A Simple Model for Algal Photosynthesis for Better Light Utilization with Flashing Light

A simple mathematical model of the algal growth in a flashing light regime is proposed. The model is based on reduction state of fast and slow charge carriers in PS-II. The model constants were fitted to the published experimental data. Chlorophyll a fluorescence measurement was used for estimating the reduction time scale of the fast charge carriers. The data suggest that the reduction time scale of fast charge carriers exponentially drops with an increase in the light intensity. For the studied light intensities, the fast carrier’s reduction time varies between 1 and 16 ms. The reduction time constant of slow charge carriers and average re-oxidation time constant of all charge carriers are estimated to be 300 and 250 ms respectively. First time the model was able to predict the photosynthesis rate for a wide range of light duty values with reasonably good accuracy. The model also accurately predicted 3X improvement in the light utilization efficiency observed in the experiments. In the second part, a simple modelling approach to simulate algal growth in a flat panel photo-bioreactor is presented. Incident light intensity and cell density were used for estimating the thickness of light region and the lateral velocity was used for modelling the movement of cells from one region to another. The model highlighted the impact of boundary layer thickness on the PBR performance and demonstrated that increasing the lateral velocity beyond 0.1 m/s does not help in further improving the productivity.

Fraction of fast charge carriers available for reduction at the start of the light period X FD Fast charge carriers occupied at end of dark time or before the start of light time X FL Fast charge carriers occupied at end of light time or before the start of a dark time X L Fraction of reduced charge carriers at the end of light time, t L X S Fraction of slow charge carriers available for reduction at the start of the light period X SD Slow charge carriers occupied at end of dark time or before the start of light time

Introduction
Fossil energy sources have been greatly exploited for meeting the world's energy demand. This has created many serious environmental, economic, and social challenges. In the last few years, great efforts have been spent on commercializing the cultivation of microalgae for the production of biofuel [1]. In addition to biofuel application, algae have been used as food, feed supplement, and it provides a benefit in capturing and utilizing CO 2 [2]. In comparison to other crops, algae have higher biomass yields per acre of cultivation, can be cultivated on nonarable land with seawater and more importantly it does not compete with the food resources. These advantages make algae a leading choice as a feedstock for the thirdgeneration biofuel industry. Despite all these advantages, techno-economic challenges have prevented the immediate commercialization of the microalgae-derived oil [3]. Research carried out in the last two decades emphasized the importance of achieving higher areal productivity (biomass productivity per unit cultivation area) and better light utilization efficiency. In order to achieve higher areal productivity and better light utilization efficiency, microalgae have to grow rapidly in the dense culture. Dense culture helps in complete absorbance of impinging light along the light path of the cultivation system. Such systems observe a steep light gradient. This steep light gradient creates light and dark zone within the system. The length scale of these zones is dependent on the factors such as algal concentration, light intensity at the surface, light path, and light attenuation coefficient of algal specie. In order to effectively utilize the incident light, the algal cells have to be efficiently moved from light to dark zone and vice versa with an optimum frequency. In the absence of optimum frequency, algal cell might observe photooxidation and/or photo-inhibition [4] due to overexposure to high light irradiance or negative growth (enhanced respiration) due to a longer stay in a dark zone. A combination of both photo-oxidation and excessive respiration can lead to nonobservance of sustained growth [5].
It is believed that improving the mixing rate inside the photo-bioreactor will greatly influence productivity through better light/dark cycle management [6]. However, quantifying the effect of the mixing rate through light/dark cycles on the rate of production is still an open question. Operating a photo-bioreactor (PBR) at the correct light/dark cycle combination to achieve the maximum possible dynamic photosynthesis rate and obtain the highest light integrated biomass productivity is a goal for every PBR design engineer. Experimentally studying the complete parametric space of the problem and then designing a novel PBR system will take significantly higher time and resources. It is therefore necessary (in a practical sense) to build a mathematical model which can integrate complex interaction between the frequency of flashing light, light intensity, and saturating light intensity of individual cells and accurately predict the dynamic rate of photosynthesis. Such a model, validated by experimental results, has potential not only to contribute to the development of new PBR design concepts, but also to reduce the development time and cost.
Dynamic photosynthesis models have been proposed in past for characterizing the effect of light/dark cycles on the photosynthesis rate. The models proposed by Eilers and Peeters [7] and Rubio et al. [8] utilize the concept of a photosynthetic factory (PSF). The PSF is defined as the sum of the light-trapping system, reaction centers and associated apparatus, which are activated by a given amount of light energy to generate charge at PS-II. Fernandez-Sevilla et al. [9] simplified the model for analytically studying the effect of a short time flashes with a very small light duty cycle on the rate of photosynthesis. The assumption of short time flashes with very small light duty cycles allowed them to safely assume an absence of photo-saturation of the PSF. All the above-discussed models [7][8][9] were formulated assuming PSF as a single charge carrier having a single reduction and re-oxidation time.
The fluorescence kinetics model [10] highlighted the presence of different types of charge carriers in PS-II having distinctly different reduction and re-oxidation time scales. Single charge carrier model has limitation in explicitly modelling the charge transfer through these different types of charge carriers. Hence, its applicability will be only limited to the conditions having a short time flashed with very small light duty cycles for which none of the charge carriers are fully reduced in light zone or in light time. A real photobioreactor (PBR) will have very wide distribution of light/ dark cycles combination. Single charge carrier model will have limitation in accurately predicting rate photosynthesis for varied light/dark cycle combinations. Zarmi et al. [11] proposed a four-step model to simulate the four stages of photon capture by the PS-II for every cycle of water splitting. While this is a more elaborate model, but it has too many parameters to be tuned for fitting the experimental data.
This work presents a simple two-charge carrier photosynthesis model that allows predicting the effect of the frequency of the flashing light on the photosynthetic response of microalgae at wider light duty cycle values and for different frequency of flashing light. PS-II fluorescence kinetic model equations proposed by Pospisil and Dau [10] are used for simulating the dynamic state of charge carriers. Experimental data of Zarmi et al. [12] was used for tuning the model and experimental data of Vejrazka et al. [13] was used for additional validation. In the final section, a simple reactor model is presented for predicting the performance of a flat panel PBR (used in experimental study of Qiang et al. [14]) by integrating the proposed kinetic model with the assumed simple flow depiction in a PBR. The reactor model was explored to build understanding of the rate of mixing and the boundary layer thickness on the performance of PBR.

Microorganism
An algal strain of the Nannochloris species isolated from the west coast of Maharashtra, India was used for growth studies. The alga was cultivated in a commercial medium prepared in an artificial sea water (urea 3.33 mM, phosphoric acid 0.21 mM and trace element [15]). The artificial seawater had a salinity of 4% by weight. The cultivation chamber was maintained at a constant temperature of 27 °C and provided with a light intensity of 300 × 10 −06 mol photons/(m 2 .s) in a standard day/night schedule. Regular harvesting of algae culture was done to ensure that the culture was in the logarithmic growth phase. Air-CO 2 mixture with 2% (by volume) CO 2 concentration was continuously fed to the cultivation chamber. Growth was characterized by measuring the optical density (OD) of an algae culture. OD of culture was measured at a wavelength of 750 nm using UV Spectrophotometer (UV 1800, Shimadzu). The culture was not allowed to cross optical density (OD) of 2 during all the experiments. All PI (growth rate) curve experiments [12], dynamic photosynthesis rate measurements and fluorescence kinetics measurements were performed on the sample during the logarithmic growth phase.

Dynamic Photosynthesis Rate Measurement
In the present work experimental setup ( Fig. 7 in Appendix I in supplementary information) described in Zarmi et al. [12] was used for collecting additional data required for detailed validation of the model. For pulsed-light experiments, four PSI light sources (Model SL-3500, with an LC-100 PSI light controller) permitted independently tuning the irradiation and dark times from 1 ms to 999 ms. The small glass reactors had the optical path of 3 cm, width 10 cm and height 15 cm. An operating height of 10 cm was used with the total fluid volume of 300 ml. Air-CO 2 mixture with 2% (by vol) CO 2 concentration was continuously fed to the glass reactor.
For both continuous and pulsed light regimes, LEDs were used for illuminating the culture. Photon flux density for photosynthetically-active radiation was measured using Apogee Instruments' quantum meter model MQ-200. The 13 cm × 13 cm LED panel was sited less than 1 cm from the cultivation glass reactor. Light intensity was measured at the center of each of nine equal-area regions comprising the reactor's illuminated surface, and the reported light intensity represents the average over these nine sections.
Growth of the culture was measured over an illumination period of 6 h, followed by a period of 6 h of a dark time, followed by a fresh run with these cycles repeated for 24 h. Each day, the culture was harvested and brought to the desired starting operating optical density of 0.08. Optical density was measured at a wavelength of 750 nm at the start (OD1) and end of irradiation (OD2) to get the specific growth rate, µ (1/h) over time Δt (h). All runs were repeated ~ 20 times, from which average and standard deviation values were determined.

Fluorescence kinetics measurement
The transients were measured using dark-adapted cells corresponding to an OD of 1 measured at 750 nm. Fluorescence induction kinetics for alga was recorded using Dual pulse amplitude modulation (PAM)-100 (Heinz Walz, Germany). All the measurements were performed on the dark-adapted sample for which the cells were incubated in darkness for 10 min immediately after sampling [16][17][18]. The sample was illuminated with a modulated light < 30 × 10 −06 mol photons/ (m 2 .s) from a light-emitting diode (with a peak wavelength at 655 nm); and minimal fluorescence (F O -corresponding to 'O' phase) was determined at 50 µs [19]. Maximal fluorescence yield (F m ) was determined by exposure to far-red illumination (at 710 nm) at the end of 100 ms for different light intensities (50, 100, 300, 500, 1000, 3000 × 10 −06 mol photons/(m 2 .s)) with intermittent 'J' and 'I' phase followed by a peak at 'P' phase. Fluorescence emission data were acquired with a sampling frequency of 10 kHz. The transient was calculated by averaging 20 replicates for each light intensity. The obtained averaged curve was then smoothed using the tailing moving average method.

Previous Dynamic Model
Fernandez-Sevilla et al. [9] simplified the model of Rubio et al. [8] by assuming its application for simulating photosynthesis for short time flashes with a very small light duty cycle. Short time flashes with a very small light duty cycle generates a small fraction of excited intermediates and makes the photo-inhibition process irrelevant. Figure 1a shows the comparison of model predictions obtained with a model presented by Fernandez-Sevilla et al. [9] with the experimental data of Zarmi et al. [12] for a constant light intensity of 1000 × 10 −06 mol photons/(m 2 .s) and light flash time of 10 ms. It can be seen from Fig. 1a that the model reasonably predicted the overall trend in the data. A Δt closer look at the comparison highlights a few clear deviations. First, at a relatively higher duty cycle (smaller dark time), the model over-predicted the photosynthesis rate. The assumption of a very small fraction of excited intermediates generated during the light/dark cycle, made during this model development, could be the possible reason for such an over-prediction. This assumption does not capture the possible effect of flux saturation observed with relatively higher duty cycles. Second, the experimental data shows an interesting feature which is better highlighted when the growth rate data is plotted in a light normalized manner. Fig. 1b shows the comparison of light normalized growth rate data of Zarmi et al. [12] with predictions obtained from Fernandez-Sevilla et al. [9] model. It can be seen that the model is not able to correctly predict the maxima pattern observed in the experimental data. This highlights the   [9] with experimental data of Zarmi et al. [12] for flash time of 10 ms and I = 1000 × 10 − . 06 mol photons/(m2.s). a Comparison of model predictions for different values of β with experimental data [12] (β is the characteristic frequency as defined by Sevilla et al. [9]). b Comparison of predicted value of light normalized specific growth rate obtained for β = 50 1/s with experimental data [12] importance of developing a kinetic model framework using multiple charge carriers.

Model Equations
The photosynthesis-irradiance (PI) curve is a kinetic response of all the terrestrial and marine plants photosynthetic responses to changes in light intensity. In the absence of the photo-inhibition term, the PI curve is mathematically represented as And where P is the specific growth rate (1/h), P max is the maximum specific growth rate (1/h) at higher than saturating light intensity, X C is the fractional activity of the photosynthetic unit (PSU) under continuous light, R M is the growth loss rate due to respiration (1/h), I is the incident light intensity and I K is the saturation light intensity. At very high light intensity (I > > I K ), X C ≈1 and PSUs are fully active (P≈ P max ). For lower light intensity (I ≤ I K ), X C < 1, indicating that PSUs are only partially active (P < P max ). The formulation of Eq. (3) is similar to the equation used by Wu and Merchuk [20], where they assumed the specific growth rate to be proportional to the fraction of closed PS-II. The current model formulation also assumes a similar proportionality of specific growth rate with the fraction of reduced (occupied) charge carriers in the PSII electron transport chain.
X C used in Eq. (3) is the fractional activity of PSU under the continuous light. However, in flashing light conditions, the PSUs will essentially show the dynamic behavior under varying light conditions. Hence, it is necessary to obtain the time-averaged activity of the PSU under flashing light conditions ( X ) to be used in Eq. (3) for obtaining the growth rate. In the following section, we present a model formulation capable of simulating dynamic photosynthetic responses.
PS-II fluorescence (FL) kinetics closely represent the state of acceptor and donor side charge carriers (Kautsky effect) [19,21]. PS-II FL kinetics have been widely utilized for quantifying the potential performance of PSU under given operating and/or stress conditions. The FL kinetics under saturating light conditions is usually represented by the OJIP curve. The OJIP phase of the Chl a fluorescence induction curve is referred to as fast transient. It is obtained when a dark-adapted algae is exposed to a saturating light. In the OJIP curve, 'O' stands for 'origin' (minimal fluorescence) and 'P' for 'peak' (maximum fluorescence). 'J' and 'I' are the inflection points between 'O' and 'P' levels.
All the three O-J, J-I and I-P phases are affected by the sequential reduction of electron transfer intermediates in PSII. Pospisil and Dau [10] proposed a mathematical formulation based on the sum of exponentials to simulate the OJIP curve. Since FL is a function of the state of the charge carrier, the current model framework uses a similar sum of exponentials to simulate the dynamic behavior of PS-II charge carriers.
In the current approach, all the PS-II charge carriers are categorized as either fast or slow carriers. All the static charge carrier sites are termed as fast charge carriers (i.e., Reaction center R C , Pheophytin Pheo, Quinone A Q A , Quinone B Q B ). The mobile carriers are termed as slow charge carriers (e.g., Plastoquinone, PQ). Light excitation of R C leads to a reduction of available (re-oxidized) charge carriers. Once all the fast carriers are occupied (reduced), then the slow carriers subsequently get occupied. The time constant of the OJ phase, in the OJIP curve, is dependent on light intensity [22]. Hence it was hypothesized that the rate of reduction of fast carriers (τ FL ) is also a function of light intensity. The time constant of the JIP phase is not dependent on the light intensity, but on temperature [22]. Based on these learnings, the rate constant of reduction of the slow carriers is assumed to be a constant value (τ SL ) and is independent of light intensity. In the photosynthesis process, charge carriers are re-oxidized after delivering their charge to the downstream process. In the present formulation, the rate constant of re-oxidation of both fast and slow charge carriers was assumed to be the same and denoted as a constant value (τ D ). This suggests that charge carriers having the highest re-oxidation time will determine the overall timescale of the re-oxidation process. Figure 2 schematically explains the proposed model framework.
The reduction rate dynamics of charge carriers are based on a similar formulation (sum of exponentials) for FL kinetics as proposed by Pospisil and Dau [10]. For dark-adapted cells the fraction of reduced charge carriers at the end of light time (t L ) given as: where X L , X FL , and X SL are the fractions of the reduced total, fast, and slow carriers (respectively) at the end of the light time (t L ). τ FL and τ SL are reduction time constant for fast and slow charge carriers respectively. The exponential term with a ratio of t L and τ FL (or τ SL ) represents the first order (4a) reduction kinetics of the carriers. X F and X S are the fractions of fast and slow carriers, respectively, available for reduction at the start of the light period. Their exact fraction depends on light exposure history, light intensity, and the fraction of fast carriers (ε), and is given as Equations (5a) and (5b) is formulated to ensure that fast carriers reduce before slow carriers. It also ensures that at low light intensity X C is less than ε. In this condition, only fast carriers get reduced (X F = X C ) and slow charge carriers remain unoccupied (X S = 0). At high light intensity, all the fast charge carriers are in a reduced state (X F = ε) and only excess charges will reduce the slow carriers (X S = X C -ε).
During the dark period, the reduced charge carriers get re-oxidized. The extent of re-oxidation of charge carriers is related to fluorescence decay kinetics (in the dark) and is simulated via a sum of exponentials formulation [10]. The current model utilizes similar formulations to obtain the extent of carriers that stay reduced at the end of the dark period (t D ) as: where X D , X FD and X SD represent the total, fast, and slow carriers, respectively, remaining reduced at the end of the dark time. The re-oxidation time constant ( D ) is the same for both charge carriers. The exponential term with the ratio of t D and D represents first-order re-oxidation kinetics. For simulating fluorescence decay two or three-stage decay is usually considered [10]. It is also intuitive to assume that fast carriers would have comparatively smaller re-oxidation times as compared to slow carriers. However, when the current model was fitted to experimental data, it was found that having the same time constant ( D ) for all carriers, both fast and slow, gave a good fit (not presented here with brevity). This indicates that re-oxidation of the slow carriers controls the overall re-oxidation of PSUs.
The slower re-oxidation of all carriers makes the model respond as if fast carriers deliver charges to slower ones, and the slower carriers deliver the charges to the biomass generation machinery.

Final Model Equations
In dark-adapted cells (t = 0) all the charge carriers are in the re-oxidized state (X SD = X FD = 0). This assumption is only valid for the first light-dark (L-D) cycle of the culture. However, in reality, at the end of each L-D cycle, the carriers might not get completely re-oxidized. The Eqs. (4) are accordingly modified to include the prereduced fraction of charge carriers from previous L-D cycles. This arrangement enables the model to consider the flashing light history in determining productivity. Modified forms of Eqs. (4b) and (4c) are presented below in the final form: After several L-D cycles, a pseudo-steady state is reached. At pseudo-steady state, the reduced fraction of carriers at the start and end of L-D cycles is similar. The time-averaged reduced fraction of carriers is found as: The specific growth rate is then obtained by using the time-averaged charge carrier occupancy in the photosynthetic activity-intensity rate equation: Equations (7a) and (6b), (7b), and (6c) are solved iteratively until a pseudo steady state is reached. The pseudo steady state is determined by < 0.1% change in the calculated X over consecutive L-D cycles. The final biomass concentration at the end of the growth period is obtained as where C initial and C final are the algae biomass concentration in the PBR at the start and end of the growth experiment, respectively, and Δt is the time of the growth experiment (h).
This model formulation of (time averaged pseudo equilibrium reduction state (Eq. (8)) is useful for determining algal growth happening under artificial flashing lighting situation (where light-dark cycle times are repetitive and identical). The same formulation can also be used for real PBR case, where light-dark cycle times are repetitive but not identical. In such case biomass growth (Eq. (9)) is averaged over several cycles.
Zarmi et al. [12] carried out flashing light growth experiments under very dilute algal densities to ensure all the cells see a similar light intensity. In their experiments, a significant amount of light (> 90% of incident light) was transmitted across the PBR during the whole growth period. For correctly quantifying the improvement in light utilization under flashing lights, the photosynthetic efficiency (PE) is calculated. In the present work, the PE calculations are made based on the amount of light absorbed by the algal cells, rather than using incident light. Following the Beer-Lambert Law, under dilute concentrations, the light absorbed (I abs ) is based on the average biomass concentration obtained as the difference between incident intensity (I 0 ) and transmitted intensity (I trans ) as follows: where α is the light attenuation coefficient having a value of 1.1 l/(g.cm) [13], and z is the thickness of a PBR (cm). Based on the absorbed light intensity, the total moles of photons (I moles ) absorbed in one hour is obtained as where A is the illuminated area of a PBR (m 2 ). Photosynthetic efficiency (PE) based on absorbed light is estimated as where V is the volume of PBR in liter (l), C V is the calorific value of algal biomass equal to 22 kJ/g, and E V is the average energy per mole of a PAR photon equal to 219 kJ/mol [23].

Model Parameters Estimation
A total of seven unknown parameters viz. P max , I K , R M , τ FL , τ SL , τ D and ε are present in the proposed model. The experimental data of algal growth in continuous light from Zarmi et al. [12] was used for estimating the values of P max , I K and R M . The experimentally measured continuous light growth data were fitted with a standard PI curve equation without photo-inhibition (please refer Fig. 8 in APPENDIX I in supplementary information). Equations (2) and (3)  O-J phase determines the charge separation from PS-II reaction center and subsequent reduction of Q A (fast charge carrier). Therefore, the O-J phase represents primary photochemical reactions that are light-dependent and independent of temperature. The J-I-P phase represents the thermal phase of photochemistry. Variable fluorescence is measured for dark-adapted algal suspension using Dual-PAM-100 (from Heinz Walz GmbH) for different light intensities and at up to 100 ms (width). Figure 3a presents the smoothened transient of all the five light intensities. The O-J time was identified for each intensity as the first point of inflection. It was observed that the O-J time is very clearly related to the light intensity and can be well correlated using a power-law Eq. (14).
The O-J time in the current model configuration represents the reduction time constant of fast charge carriers (τ FL ) and is related to incident light intensity (I) as, where 'B' and 'm' are constants and are found to be 165 and -0.56 respectively (see Fig. 3b The presented model relies on the categorization of PS-II charge carriers as fast and slow. It needs some estimate of the relative fraction of fast and slow charge carriers per PS-II. For estimation purposes, an assumption of 3 fast charge carrier sites per PS-II is made, i.e., one each for a R C , a Pheo and a Q A . For a slow charge carrier, the number of PQs per PS-II is needed. Several experimental studies used Chl a variable fluorescence measurement and estimated 4-10 PQ molecules per PS-II [24][25][26][27]. Each PQ molecule carries 2 charges. Therefore, the current model framework guesses the presence of 8-20 slow charge carrier sites per PS-II. Thus, the fraction of fast charge carriers to total charge carrier sites per PS-II (ε) would be in the range of 0.13-0.28.
Two unconstrained parameters (τ SL and τ D ) and one constrained parameter (ε) were fitted to the flashing light growth data [12] using simplicial homology global optimization (SHGO) in Python (Scipy). Table 1   An effort was made to have a model formulation with as many parameters as possible by direct experimental measurement (5 out of 8) and minimize the number of model parameters needing numerical fitting to experimental data (only 3 out of 8). This would allow the model to be used for any other algal species once the relevant experimental measurements are done.
A comparison of predicted photosynthetic efficiency with the experimental data for light times varied between 5 to 35 ms and dark time varied between 0 to 700 ms is shown in Fig. 4. Figure 4 shows good agreement of model prediction with the experimental data from Zarmi et al. [12].
The model is then employed for simulating specific light/dark cycle combinations where light flash time varied from 5 to 35 ms by keeping averaged light intensity i.e. ϕI 0 constant. Table 2 presents the comparison of predicted and experimental values of the photosynthetic efficiency estimated for different combinations of L-D cycles having the same averaged light intensity. The two carrier model presented in this work predicted the effect of flash time on the photosynthetic efficiency. Thus, it can be said that the proposed two-carrier model was able to predict the effect of flash time on the rate of production and hence can help in identifying the optimum flash time essential for maximizing the light integration.

Additional Validation of Model
The kinetic model is further utilized for predicting the growth rate at different light intensities with different multiple L-D cycles. It can be seen from Fig. 5a that for all the studied light intensities, the model predictions are in good agreement with the experimental data [12], for both continuous light as well as for the L-D cycle having a light time of 150 ms and dark time of 50 ms. It must be noted that these experimental data (growth rate under flashing light of light intensity other than 1000 × 10 −06 mol photons/ (m 2 .s)) were not used for the model parameter tuning. The model significantly under predicts the PE value for the light intensity of 500 × 10 −06 mol photons/(m 2 .s) with an L-D cycle of 10 ms light time and 290 ms dark time. The model predicted ~ 22.5% PE as compared to 30% observed in the experiments with the same light conditions. This highlighted the necessity of including more experimental data at a lower light intensity to effectively tune the model.
The model was then extended further for predicting the flashing light growth data presented by Vejrazka et al. [13]. This experimental dataset was chosen for the current model validation because (a) it used a wide range of realistic combinations of light times and dark times for flashing light growth conditions; (b) the PI curve characteristics of Chlamydomonas reinhardtii used their work have similar characteristics as of Nannochloris species used in Zarmi et al. [12]; (c) the growth rate data for continuous and flashing light were available under similar light conditions. Vejrazka et al. [13] had used a 12.3 ml PBR with a light path of 1.5 cm. They measured oxygen evolution rate with a total flashing light intensity of 1080 × 10 −06 mol photons/(m 2 .s) provided from both sides of a PBR.
The net specific oxygen production rate ( P O 2 ,n,LD ×10 -06 mol O 2 /(g.s)) reported in the original article is correlated to the biomass-based specific growth rate (P, 1/h)   based on elemental analysis and growth stoichiometry as presented by Vejrazka et al. [28]: The parity plot to compare predicted and measured specific growth rates for the flashing light conditions is shown in Fig. 5b. The model predicted the flashing light growth rates with reasonably good accuracy. In the presented comparison, experimental data for 100 Hz frequency were not used as the model was not tuned with data obtained with such high flashing light frequency conditions. Also, for a real PBR system with high algal density, a realistic turbulent frequency would be smaller than 25 Hz [29].
Overall, it can be said that the proposed two-carrier kinetics model improved the capability in predicting the dynamic rate of photosynthesis for a very wide range of light/dark cycle time combinations. It must be noted that the data used for model tuning was obtained with white light. It is possible that these tuning parameters might differ for red and/or blue light. In such situation, it is appropriate to collect data for determining the model parameters.

Photobioreactor (PBR) Model Development
The main motivation for building the kinetics model was to establish a tool that would significantly reduce the time and resources for developing highly efficient novel PBR designs. In this section, the capability of the present kinetic model in predicting PBR productivity by integrating it with simplified fluid dynamics is demonstrated. It must be noted that the authors are already working on a more comprehensive reactor model by integrating CFD-predicted fluid dynamics with the proposed kinetics model; those efforts are beyond the scope of the present paper. Qiang et al. [14] demonstrated the cultivation of algae (Spirulina platensis) in the range of 5-50 g/l concentration for 6 different light intensity values up to 8000 × 10 −06 mol photons/(m 2 .s) in a flat panel vertical PBR having a small optical path of 1.4 cm. This PBR design -at a very high algae biomass concentration and high mixing rates -allows for the generation of the flashing light effect. They measured the biomass growth rate every 4-8 h for a range of light intensities and initial biomass density and included replicate experiments. The experimental data of Qiang et al. [14] was used to validate the simple reactor model presented in this paper. This experimental dataset was selected for validating the model because (a) it demonstrated high photosynthetic efficiency of 15-16% as observed in Zarmi et al. [12], (b) experiments were conducted under controlled temperature conditions, and (c) productivity data were obtained for a very wide range of biomass density (5-50 g/l) and 6 different light conditions. Although the final goal is to integrate photosynthesis kinetics with the prevailing fluid dynamics, it is still a farfetched goal due to a limited understanding of PBR hydrodynamics with dense cultures. It is very well known that with an increase in cell density/concentration the thickness of the light zone reduces and potentially approaches the laminar boundary layer thickness. Resolving turbulent flow near the wall/ light zone thus becomes a separate research problem. It must be noted that this work aims to present a novel kinetic model capable of predicting dynamic photosynthesis rate for different light/dark cycles. However, for demonstrating its capability in adequately predicting the PBR performance, it was integrated with the assumed simplified flow pattern in a PBR. Algal cells are very small in size (potentially even smaller than the Kolmogorov length scale of turbulence; see APPENDIX II in supplementary information for sample calculation of Kolmogorov length scale) and have density nearly equal to water. It is, therefore, safe to assume that the  [12] for different light intensities; symbol denotes data. b Comparison of predicted specific growth rate with experimental data [13] cells would move in a PBR with fluid velocity, and averaged fluid velocity can be used for estimating the average residence time of cells in light and dark zones of a PBR.

Reactor Model Development
Qiang et al. [14] conducted biomass growth rate experiments in a flat panel vertical PBR. The air-CO 2 mixture was continuously sparged from the bottom of the reactor with a velocity of 3 vvm (volume of air per minute per culture volume). At a sparging rate of 3 vvm, a highly transient gas-liquid flow gets generated within a PBR. However, a simple flow pattern was assumed for estimating the average residence time of cells in light and dark zones.
Readers are requested to refer Fig. 9 in APPENDIX I in supplementary information for visualizing the discussed flow pattern. The flow pattern also includes the presence of a boundary layer near the PBR wall. The boundary layer will have very low fluid velocities, which will lead to very slow transport of algal cells from the boundary layer into the bulk fluid and vice versa. It is also well known that particles smaller than turbulent eddies significantly damp the prevailing free stream turbulence due to their interaction with them [30]. Such a dampening of turbulence will also contribute to an increase in the boundary layer thickness. Therefore, for the same mixing rate, the observed boundary layer thickness will increase with an increase in biomass concentration. With all these fluid dynamic attributes, the developed photosynthesis kinetics model is integrated with the assumed flow pattern to predict the experimental data of Qiang et al. [14]. The laminar boundary layer (BL) present next to the wall has a thickness denoted by 'δ BL '. Since the mass transport across the boundary layer thickness is very slow, from the model simplification perspective, assumed prolonged light exposure of the algal cells present in the boundary layer. The remainder of the volume (z-2 × δ BL ) of the PBR has a rapid movement of algal cells due to turbulent flow. In this bulk volume, the algal cells were hypothesized to rapidly move from one boundary layer to another with a lateral velocity (Z vel ), which is generated due to the bubble passage. The light attenuates within the PBR to generate a light zone (LZ) and a dark zone (DZ). Depending on the algal density and light intensity, the light zone thickness can extend beyond the BL into the bulk. The average light intensity in BL is always higher than the average light intensity in the LZ. Hence, the specific growth rate in BL is going to be always higher than the specific growth rate in the LZ. One must keep in mind that the timescales of biomass growth are much larger than the timescales of mass exchange between BL and bulk PBR volume. Therefore, potentially no build-up of algal concentration in BL owing to a higher growth rate in BL can occur, and hence it is safe to assume a constant biomass density across the thickness of the PBR. Qiang and Richmond [29] roughly estimated Z vel to be in the range of 0.1-0.3 m/s when operated at a 4 vvm gas sparging rate. In the present study, simulations were carried out with a Z vel of 0.2 m/s for a gas sparging rate of 3 vvm.
The estimation of the laminar boundary layer thickness in a PBR is complex. Generally, the boundary layer thickness is a function of Reynolds number (i.e., viscosity and density of the algal slurry, velocity, and characteristic length). However, the presence of algal cells dampens the turbulence and thus contributes to further extending the laminar boundary layer thickness. In the current analysis, the laminar boundary layer thickness (δ BL ) is proposed to be a function of algal concentration only (Table 3). Thus, this incorporates the combined effects of viscosity and turbulence modulation due to the presence of algal cells on the laminar boundary layer thickness.
Using lateral velocity (Z vel ), the incident light intensity from one side (I 0 ), the light path in the PBR (z) and algal concentration (C), the other parameters (given in Table 3) were obtained to be used in the photosynthesis kinetics model for predicting the biomass productivity. The constants 'K' and 'n' were fitted to adequately match the experimental data of Qiang et al. [14]. Those values are K = 9.8 × 10 −03 and n = 0.48, for C in g/l and δ BL in cm. The estimated boundary layer thickness (δ BL ) for algal density in the range of 10 to 50 g/l varies from 0.1 to 0.5 mm. As depicted in Fig. 9 in APPENDIX I in supplementary information, the light zone extending beyond the laminar boundary layer has a thickness of 'δ L '. The thickness of LZ (δ BL + δ L ) at one side of the PBR is calculated using incident light (I 0 ) and compensation light intensity (I C = 70 × 10 −06 mol photons/(m 2 .s)) as obtained from the PI curve ( Fig. 8 in APPENDIX I in supplementary information).  .C. L 1 3

Comparison of Reactor Model Predictions with Experimental data
A comparison of model predictions with the experimental data of Qiang et al. [14] is presented in Fig. 6a. Except for incident light intensity of 2000 × 10 −06 mol photons/(m 2 .s), the model was able to adequately predict the volumetric production rate in a PBR. The model was also able to capture the key features such as the existence of an optimal biomass density for given light intensity and the increase in optimal density with an increase in the incident light intensity. The exact reason for under-predicting the volumetric biomass productivity for an incident light intensity of 2000 × 10 −06 mol photons/(m 2 .s) is not known. For this data set, even Greenwald et al. [31] observed deviation while correlating optimal cell density with an incident light intensity. The ultra-high biomass productivities demonstrated by Qiang et al. [14] were essentially achieved for two reasons. First, they operated their PBR at an extremely high biomass density to achieve effective light intensity less than saturation light intensity. Second, they limited the light exposure time by creating L-D zones, thus avoiding photo-saturation. Due to this, the charge carriers (plastoquinone pool) were continuously under redox poise i.e., the PQ pool was reduced in the light time, and re-oxidized during the subsequent dark time. This ensured minimal wastage of light by the PSUhence resulting in higher PE at a light intensity much higher than the saturated light intensity I K .
Even with the assumption of a simple flow pattern, the model has predicted the experimental data with reasonable accuracy. To further improve the predictive capability of the model, one has to include a more realistic flow pattern with kinetics. There are several publications [32][33][34] on CFD simulations of gas-liquid flows in a flat bubble column reactor that approximates a flat panel PBR. Coupling kinetics with such CFD models will help in accurately predicting a PBR performance. Since the current model is only based on algebraic equations (without ODEs/PDEs) it has the added advantage of being computationally less demanding. Despite this simplification of the PBR flow pattern description, the current model framework provided an opportunity to get more insights into building a preliminary mechanistic understanding of PBR performance. In subsequent sections, the reactor model is used to build our understanding of the effects of the magnitude of Z vel and the boundary layer thickness δ BL on biomass productivity.

Effect of Lateral Velocity and Boundary Layer Thickness
In the present reactor model framework, Z vel was tuned to match the experimental data. However, it is also worthwhile to build our understanding of the impact of Z vel on biomass productivity. Model simulations were carried out for a range of Z vel values varying from 0.01 m/s to 0.2 m/s. Simulations were conducted with a light intensity total of 1200 × 10 −06 mol photon/(m 2 .s) provided from both sides of a PBR. It must be noted that for the Z vel value of 0.01 m/s L-D cycle time (t C ) corresponds to 1400 ms and for 0.2 m/s it corresponds to 70 ms.
The model predictions are shown in Fig. 6b. For very low biomass density (≤ 3 g/l) there is no effect of t C on productivity. This is because the light zone covers the light path length (1.4 cm). In such a situation, the algal growth occurs under continuous illumination conditions; hence, for all cycle times, the model predicted similar productivity. The model predicted significant productivity improvement for t C values ≤ 280 ms having lateral velocity (Z vel ) ≥ 0.05 m/s. The model also predicted marginal improvement in productivity beyond the lateral velocity (Z vel ) value of 0.1 m/s or cycle time (t C ) ≤ 140 ms. These predictions are found to be consistent with the experimental observations reported by Qiang et al. [14].
Algal biomass alters rheology, boundary layer thickness, and turbulence in the PBR. Although boundary layer thickness is a very small fraction of the PBR volume, it influences the biomass productivity to a significant extent as it impacts the residence time of cells in the light zone. Simulations were conducted with a light intensity total of 4000 × 10 -06 mol photons/(m 2 .s) provided from both sides of a PBR. In these simulations boundary layer thickness (estimated using equation (a) in Table 3) was varied from the base case by ± 20%. Figure 6c presents the effect of boundary layer thickness on the predicted biomass productivity. The model predicted higher biomass growth for the thinner boundary layer and the productivity drops as the thickness of the boundary layer increases. This highlights the importance of the boundary layer in accurately predicting biomass productivity. It must be noted that the boundary layer contributes only 1-6% of a PBR volume but it still has a substantial impact on biomass productivity. The commercial algal PBRs of Subitec GmbH [35], creates intermittent circulation cells with higher liquid velocity to not only create L-D cycles but also help in reducing the boundary layer thickness. Therefore, during the development of novel PBRs, development engineers should focus on creating innovative internal structures in the PBR for reducing the effective boundary layer thickness for improving biomass productivity.
Finally, it can be said that greater applicability of the kinetics model was demonstrated by coupling it with an assumed prevailing simple flow in a PBR. The PBR reactor model was able to predict the biomass productivity data of Qiang et al. [14] with reasonable accuracy. The model also provided the greater insight into the degree of mixing required for achieving higher productivity. The model has also helped in highlighting is the role of boundary layer thickness on biomass productivity which has never been

Conclusions
A simple photosynthetic kinetic model is presented in this work. Model equations were formulated assuming the presence of two types of charge carriers in PS-II having different timescales of reduction and re-oxidation reactions. A nice agreement between the experimental data and model predictions was obtained for continuous light as well as for flashing light experiments. Definition of light flash time and its relationship with incident light intensity was for the first time demonstrated in this work. Its integration in a kinetic model allowed us to predict the rate of dynamic photosynthesis process for a very wide combination of light/dark cycles. The study also highlighted that the same model parameters can be used with confidence for other algal species having similar PI curve characteristics. The model was then extended to simulate the performance of a flat panel PBR. By including simple flow characteristics such as lateral velocity and the boundary layer thickness, the model was able to predict the performance of a flat panel PBR with reasonably good accuracy. The model explained and quantified the critical role of the boundary layer on biomass productivity. Such a comprehensive model framework offers great potential as a tool for conceptualizing, designing, and evaluating novel photo-bioreactor systems with reduced development cost and time.