experiments 2 – 4). We chose LI₀ = 18 W/m² to keep the initial LI_SA at around 16 W/m², which is in the range where LI_SA does not have a sensitive effect on μ_syn (Fig. 1). In each experimental series, we computed the LI-adjusted μ_max value from Eq. (1) and the actual LI_SA value; we call this value μ_syn, LI.

Experiment 2 evaluated the kinetics for N_i limitation, and Fig. 2 shows μ_syn for N_i varying from 0 to 124 mgN/L. The results clearly exhibit a Monod-type saturation pattern with μ_syn saturating at μ_syn,LI = 1.6/d (based on LI limitation from experiment 1). Non-linear regression (R² = 0.97) yields K_S,Ni = 1.4 mgN/L. This value means that the NO₃⁻ concentration in a PBR should be greater than 13 mgN/L to maintain 90% of the maximum specific growth rate, but the rate is slowed to 10% of the maximum value when the NO₃⁻ concentration falls below 0.16 mgN/L. Serious N_i-limitation of cyanobacteria is highly undesirable, because it can cause visible chlorosis, or yellowish coloring of the cells due to the degradation of thylakoid membranes [20] or a decrease in their chlorophyll content [21]. Keeping N_i over 2.3 mgN/L in a PBR is necessary for growing PCC6803 at a benchmark μ_syn of at least 1/d (with LI_SA~16 W/m²).

Fig. 3 shows the photosynthetic specific growth rate as a function of the total-P_i concentration in experiment 3. With the pH held constant at 8±0.2, the dominant P_i species were H₂PO₄⁻ and HPO₄²⁻, present at a molar ratio of 7:43. Thus, the corresponding ranges for H₂PO₄⁻ and HPO₄²⁻ concentrations were 0.14–3.8 mgP/L and 0.86–23 mgP/L, respectively. The experiments showed an almost constant μ_syn value (1.6/d) for total-P_i concentrations of 3 mgP/L and higher. Only when the total-P_i concentration was less than 1 mgP/L, the lowest concentration tested except the control, did μ_syn show a slightly lower value, 1.4/d.

Applying μ_syn,LI (1.6/d) gave a good match of the Monod function to the experimental results and led to an estimated K_S,Pi of 0.06 mgP/L for total P_i (R² = 0.99). This low K_S,Pi value indicates very high affinity for P_i and that it is not necessary to maintain a high concentration of P_i in the PBR, compared to what is required for N_i. For example, maintaining μ_syn ≥ 1/d demands that the total-P_i concentration be at least 0.1 mgP/L.

Although a high specific growth rate does not require a high P_i concentration, it is very important to monitor the P_i concentration and ensure that it does not become seriously limiting. Previous studies reported that a deficiency of P_i may have an important impact on photosynthetic activity, since P is a key component in energy transfer and signal transduction, as well as lipid biosynthesis for algae and cyanobacteria [22]. Once P_i-limitation sets in, it may cause severe change of photosynthesis in PCC6803, as has been noticed for other cyanobacteria [23]. Problems of P_i depletion can be exacerbated when complexation of HPO₄²⁻ and PO₄³⁻ with cations such as Ca²⁺ and Mg²⁺ reduces the availability of P_i by forming octacalcium phosphate (Ca₈(HPO₄)₂(PO₄)₄·5H₂O) and hydroxyapatite (Ca₁₀(OH)₂(PO₄)₆), particularly as the pH increases [24].

In experiment 4, we varied the C_i concentration using NaHCO₃ as the C_i source, not by aeration. This ensured that our results were not confounded by mass transfer resistance from CO_2(g) to CO_2(aq). The pH was maintained at 8 with a phosphate buffer; thus, the mass distribution between CO_2(aq) and HCO₃⁻, the C_i sources for PCC6803 [25], was fixed at a molar ratio of 1:49. The corresponding test ranges for CO_2(aq) and HCO3- were 0–5.6 mgC/L and 0–279 mgC/L, respectively.

Fig. 4 illustrates a clear Monod-like relationship between μ_syn and the C_i concentration. The experimentally observed value peaked at around 1.5/d for C_i. Setting the maximum specific growth rate as μ_syn,LI = 1.6/d (from LI_SA and experiment 1) gave an excellent match with the experimental results, and nonlinear regression estimated K_S, C_i = 0.6 mgC/L for total C_i. Thus, PCC6803 had a high affinity for C_i when HCO₃⁻ was the dominant species at pH around 8. Due to its large dominance over CO_2(aq) in solution, HCO₃⁻ probably was the species taken up by PCC6803 in these experiments [26], and K_S,Ci, = 0.6 mgC/L probably corresponds to the K_S value for HCO₃⁻. The C_i concentration needed to achieve μ_syn of 1/d is approximately 1.0 mgC/L, which is intermediate between 0.1 mgP/L and 2.3 mgN/L.

Table 2 compares our LI parameters to literature values obtained for other cyanobacteria and algae. Our K_L value was similar to other results (6 – 18 W/m²), but K_L,I was considerably smaller than previously reported values. Most notably, Synechococcus sp. PCC7942 had a reported K_L,I of 97 W/m² [27], although light attenuation was not considered in the estimation. Using LI_SA instead of LI0 would make their K_L,I smaller: ~77 W/m² based on the analysis presented in Supporting Information (Table S3 and Fig. S5). Thus, the comparison of K_L,I values suggests that PCC6803 is relatively more sensitive to photo-inhibition than are other phototrophs that have been evaluated. However, significant negative impacts of LI self-inhibition require a substantially greater LI_SA value than ~62 W/m² (Fig. 1).

Table 3 also shows all of our K_S and μ_max values together with literature values obtained with other algae or cyanobacteria. Compared to ranges for algal species with reported μ_max values (0.7 – 1.7/d), PCC6803 had by far the highest μ_max (2.8/d), although the highest observed μ_syn,LI was 1.6/d due to photo-inhibition. The relatively high μ_max values suggests that PCC6803 behaves as an r-strategist, or a copiotroph that employs a rapid-growth strategy [28] that gives it a competitive advantage when conditions allow it to avoid significant limitation by LI or any of the inorganic nutrients.

The values of K_S,Ci, reported for algae (0.002–0.005 mgC/L) are significantly lower than for PCC6803. This means that the growth rate of PCC6803 can be more easily limited by C_i, compared to algae, when an imbalance between supply and utilization of CO₂ occurs. Thus, PCC6803 needs to have a sufficient C_i supply rate to avoid HCO₃⁻ limitation, which would counteract it growth-rate advantage over algae.

Due to PCC6803’s relative sensitivity to the C_i concentration, it would be possible that high pH may slow the C_i-uptake kinetics by changing the dominant C_i form to CO₃²⁻, with CO₂(aq) becoming negligible because the concentrations of available C_i species - CO_2(aq) and HCO₃⁻ - are determined by the pH [25]. The pH also affects the complexation equilibria of cationic and anionic species that can precipitate with C_i. High pH by active photosynthesis can lower the available C_i by precipitating it as part of CaCO₃.

Similar to C_i, PCC6803 is more easily limited by low N_i, compared to other phototrophs for which we have reports [16, 29]. Although our estimated K_S,Ni (1.4 mgN/d) is not far from a literature value (5.3 mgN/L) obtained for Spirulina platensis [9], which, like Synechocystis, is a non-diazotrophic cyanobacterium [20], several algae show K_S,Ni values about one order of magnitude lower (0.02~0.16 mgN/L).

Unlike for C_i and N_i, PCC6803 is not more severely limited by P_i than are other phototrophs. A study using the cyanobacterium Synechococcus sp. PCC7942 reported K_S,Pi = 0.07 mgP/L [30], almost identical with our result (0.06 mgP/L). In addition, the K_S,Pi values of several algae were in a similar range, 0.02 to 0.12 mgP/L, although one report was much higher [31]. Such similarity may indicate that utilization mechanisms of P_i are invariant.