# Scalable model development of carbon photosynthetic assimilation and partitioning in a green microalga during nitrogen starvation

Paul Chambonniere, Evita Dollon, Alexandra Dimitriades-Lemaire, Jean-François Sassi, Florian Delrue

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## To cite this version:

Paul Chambonniere, Evita Dollon, Alexandra Dimitriades-Lemaire, Jean-François Sassi, Florian Delrue. Scalable model development of carbon photosynthetic assimilation and partitioning in a green microalga during nitrogen starvation. Bioresource Technology, 2026, 441, pp.133585. 10.1016/j.biortech.2025.133585. cea-05424045

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# Scalable model development of carbon photosynthetic assimilation and partitioning in a green microalga during nitrogen starvation

Paul Chambonniere<sup>a,b,*</sup>, Evita Dollon<sup>b,c D</sup>, Alexandra Dimitriades-Lemaire<sup>b D</sup>, Jean-François Sassi<sup>b D</sup>, Florian Delrue<sup>b</sup>

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## HIGHLIGHTS

* Nitrogen starvation caused approx. 2000-fold photosynthetic activity decrease.

* Decrease of photosynthetic activity was faster at higher light density.

* Photosynthetic carbon assimilation was predictable from biomass C:N ratio.

* Carbon partition between protein, lipid, and carbohydrate pools was described.

* A scalable model of carbohydrate/lipid content during N starvation was calibrated.

## GRAPHICAL ABSTRACT

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## ARTICLE INFO

**Keywords:**

Microalgae

Lipid

Biofuel *Chlorella vulgaris*

Mathematical modelling

Photosynthesis

## ABSTRACT

Lipid accumulation in green microalgae is induced by stresses (e.g. nitrogen starvation) which compromise photosynthetic activity resulting in significantly lower biomass productivity than under nutrient replete conditions. While algae photosynthetic growth has been well characterized and modelled under nutrient replete conditions, the loss of photosynthetic activity during nitrogen starvation lacks specific studies to determine suitable parameterisation. The loss of photosynthetic activity of the lipid-accumulating microalgae *Chlorella vulgaris* NIES 227 was studied under varying light intensities during nitrogen starvation. Partition of assimilated carbon between the different macromolecules pools (carbohydrates, lipids, and proteins) was concomitantly monitored. The results showed that the decrease of photosynthetic activity correlated well to the increase of cell C:N ratio (R² = 0.883, N = 65) enabling to develop a model of microalgae growth and carbon partition under nitrogen starvation. Biomass dry-weight increase could be predicted with good accuracy (R² = 0.940, N = 66), as total lipid and carbohydrate production could also be predicted with fair accuracy (R² = 0.841 and 0.618, respectively). The present study henceforth showed that modelling microalgae productivity based on photosynthetic activity inferred from local light intensity, as done in scalable models under nutrient replete conditions, may be extended to nitrogen starvation conditions and enabled the prediction of lipids and carbohydrates

<footnote><sup>a</sup> Aix Marseille Université, Centrale Med, CNRS, M2P2 Marseille, France</footnote>

<footnote><sup>b</sup> MicroAlgae Processes Platform—CEA, CEA Tech Région Sud, 13108 Saint Paul lez Durance, France</footnote>

<footnote><sup>c</sup> INSERM U1338, Laboratoire de Biologie Computationnelle et Quantitative CNRS UMR 7238, Sorbonne Université, Paris, France</footnote>

<footnote>* Corresponding author at: Aix Marseille Université - M2P2, Europole de l'Arbois, Bat. Laennec, Hall C, BP 80, 13545 Aix en Provence cedex, France.
E-mail address: paul.chambonniere@univ-amu.fr (P. Chambonniere).</footnote>
productivity. The model proposed should thus prove useful in optimizing photobioreactors design for the pro-
duction of important energetic molecules based on light distribution knowledge.

# 1. Introduction

Microalgae are fast-growing photosynthetic microorganisms which have been proposed as an interesting solution for carbon dioxide (CO2) capture (Daneshvar et al., 2022). Doubling times as low as 7.4 h have for instance been reported for the microalgae *Chlorella vulgaris* grown in photoautotrophy (Li et al., 2023). Because some strains have also been evidenced to accumulate large quantities of lipids (Arif et al., 2020; Udayan et al., 2022), microalgae are perceived as a good candidate for the generation of a biofuel feedstock (Amjith and Bavanh, 2022; Chisti, 2007; Gouveia, 2011). This is particularly the case for the strain *Chlor-ella vulgaris* NIES 227 which has been shown to store photoautotrophically 54 % of its dry-weight in the form of lipids (Ramírez-Romero et al., 2022). Lipid accumulation in microalgae is driven by stresses inducing an imbalance in the energy harvested by the photosynthetic apparatus which overpasses the reducing power demanded by the cell anabolism for carbohydrate generation through Calvin cycle (Morales et al., 2021; Ördög et al., 2016). The excess reducing power is directed toward the synthesis of lipids to avoid reductive damage to cell components. Lipids synthesized become in return an energy and carbon source available when the imbalance is corrected (Roessler, 1990). Among a range of conditions under which microalgae have been found to accumulate lipids, nitrogen (N) deprivation is the most commonly adopted due to being cost-effective and easy to implement (Morales et al., 2021). However, because the stress applied leads to a decrease in the growth rate of the microalgae, it is challenging to achieve high lipid productivity at high lipid content (Chambonniere et al., 2022) which is needed to lower overall microalgae lipid production cost (Rogers et al., 2014). Developing economically competitive microalgae cultivation for lipid production at industrial scale will therefore demand optimized cultivation conditions, which could be achieved using scalable lipid production models.

Numerous predictive models for microalgae production have been developed (Béchet et al., 2013; Bekirogullari et al., 2020) consisting in phenomenological models calibrated from regressions on experimental data. Recent development in analysis and computational capacity lead to the rise of metabolic network models (Baroukh et al., 2015) and machine learning approaches (Syed et al., 2024). Such modelling approaches provide new perspectives for the prediction of lipid productivity (Baroukh et al., 2022; Liyanaarachchi et al., 2021; Ren et al., 2019). Machine learning models however require standardized training experiments which can only be achieved at laboratory scale meaning some scale-dependent factors remain unaccounted for (e.g. transport phenomena). Consequently, scaling-up such models to industrial scale remains hitherto highly challenging (Coşgun et al., 2023; Helleckes et al., 2023). Metabolic models remain as of today inefficient for accurate predictions of lipid production due to their non-dynamic nature (Tibocha-Bonilla et al., 2018). Consequently, dynamic regressive phenomenological models of microalgae growth appear, for now, more suited to the development of a scalable model being capable of integrating scale-dependent phenomena (e.g. light distribution, transport phenomena) through discretization. In a review including such models, Béchet et al. (2013) concluded that models which determined local light intensity in the culture broth and summed the local productivities to obtain global photobioreactor productivity offered the best compromise between scalability and practicability for engineering applications. In following studies by the same research team, such model of microalgae production in nutrient replete conditions was calibrated at laboratory-scale under varying light and temperature growth conditions in order to determine photosynthetic carbon absorption through the use of respirometry. The model derived from this approach was first validated

at bench-scale (Béchet et al., 2015), followed by validation up to pilot-scale (Béchet et al., 2014) demonstrating the remarkable scalability of such phenomenological approach with independent calibration of the microalgae photosynthetic response. However, the model from Béchet et al. only applied to biomass production in nutrient-replete conditions hence not applying for secondary metabolite production such as lipid production.

Among models including lipid productivity, reviewed by Bekirogullari et al. (2020), most were calibrated from the minimization of a model error function, accounting from a dozen to up to twenty-seven parameters. Although all models presented are mechanistically justified, and exhibit fair predictions, model identifiability is challenging to ensure in view of the large amount of parameters, presenting at times correlations. Significant parameter error may therefore be expected meaning upscaling of the models could prove problematic. Having identified this pitfall, Ryu et al. (2018) implemented the identifiable parameter method to reduce parameter uncertainty. Mairet et al. (2011), proposed a model calibrated from compartmented calibrations (an independent calibration for carbon assimilation prediction, and an independent calibration for intracellular carbon flow prediction) but the photosynthetic response of the microalgae during nitrogen starvation was unfortunately outside of the scope of this study. To our knowledge, no follow-up studies aiming at upscaling the models cited by Bekirogullari et al. (2020) exist. Building from these observations, we advocate that model parameters values, and variations according to environmental variables should be determined as much as practically possible from independent calibration experiments or from identifiable observations. This was the approach adopted by Béchet et al. (2015) with the independent determination of PI-curve parameters prior to implementation in a global model in separate experiments from laboratory to pilot scale (Béchet et al., 2015, 2014). The present study aimed at expanding this modelling approach in the context of lipid accumulation during nitrogen starvation.

Under the hypothesis that carbon photosynthetically assimilated could be predicted from the independent evaluation of the photosynthetic activity according to light conditions in the reactors, a model was developed by calibrating photosynthetic inorganic carbon assimilation and the dynamics of internal carbon allocation. To achieve so, lipid accumulation in the microalga *Chlorella vulgaris* NIES 227 was studied in batch conditions under N-starvation and varying light densities. The microalgae were controlled for the evolution of photosynthetic productivity (determined by separate measurements of oxygen productivity according to light intensity), algae biomass growth (dry-weight and cell count), and the cell content in proteins, lipids, carbohydrates, and pigments. An overall mathematical model of biomass and cell growth, and cell content under nitrogen starvation is proposed predicting lipid accumulation.

# 2. Materials and methods

## 2.1. Microalgae stock cultivation and preparation of the experiment

The microalgae *Chlorella vulgaris* NIES 227 was pre-cultivated in nutritive medium (1622 mg·L⁻¹ of NaNO₃, 233 mg·L⁻¹ of KH₂PO₄, 143 mg·L⁻¹ of MgSO₄, 7H₂O, 1.5 mg·L⁻¹ of CaCl₂, 2H₂O, and 1 mL·L⁻¹ of Hutner solution (Hutner et al., 1950, for trace elements). Aiming at constituting sufficient biomass in exponential growth phase for the start of the experiment, a first pre-culture was started 10 days prior to the experiment in a 1L E-flask. 3 days prior to the start of the experiment, this first pre-culture was diluted into 4x600 mL nutritive medium in 1L Erlenmeyer flasks (E-flasks). Both inocula were grown in an incubator
ZWYC-290A (Labwit Scientific, Melbourne, Australia) under 2 % CO₂ enriched atmosphere, at 25 °C, illuminated with approx. 220 µmol·m⁻²·s⁻¹ light intensity (100 % light intensity setting), and 20 h/4h light/dark cycles (start of daylight at approx. 11 A.M). On the day the experiment was started, all four stock solutions were mixed together, centrifuged for 5 min at 4,400 rpm with a high speed centrifuge Avanti J-26S XP (Beckman-Coulter, Brea, CA, USA), rinsed with autoclaved N-free medium at 50 % concentration for other nutrients to remove traces of inorganic nitrogen, centrifuged a second time (same parameters), and resuspended with a small amount of autoclaved N-free medium (at 50 % concentration for other nutrients). The algal slurry hence constituted was controlled for dry-weight, and adapted volumes were resuspended in autoclaved 50 % and N-free media aiming at respecting target initial biomass concentrations (see next paragraph). Samples of the algal slurry were stored at 4 °C until processing for the analysis of biomass content in lipids, proteins, carbohydrates, and pigments as performed for the rest of experimental samples (description in the following).

## 2.2. Cultivation conditions

All tests were performed in 1L E-flasks. 9 reactors were prepared of which 3 reactors were covered with 75 % PAR uniform light cutting filters (ref. 210 0.6ND, LEE Filters, Andover, United Kingdom), and 3 reactors were covered with 50 % PAR uniform light cutting filters (ref. 209 0.3ND, LEE Filters, Andover, United Kingdom). The last 3 reactors were left uncovered. Algae cells were resuspended in order to obtain three starting dry-weight (DW) concentration 0.101 ± 0.004, 0.303 ± 0.012, or 1.11 ± 0.04 g DW·L⁻¹. Each of the three tested concentrations were submitted to one of the three illumination conditions. 9 reactors of 550 mL cultivation broth were henceforth obtained, each presenting a specific set of incident light intensity and initial concentration conditions (Table 1). The reactors were cultivated under the same incubation conditions as the stock solutions, being introduced in the incubator at approx. 3:30P.M. on the first day of the experiment. Incident PAR light intensity for each reactor was measured with a Quantum Sensor MQS-B (Walz, Effeltrich, Germany) positioned in place of each reactor center (light meter ULM-500 Walz, Effeltrich, Germany). A conservative 7.5 % measurement relative error was assumed for incident light intensity measurement (cumulating a 5 % measurement error indicated by the constructor and the variability seen during measurement).

<table><caption>**Table 1**
Main characteristics of cultivation conditions in the 9 experimental reactors.</caption><thead><tr><th>Initial concentration (g DW L<sup>-1</sup>)</th><th>Initial cell density (cell L<sup>-1</sup>)</th><th>Incident light intensity (µmol m<sup>-2</sup> s<sup>-1</sup>)</th><th>Dry-weight specific light density (µmol (g DW s)<sup>-1</sup>)</th><th>Cell specific light density (µmol (cell s)<sup>-1</sup>)</th></tr></thead><tbody><tr><td>(1.01 ± 0.04)·10<sup>-1</sup></td><td>(2.49 ± 0.19)·10<sup>9</sup></td><td>52.5 ± 2.0</td><td>6.00 ± 1.77</td><td>(1.38 ± 1.17)·10<sup>-10</sup></td></tr><tr><td>(1.01 ± 0.04)·10<sup>-1</sup></td><td>(2.49 ± 0.19)·10<sup>9</sup></td><td>112 ± 2</td><td>11.8 ± 4.0</td><td>(3.56 ± 2.39)·10<sup>-10</sup></td></tr><tr><td>(1.01 ± 0.04)·10<sup>-1</sup></td><td>(2.49 ± 0.19)·10<sup>9</sup></td><td>198 ± 2</td><td>20.3 ± 7.3</td><td>(7.20 ± 3.53)·10<sup>-10</sup></td></tr><tr><td>(3.03 ± 0.12)·10<sup>-1</sup></td><td>(7.47 ± 0.57)·10<sup>9</sup></td><td>55.5 ± 2.1</td><td>2.26 ± 0.66</td><td>(4.52 ± 3.77)·10<sup>-11</sup></td></tr><tr><td>(3.03 ± 0.12)·10<sup>-1</sup></td><td>(7.47 ± 0.57)·10<sup>9</sup></td><td>117 ± 2</td><td>4.15 ± 1.54</td><td>(1.04 ± 0.75)·10<sup>-10</sup></td></tr><tr><td>(3.03 ± 0.12)·10<sup>-1</sup></td><td>(7.47 ± 0.57)·10<sup>9</sup></td><td>212 ± 2</td><td>7.17 ± 2.60</td><td>(2.12 ± 1.2)·10<sup>-10</sup></td></tr><tr><td>1.11 ± 0.04</td><td>(2.73 ± 0.21)·10<sup>10</sup></td><td>56.0 ± 2.1</td><td>0.820 ± 0.140</td><td>(2.01 ± 0.69)·10<sup>-11</sup></td></tr><tr><td>1.11 ± 0.04</td><td>(2.73 ± 0.21)·10<sup>10</sup></td><td>115.5 ± 2.2</td><td>1.48 ± 0.36</td><td>(3.86 ± 1.53)·10<sup>-11</sup></td></tr><tr><td>1.11 ± 0.04</td><td>(2.73 ± 0.21)·10<sup>10</sup></td><td>212 ± 2</td><td>2.51 ± 0.68</td><td>(6.19 ± 3.24)·10<sup>-11</sup></td></tr></tbody></table>

## 2.3. Measurements

### 2.3.1. Sampling

On days 1, 2, 4, 8, 10, 14, and 16, between 50 mL (for reactors having higher biomass concentration) and 80 mL (for the three reactors of lowest biomass concentration) cultivation broth was sampled at approx. 9 A.M. More important volumes were needed to obtain sufficient biomass and achieve precise measurements for the three reactors of lowest algal densities, leading to ending the experiment on day 10 when insufficient volume was reached. Aliquots were kept at 4 °C until processing for analysis (up to 3 h). Part of the liquid samples were analysed for cell count, dry-weight, and pigment content. Known volumes of the rest of the samples were introduced in several Falcon tubes (Fisher Scientific, Thermo Fisher Scientific Inc., Waltham, MA, USA), and centrifuged (4500×g for 10 min in an Allegra X-15r centrifuge, Beckman Coulter™, Brea, CA, USA). The supernatant was discarded, and the pellet was resuspended in ultrapure water to rinse traces of supernatant. This protocol was repeated a second time before a third round of centrifuge. The pellets were then analysed for biomass content in total carbon, total nitrogen, total lipids, and total carbohydrates according to the protocols described below.

### 2.3.2. Biomass growth monitoring

Biomass growth in the photobioreactors was monitored both in terms of dry-weight concentration, cell count in the microalgae culture, and optical density of the culture. Dry-weight was determined according to the standard method 2540. D (Eaton et al., 1998), using GF/A filters (Whatman, Maidstone, UK). An 8 % measurement relative error was considered for dry-weight measurement (Béchet et al., 2015). Biomass cell concentration was measured using a cell counter Multisizer 4 (Beckman Coulter, Brea, California, USA): briefly, approx. 20 µL of the culture (volume adjusted depending on the expected cell concentration) was suspended in 20 mL of Isoton II Diluent (Beckman Coulter, Brea, California, USA). A 15 % relative error was considered for the values measured (Chambonniere et al., 2022). Optical density at 683 nm and 880 nm were controlled by spectrophotometry (Epoch 2, Biotek™, Winooski, Vermont, USA). A 1 % error on optical density was assumed based on the constructor specifications.

### 2.3.3. Biomass metabolic status

The metabolic status of the microalgae is discussed in the present study in terms of quota in proteins, quota in lipids, and quota in carbohydrates, i.e., the mass of proteins, lipids, or carbohydrates divided by 1) the corresponding mass of microalgae (henceforth defined as mass quota, noted Q), or 2) the number of cells of microalgae (henceforth defined as cell quota, noted q).

**Proteins:** The protein content was measured using the nitrogen content of the microalgae as a proxy (Laurens et al., 2012; Lourenço et al., 1998). The nitrogen content of the biomass was determined from a pellet of a known volume of the culture, immediately resuspended in a known volume of ultrapure water and analysed for nitrogen content using an analyzer TOC-L (Shimadzu, Kyoto, Japan). The protein quota of the algal biomass was computed assuming a nitrogen-to-protein ratio of 5.04 ± 0.59, as reported for *Chlorella vulgaris* (Templeton and Laurens, 2015). A 2.4 % relative error was considered for the measurement of TN content of the biomass (Li et al., 2019). Determination of the error on protein quota also assumed a 1 % relative error for dilution factors. Total carbon in the biomass was likewise determined from the run of the analyser TOC-L.

**Carbohydrates:** Centrifuged pellets of known volume of cultivation broth were hermetically sealed and stored in the freezer until analysis. Total carbohydrates were measured using the protocol developed by Dubois et al. (1956). Briefly, lyophilized biomass was digested in 2.5 M HCl (0.5 mL/g DW) at 100.5 °C for 3 h. The digestate was diluted in ultra-pure water to obtain measurable levels of total carbohydrates, and mixed with 2.5 mL of 95 % H₂SO₄ and 500 µL of 5 % (w/v) phenol
solution. The total carbohydrate in the suspension obtained was determined using colorimetry, by comparing the light absorption of the solution at 483 nm with a calibration curve derived from a glucose standard solution (spectrophotometer Epoch 2, Biotek, Winooski, Vermont, USA). The samples were typically measured in duplicates. Uncertainty as to the total carbohydrates in the biomass was based on the standard deviation of replicates. When no replicate was available, 20 % relative error was assumed as proposed in Chambonniere et al. (2022).

**Lipids:** Total lipids were measured from freeze-dried pellets prepared according to the same protocol as used for total carbohydrates analyses, and determined by GC-FID (GC-2010 Pro AOC-20i/AOC-20 s, Shimadzu, Kyoto, Japan) following extraction and *trans*-methylation of the lipids. Briefly, the lyophilized biomass was mixed with 3 mL of a solution of 1.25 M hydrogen chloride in methanol (reference 17935 Supelco, Sigma-Aldrich, Saint Louis, Missouri, USA), 0.2 mL of a 3 mg·L⁻¹ solution of TAG C15:0 (reference T4257, Sigma-Aldrich, Saint Louis, Missouri, USA) in hexane 95 %, anhydrous for internal calibration of the measurement, and incubated at 85 °C for 1 h. A total of 3 mL of HPLC grade hexane was added (reference 043263.AK, Thermo Scientific, Thermofisher Scientific Inc., Waltham, MA, USA); 1 mL of ultra-pure water was added to the sample, and the solution was briefly vortexed before centrifugation (1500 rpm, 5 min). The supernatant hexane phase was introduced into vials for GC analysis. The total lipid quota was determined from the sum of the areas of all peaks detected, translated into a concentration based on the ratio of the area to the known quantity of C15:0 initially introduced. Uncertainty regarding the lipid content in the biomass was based on the standard deviation of replicates. When no replicate was available, 11 % relative error was assumed as proposed in Chambonniere et al. (2022).

## 2.4. Pigment content

Microalgae pigment content was determined by extraction followed by colorimetric measurement following the methodology of Wellburn (1994). Briefly, 1 mL of experimental broth, diluted by a factor 2 and 10 for reactors of medium and high concentration, respectively, was placed in a 1.5 mL tube (Eppendorf, Hamburg, Germany), centrifuged (4500×g for 10 min in a centrifuge Mini-Spin Plus, Eppendorf, Hamburg, Germany), resuspended in 1 mL of dimethyl sulfoxide (DMSO), incubated approx. 45 min in the dark at 60 °C (block heater Stuart SBH130D, Cole-Parmer, Vernon Hills, IL, USA), centrifuged a second time in the same conditions. The supernatant was collected and placed in a 1-cm light-path cuvette (ref. 11602609, Thermofisher Scientific Inc., Waltham, MA, USA). If the biomass pellet was visibly holding green colour, it was resuspended in DMSO to repeat a second extraction and mix both pigment extract. A minimum of two and up to three extractions, if deemed necessary, were performed. Optical density of the suspension obtained was determined for the wavelength 450, 649, and 665 nm in order to determine chlorophyll-a, chlorophyll-b, and total carotenoids concentration as performed in Six et al. (2024a).

### 2.4.1. Biomass productivity status

Biomass photosynthetic productivity was determined from measurements of oxygen productivity under illumination using a respirometry cell OX1LP-6 (Qubit™, Kingston, Ontario, Canada), maintained at 25 °C by a recirculatory water bath (Haake SC100-A10, Thermo Fisher Scientific, Waltham, MA, USA). 10 mL, 2 mL, or 1 mL were sampled from low, intermediate, and high (respectively) biomass concentration reactors, centrifugated and resuspended in 10 mL N-replete fresh medium as performed in Chambonniere et al. (2022) aiming at obtaining similar optical properties for all respirometry tests. Dissolved oxygen in the tested resuspension was stripped by bubbling N₂ gas for approx. 30 s, and immediately introduced into the OX1LP-6 cell. The cells were kept in the dark for 10 min, also enabling the temperature in the tested volume to equilibrate at 25 °C ± 1 °C. At time 0, data logging was started, to record dissolved oxygen concentration variations during

two 10-minutes sequences of 5 min in the dark followed by 5 min under illumination. The run in light conditions were performed at different light intensities, one being set at 100 µmol·m⁻²·s⁻¹, the other being set at 50 µmol·m⁻²·s⁻¹ in the constructor software, in no particular order. The contribution by the biomass to oxygen production in light conditions and oxygen consumption in the dark (henceforth referred to as respiration) were computed correcting the measured evolution of oxygen concentration for the exchange with the atmosphere determined in a separate experiment (see SI-1). Biomass final rates of O₂ production (respectively consumption) were finally computed by linear regression of the amount of oxygen computed to be produced (respectively consumed) against time, under light or dark conditions. Biomass net photosynthetic-oxygen-productivity was then obtained for the two tested light intensities as the rate of oxygen production compensated by the respiration rate recorded in the dark phase immediately prior to the illumination phase. Total light available during respirometric tests in the OX1LP-6 set-up was determined from a calibration experiment as described in Chambonniere et al. (2022).

## 2.5. Modelling of lipid accumulation by Chlorella vulgaris NIES 227

### 2.5.1. Model structure

With the hypothesis that photosynthetic activity of *Chlorella vulgaris* NIES 227 was correlated to the cell metabolic status as suggested by Chambonniere et al. (2022), a predictive model of lipid accumulation was developed in the present study based on the structure described in the following, also summarized in Fig. 1.

From an initial microalgae culture of known metabolic status and concentration:

Knowing incident light flux and culture optical conditions (expected to depend on the microalgae metabolic status Kandilian et al., 2014), the photosynthetic activity of the cells can be determined in order to predict total sequestered carbon from photosynthesis stoichiometry (Model block 1).

* The dynamics of the repartition of assimilated carbon between the different pools of macromolecules must be determined, which the present study also aimed to tackle (Model block 2).

* The amount of newly created cells is determined from a predicted cell division rate, expected to be a function of the cell quota of the limiting nutrient, here being nitrogen (Droop, 1974; Hellweger and Kianirad, 2007) (Model block 3).

* From the new allocation and new cell density in the broth, a new metabolic status can be determined, and the calculation can be reiterated in the new conditions.

### 2.5.2. Model development

**Determination of photosynthetic carbon captured (Block 1):** The PI-function expressing local specific photosynthetic O₂ production (p⁷⁰⁰O₂) according to light intensity in the local volume (I⁷⁰⁰c) was adapted from the model developed by Béchet et al. (2015), accounting for light inhibition since inhibition was observed in a calibration experiment performed separately (SI-2). The final expression is given in Eq. (1) where Pₘ is maximum specific rate of O₂ production by the microalgae, σ₆₈₃ is the mass extinction coefficient of the culture broth at 683 nm, Xₐ₉₆₄ is the algae concentration in the local volume, K₁ and K₂ are respectively the half-saturation constant and the inhibition constant with regard to light intensity, and λ is the rate of algal O₂ consumption for cell maintenance. The baseline *Chlorella vulgaris* NIES 227 PI-curve at 25 °C first calibrated during exponential growth phase in a separate experiment (SI-2) enabled to determine a first set of values for Pₘ, K₁, and K₂ respectively equal to (9.10 ± 0.32) · 10⁻⁴ gO₂ · (gDW·s)⁻¹, 156.0 ± 4.7µmol · (L·s)⁻¹, 3.09 ± 0.11µmol · (L·s)⁻¹.
<figure><img src="image_6.png" /><figcaption>Fig. 1. Conceptual structure for the predictive model of lipid accumulation in the microalgae *Chlorella vulgaris* NIES 227 under nitrogen starvation.</figcaption></figure>

$$P_{O_2}^{loc} = \frac{P_m \cdot \sigma_{683} \cdot X^{algae} \cdot I^{loc}}{K_1 + \sigma_{683} \cdot X^{algae} \cdot I^{loc} + \left( \frac{\sigma_{683} \cdot X^{algae} \cdot I^{loc}}{K_2} \right)^2} - \lambda \quad (1)$$

During N-starvation experiment, *Chlorella vulgaris* NIES 227 photosynthetic productivity of O₂ was assessed assuming the ratio P_m to K_1 was constant (value noted α<sub>PI</sub>) as the study of chlorophyll fluorescence by Kruskopf and Flynn, (2006) suggested. K_2 was considered constant, this hypothesis having little impact in the present study since the light intensities experienced were not in the range of light inhibition based on the PI calibration experiment. These two hypotheses, admittedly strong, enabled to compute P_m as the single parameter of the optimisation problem from the two PI measurements performed for each sample, thus lowering final uncertainty on P_m. P_m parameter was determined for all samplings of each reactor by fitting Eq. (1) to the data obtained during oxygen productivity measurements, based on the minimisation of the squared residuals by Nelder-Mead method (optim function in R Version 4.3.2 software). More details on the determination and uncertainty analysis for P_m parameter are given in SI-3. Variations of the coefficients σ<sub>683</sub> and λ according to the metabolic status of the microalgae were also investigated from the data generated in the present study.

**Carbon partition model (Block 2):** carbon assimilated as CO₂ by the microalgae is expected to be initially transformed into carbohydrates, subsequently converted toward proteins or lipids (Mairet et al., 2011; Ryu et al., 2018). Therefore, carbon allocation was determined from a rate of conversion of carbohydrates to lipids evaluated based on lipid, carbohydrate, and protein content dynamics observed. The loss of cell materials through cell endogenous respiration was also investigated as part of this modelling block.

**Cell division rate (Block 3):** Cell division rate (μ<sup>div</sup>) was calculated as 1/n • dn/dt, where n is algae cell concentration in the reactor. μ<sup>div</sup> was computed based on the values of Δn/Δt for consecutive measurements, and the average of consecutive values for n at the denominator. Correlations between μ<sup>div</sup> and other measured parameters as well as mathematical relationships with such parameters were investigated.

### 2.5.3. Verification of model performance

Having fully calibrated the model by individual blocks, it was necessary to verify each block could function jointly. This was verified by comparing predicted variables of biomass growth (concentration and cell count) and biomass content (protein, lipid, and carbohydrates) for

initial biomass and light conditions matching experimental data.

## 2.6. Data analysis

### 2.6.1. Data and code availability

All the data associated to the present study are available at the online repository (Chambonniere et al., 2025). Unless specified otherwise, numerical analyses and figures shown in this work were performed using R Statistical Software (v4.3.2; R Core Team 2023). The scripts developed have been made available from the repository cited in previous sentence. R-packages used to complete analyses are referenced in SI-17.

### 2.6.2. Correlation analysis

A large amount of parameters and variables were tracked in this study (see SI-4) and it could be challenging to determine the best predictors for a given variable. When no clear phenomenological law could be derived from the data, best predictors were searched for by ranking the dependence of the variable to all other measured parameters through Spearman rank analysis. A mathematical dependence was then searched for with the parameters which exhibited the most significant relationship (i.e. lowest p-values). This routine was repeated for the residuals from the fit henceforth developed, which was again repeated until minor improvement of the fit could be obtained, or no parameters ranked significantly with the residuals.

### 2.6.3. Confidence interval and error analysis

Confidence interval in text are given with sample standard deviation or statistical standard error. Error bars in figures represent 95 % confidence interval, calculated as 1.96 times the standard error assuming normal distribution unless mentioned otherwise. Error on main measurements were determined as specified in the Materials and Method sections, and error on calculated variables was evaluated from error propagation analysis, or as specified in the text.

## 3. Results

### 3.1. Culture monitoring

#### 3.1.1. Light density in the reactors

Light density in the reactors was estimated by dividing incident light
supply by total dry-weight or total cell count in the reactor (SI-5). As sampling diminished the total volume in the reactors, it contributed to increase light density in the volume. Sampling therefore had an antagonist effect compared with the impact of biomass growth which drove a global decrease of light density in the reactors. This lead to a complex behaviour in the total light density (Fig. S5-1) which can be summarised:

Light absorbed specifically to biomass dry-weight was fairly constant albeit somewhat decreasing throughout the experiment (Fig. S5-1a),

Light absorbed specifically to microalgae cell initially decreased sharply and stabilised between day 2 and day 4 for the rest of the experiment (Fig. S5-1b).

This analysis nevertheless evidenced a consistent light supply hierarchy throughout the experiment: light density in the reactors expectedly increased with incident light intensity and lower biomass concentration. A summary of resulting experimental conditions for each reactor is given in Table 1.

### 3.1.2. Biomass growth

The evolution of dry-weight, cell count, and total pigment content in the reactors throughout the experiment is shown in Fig. 2. All reactors exhibited a steady increase in dry-weight (Fig. 2a). All reactors underwent cellular division at the start of the experiment to plateau at the maximum cell concentration by day 4 (Fig. 2b). Pigment content of the biomass (Fig. 2c, shown as total concentration in the reactors) globally decreased in all reactors at rates increasing with increasing light densities (SI-6). While mass specific chlorophyll-a and -b gradually decreased in all reactors throughout N deprivation, carotenoids content exhibited an initial increase in all reactors before subsequent decrease, again sharper with higher light densities.

### 3.1.3. Photosynthetic activity

The evaluation of biomass maximum specific rate of photosynthetic oxygen production (coefficient $P_m$) according to the microalgae metabolic status confirmed that N-deprivation lead to a progressive

degradation of the microalgae photosynthetic activity, quantified by a decrease of the parameter $P_m$ (Fig. 3). The present work indicates that it is possible to internalize the loss of microalgae productivity during N starvation and that the model developed by Béchet et al. (2015) may be generalized to growth under N starvation. The relevance of this

Maximum specific rate of photosynthesis
(g O₂·g⁻¹·s⁻¹)

<figure><img src="image_7.png" /><figcaption>Fig. 3. Evolution of biomass maximum specific rate of photosynthetic oxygen production quantified by the provided colour-scale according to carbohydrates, lipids, and proteins mass quota of the biomass. (NB: the percentage shown do not represent the exact quota as they are homothetically transformed to sum to 1 in the ternary diagram).</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>Fig. 2. Growth of *Chlorella vulgaris* NIES 227 for varying light conditions. Evolution of biomass dry-weight (a), algae cell count (b, log-scale was used on the Y-axis), and pigment content of the microalgae (c, log-scale was used on the Y-axis).</figcaption></figure>
debris, resulting in unexpectedly low lipid content. Functional compartment, defined as 100 % minus the added shares of carbohydrates, proteins, and lipids following Ryu et al. (2018), were initially low with a global increase despite significant variations. It must be taken into account that, due to the definition adopted, functional compartment i) also includes inert debris, and ii) aggregates the errors on the measurements of the three macromolecular pools likely explaining some of the strong variations noted. Globally, functional compartment content increased in the reactors, the final share increasing with increasing light density (p = 0.0128, see SI-6). Because the culture fitness also decreased with increasing light density (e.g. indicated by the sharper decrease of pigment content and photosynthetic activity), the increase in the functional compartment share of the cell may have been linked to the production of inert cell debris. Inert debris concentration having not been measured in this study, this hypothesis could not be verified.

## 4. Modelling algae growth and lipid accumulation during nitrogen starvation

The quantitative development of the modelling for the different blocks outlined in Fig. 1, is presented below.

### 4.1. Block 1: Carbon sequestration

#### 4.1.1. Photosynthetic carbon sequestration prediction

Aiming at demonstrating the accuracy of Eq. (1) calibrated from oxygen productivity measurements, the carbon accumulated by the microalgae in each reactor was predicted based on the values of the coefficient $P_m$ previously determined (Fig. 3), the light distribution in the reactors based on known incident light intensity, and the light absorption of the microalgae measured. A theoretical gross O₂ rate of production in the reactors was hence determined (see calculations detailed in SI-8), and then converted into a carbon assimilation rate assuming 4 mol of electrons were generated for each mole of O₂ produced, and that 24 mol of electrons were needed to subsequently generate 1 mol of glucose (Béchet et al., 2015). Likewise, endogenous consumption of glucose (respiration) was accounted for from the measurement of oxygen consumption rate in the dark, with the corresponding stoichiometry of 1 mol of glucose consumed per 6 mol of O₂ consumed (Béchet et al., 2015). However, when accounting for carbon loss based on the respiration rates measured, it was found that more carbon was used than assimilated in the conditions of the experiments. It must be noted that respirometry measurements were performed in nutrient replete conditions, while reactors consisted in N-depleted media. Hence, less reducing power was likely needed during cultivation as some of the cell anabolic pathways were shut down (primarily N assimilation). It was therefore concluded that dark decay in nutrient replete conditions is not a good measurement of carbon consumption for maintenance during nitrogen starvation. To verify the soundness of model block 1 despite lacking accurate measurement of endogenous respiration rates, respiration was first neglected to predict a theoretical gross carbon assimilation by the microalgae. Measured total functional carbon in the reactor stored by the biomass (defined as the carbon stored in the form of carbohydrates, lipids, and proteins, see SI-9) was compared to the theoretical gross carbon accumulation in the biomass henceforth predicted. A fairly linear relationship was found between predicted gross assimilated carbon and measured net functional carbon stored by the biomass (adjusted-R² = 0.925 when testing for linear regression), theoretical gross carbon being expectedly generally higher than measured net carbon (linear regression coefficient of 0.8837 ± 0.0310), indicating that some reducing power was nevertheless used by the microalgae during N starvation for other processes than carbon accumulation (Fig. 5).

Aiming at evaluating the reducing power effectively used by the microalgae in non-carbon sequestration pathways during nitrogen starvation, the difference between theoretical biomass specific gross

<figure><img src="image_9.png" /><figcaption>Fig. 5. Net measured against gross theoretical prediction of the total carbon fixed by the biomass expressed as the global concentration in the reactors.</figcaption></figure>

total carbon fixed and experimentally determined biomass specific functional carbon was computed for each reactor. Specific rate of carbon consumption was then determined from linear regression of this difference against time. The results from this calculation and from the linear regression are available in SI-10. Most reactors showed some level of consumption of reducing power, a notable exception being the reactor [1.0 g DW·L⁻¹; absence of light filter]. No particular tendency with light density could be evidenced likely due to significant uncertainty (Fig. S10-2). A global endogenous rate of functional carbon consumption was therefore computed as the average of all hence determined rates, equal to (1.44 ± 0.539)·10⁻⁷ g C·(g DW·s)⁻¹ (the standard error was here computed as $SE/\sqrt{n}$ where SE is the standard deviation over the values computed for all reactors and n is the sample size, Altman and Bland, 2005).

#### 4.1.2. Prediction of the maximum specific rate of photosynthesis ($P_m$)

In order to obtain a predictive model with the proposed structure, the maximum specific rate of photosynthesis ($P_m$) must be predictable from the knowledge of algae cell metabolic status. This predictability was explored through Spearman rank correlation analysis. Chlorophyll-a and b content, and C:N ratio were the parameters having the most significant correlation with $P_m$ (respectively $R^2 = 0.961; 0.960; -0.952, p = 2.24 \cdot 10^{-37}; 5.04 \cdot 10^{-37}; 5.28 \cdot 10^{-34}$). Being conveniently predictable with the proposed model structure, C:N ratio was retained for a predictive model of $P_m$ parameter. An exponential relationship as described by Eq. (2) was established between $P_m$ and C:N ratio ($R^2 = 0.644$, Fig. 6), where $[C:N]$ is the carbon to nitrogen ratio in the biomass, and $P_m^{max}$ is the maximum specific rate of photosynthetic oxygen production under optimal growth conditions as determined in SI-2. $[C:N]_{char}$ and $\gamma_{Pm}$ are fitting parameters, which can respectively be assimilated to a microalgae carbon to nitrogen ratio characteristic to the photosynthetic activity, and a linearity coefficient.

$$P_m = P_m^{max} \cdot \left( \frac{[C:N]}{[C:N]_{char}} \right)^{\gamma_{Pm}} \quad (2)$$

$[C:N]_{char}$ and $\gamma_{Pm}$ were determined to be respectively equal to 2.75 ± 0.32gC·(gN)⁻¹, and -2.88 ± 0.13 through linear regression after transformation of Eq. (2). The residuals of this regression were likewise tested for Spearman rank correlation analysis which showed that no other parameter had a predictive influence (see SI-11). It must be noted that the experimental apparatus available only enabled to measure two points to assess PI curves of each culture in order to determine
<figure><img src="image_10.png" /><figcaption>Fig. 6. Maximum specific rate of photosynthetic oxygen production against C: N mass ratio of biomass content. The blue line represents the log-linear regression between variables and the grey area represents the standard error inferred from Eq. (2) modelling. A log-scale was used for the Y-axis.</figcaption></figure>

Pm coefficient for each sampling day. It is therefore admitted that significant uncertainties escaping error analysis impacted Pm determination. Experimental apparatus such as used by Béchet et al. (2015) enabling to draw 6 points PI curves in one coordinated experiment would likely enable finer observations. Nevertheless, the approach clearly permitted to evidence how C:N ratio in the microalgae is a fair predictor of Pm coefficient, and that accurate estimation of Pm coefficient can be obtained from the evaluation of C:N ratio.

### 4.1.3. Prediction of light extinction coefficient

The prediction of CO₂ assimilation requires to be able to estimate and predict the light effectively harvested by the microalgae. A variety of models of light absorption by microalgae exist in the literature (Diotto et al., 2022; Krichen et al., 2021; Pruvost et al., 2002). Béchet et al. (2015) accurately predicted light used by the biomass from laboratory to pilot-scale using a Beer-Lambert law where the total broth absorbance is the product of the biomass concentration, the optical path length of the radiation, and the mass specific extinction coefficient determined at 683 nm ($\sigma_{683}$ coefficient Eq. (1)). This hypothesis was hence replicated in the present study to compute light distribution in the reactors (as detailed in SI-5). $\sigma_{683}$ was evidenced in this study to vary significantly in the course of the experiment. Predicting this evolution as nitrogen starvation progresses is therefore necessary to achieve a predictive model of lipid accumulation. Successive Spearman's rank correlation analysis evidenced that $\sigma_{683}$ was primarily correlated to the mass quota of protein ($Q_p$), and the carbohydrate cell quota ($q_{CHO}^{cell}$). Analysis and results are detailed in SI-12. Overall, $\sigma_{683}$ was modelled according to Eq. (3), where $\sigma_{max}$, $K_{qp}$, and $\lambda_{CHO}$ are fitting parameters determined through non-linear fitting by Gauss-Newton method (nls function in R 4.3.2). Excellent fit was obtained ($R^2 = 0.869$, $N = 66$, see Fig. S12-3) for the following values of the fitting parameters $\sigma_{max} = 0.956 \pm 0.029 \text{ m}^2 \cdot (\text{g DW})^{-1}$, $K_{qp} = 0.142 \pm 0.012 \text{ g P} \cdot (\text{g DW})^{-1}$, and $\lambda_{CHO} = (1.51 \pm 0.32) \cdot 10^{10} \text{ m}^2 \cdot \text{cell} \cdot (\text{g CHO} \cdot \text{g DW})^{-1}$.

$$ \sigma_{683} = \sigma_{max} \cdot \frac{Q_p}{K_{qp} + Q_p} - \lambda_{CHO} \cdot q_{CHO} \quad (3) $$

### 4.2. Block 2: prediction of carbon partitioning

The evolution of macromolecules (proteins, carbohydrates, and lipids) total concentration in the reactors (Fig. 4 a), b), and c)) helped determine that i) total protein content was stable if not slightly degraded

over the course of the experiment; ii) carbohydrates were produced immediately from the onset of N starvation by the microalgae to stabilize at a high level from day 2; iii) lipid content increased only following the stabilization of the carbohydrate concentration. When analysing the cell content in macromolecules (Fig. 4 g), h), and i)), the following two scenarios could be noticed:

- carbohydrate cell quota initially increased concomitantly to an initial increase (or somewhat stable) lipid cell quota,

- carbohydrate cell quota initially remained stable (a decrease being even noticed in some reactors) in which case lipid cell quota noticeably decreased.

In all reactors, protein cell quota decreased throughout the experiment.

Based on these observations, it is conjectured that photosynthetically assimilated inorganic carbon in the form of carbohydrates through Calvin cycle (Chow, 2003), is subsequently converted into lipids but only above a threshold value for carbohydrate cell quota (henceforth referred to as “basal” carbohydrate cell quota). Initial cell division hence enabled an increase in the carbon captured by the culture with limited lipid accumulation unless under very high photosynthetic activity when assimilated carbon overpassed the new capacity of the culture to accumulate carbohydrate. This was followed by slower cell division that lead to above-threshold carbohydrate cell quota upon new carbon assimilation. The existence of an initial carbohydrate peak also indicates that the rate of conversion of carbohydrates to lipids is limited by a maximal physiologically possible value. When maximal carbohydrate concentration in the reactors were achieved, steady lipid accumulation was observed indicating that as much carbon was converted from carbohydrates into lipids as was assimilated by the photosynthetic apparatus.

Overall, based on the observations reported here, the following dynamics for *Chlorella vulgaris* NIES 227 metabolic status during nitrogen starvation is proposed:

1) From the onset of N starvation, carbon is absorbed as enabled by the photosynthetic apparatus.

2) No significant decrease in protein global content in the biomass was noticed (only marginal protein degradation was concluded to be possible in some conditions, see SI-6). Therefore, protein degradation was neglected in the present analysis. This however lead to a decrease in protein cell content due to cell division.

3) The newly assimilated carbon is instead entirely diverted toward the formation of carbohydrates until reaching a maximum cell quota.

4) Assimilated carbon that would lead to above maximum carbohydrate cell quota is transformed into lipids, though being limited by a maximum physiologic rate of conversion, inducing an accumulation of carbohydrates when reaching this maximum.

The basal carbohydrate quota ($q_{CHO}^{basal}$) was determined to be $(4.47 \pm 0.24) \cdot 10^{-12} \text{ g CHO} \cdot \text{cell}^{-1}$, computed as the average of all the values measured excluding day 1 and 2 of the experiment when dynamic carbohydrate quota evolution was noticed. A maximum rate of conversion of lipid production ($p_L^{max}$) was determined to be $(3.30 \pm 2.31) \cdot 10^{-17} \text{ g Lipids} \cdot \text{cell}^{-1} \cdot \text{s}^{-1}$ (see SI-13 for determination). The global model henceforth described was put into equations given in Box 1.

### 4.3. Block 3: Modelling of microalgae cell division

Having evidenced the need to determine the maximum cell quota in carbohydrate to determine the onset of lipid accumulation, it is necessary to be able to predict cell division. Spearman rank coefficient analysis evidenced that cell division specific rate was primarily correlated to the protein mass quota (computed as the average over each time interval). Highly time dependent variables (e.g. reactor volume and depth for
Box 1

Set of equations for the proposed model.

Parameterisation equations used in the present model are reminded here, with parameters as defined in the main text.

$$P_m = P_m^{\max} \cdot \left( \frac{[C : N]}{[C : N]_{\text{char}}} \right)^{\gamma_{P_m}} \quad (2)$$

$$\sigma_{683} = \sigma_{\max} \cdot \frac{Q_P}{K_{q_P} + Q_P} - \lambda_{CHO} \cdot q_{CHO} \quad (3)$$

$$\mu_{cell} = \mu_{cell}^{\max} \cdot \exp\left(-\frac{q_{P}^{\text{char}}}{q_{P}}\right) \quad (4)$$

The system of differential equations to solve include equation include Equations 5 to 8, where the following variables and parameters are defined:

$P_{CHO}^{net}$: net volumetric carbohydrate productivity

V: reactor volume

$Y_{O_2}^{CHO}$: yield of carbohydrate from O₂ production for photosynthesis

$P_m$: maximum specific rate of O₂ production by the microalgae

$\sigma_{683}$: mass extinction coefficient of the culture broth at 683 nm

$X_{\text{algae}}$: Biomass concentration

$K_1, K_2$: half-saturation and the inhibition constant (resp.) with regard to light intensity

$\lambda$: rate of algal O₂ consumption for cell maintenance

n: Biomass cell count

$q_P, q_{CHO}, q_L$: cell quota in proteins, carbohydrates, and lipids (respectively)

$Q_P, Q_{CHO}, Q_L$: mass quota in proteins, carbohydrates, and lipids (respectively)

$Y_L^C$: carbon to lipid mass ratio in the biomass

$Y_{CHO}^C$: carbon to carbohydrates mass ratio in the biomass

$$P_{CHO}^{net}(t) = \frac{Y_{O_2}^{CHO}}{V} \cdot \iiint_{\text{Reactor}} \left( \frac{P_m \cdot \sigma_{683} \cdot X_{\text{algae}} \cdot I_{\text{loc}}}{K_1 + \sigma_{683} \cdot X_{\text{algae}} \cdot I_{\text{loc}} + \left( \frac{\sigma_{683} \cdot X_{\text{algae}} \cdot I_{\text{loc}}}{K_2} \right)^2} - \lambda \right) \cdot X \cdot dV \quad (5)$$

$$\frac{dn}{dt} = \mu_{cell} \cdot n(t) \quad (6)$$

$$\frac{d(q_P \cdot n)}{dt} = 0 \quad (7)$$

$$\frac{d\left(q_{CHO} \cdot n + \frac{Y_L^C}{Y_{CHO}^C} \cdot q_L \cdot n\right)}{dt} = P_{CHO}^{net}(t) \quad (8)$$

Several scenarios are then possible to compute the volumetric productivity of carbohydrates and lipids (respectively $P_{CHO}$ and $P_L$), defined in the variations of Equations 11 and 12 where the following variables and parameters are first defined:

$q_{CHO}^{basal}$: basal carbohydrate cellular quota

$P_L^{aux}$: auxiliary variable representing the potential volumetric lipid productivity

$P_L^{max}$: max cellular rate of lipid production

$P_L^{max}$: max volumetric lipid productivity

$$P_L^{aux} = \frac{Y_{CHO}^C}{Y_L^C} \cdot P_{CHO}^{net}(t) \quad (9)$$

$$P_L^{max}(t) = P_L^{max}(t) \cdot n(t) \quad (10)$$

If $q_{CHO}(t) \le q_{CHO}^{basal}$

$$P_{CHO}(t) = P_{CHO}^{net} \quad (11i)$$

$$P_L(t) = 0 \quad (12i)$$

If $q_{CHO}(t) > q_{CHO}^{basal}$:

If $P_L^{aux} \le P_L^{max}$

$$P_{CHO}(t) = 0 \quad (11ii)$$

$$P_L(t) = P_L^{aux} \quad (12ii)$$

If $P_L^{aux} > P_L^{max}$

$$P_{CHO}(t) = P_{CHO}^{net} - \frac{Y_L^C}{Y_{CHO}^C} \cdot P_L^{max} \quad (11iii)$$

$$P_L(t) = P_L^{max} \quad (12iii)$$

We solve Equation 8 by separating it in the differential equation for carbohydrates (Equation 8i) and the differential equation for lipids (Equation 8ii)

$$\frac{d(q_{CHO} \cdot n)}{dt} = P_{CHO}(t) \quad (8i)$$

$$\frac{d(q_L \cdot n)}{dt} = P_L(t) \quad (8ii)$$

The biomass dry-weight concentration is determined from the yield of dry-weight to functional carbon determined ($Y_C^X$) in this study, from the carbon content of the different macromolecules contained in the microalgae (Equation 13):

$$X = Y_C^X \cdot (Y_L^C \cdot q_L \cdot n + Y_{CHO}^C \cdot q_{CHO} \cdot n + Y_P^C \cdot q_P \cdot n) \quad (13)$$
which a direct correlation to cell division rate is not expected were excluded from the analysis. Droop model was nevertheless found to poorly predict cell division rate variations (see SI-14). Instead, a decreasing exponential relationship was noticed between cell division rate and the protein cell quota in any given reactor, and Eq. (4) was therefore fitted for all reactors where $\mu_{cell}^{max}$ is a theoretical maximum cell division rate (d$^{-1}$), and $q_{P}^{char}$ is the characteristic protein cell quota for cell division in the reactor light conditions for the microalgae strain.

$$ \mu_{cell} = \mu_{cell}^{max} \cdot \exp\left(-\frac{q_{P}^{char}}{q_{P}}\right) \qquad (4) $$

Based on error analysis, theoretical maximum cell division rate was concluded to be uniform in all light conditions with a value of $\mu_{cell}^{max} = (4.25 \pm 0.45) \cdot 10^{-5} \text{ s}^{-1}$ (the standard error was computed as $SE_{\sqrt{n}}$ where SE is the standard deviation over the values computed for all reactors and n is the sample size). The characteristic protein cell quota for cell division was found to decrease with increasing light density in the reactor although the varying light condition in the reactor made it challenging to determine a predictive relationship between light density in the reactor and the characteristic protein cell quota for cell division. A linear relationship was retained for simplicity. Detailed results from this analysis are given in SI-14.

### 4.3.1. Model verification

The model as presented in previous paragraphs was implemented in R software for initial conditions recreating the starting conditions of each reactor, and taking into account the volume change due to sampling. In addition, a ratio between functional carbon and dry-weight of $1.92 \pm 0.03gDW \cdot (gC)^{-1}$ was determined and used for the model (see SI-15). The ability of the model to reproduce the dynamics of lipid accumulation as monitored in the present experiment was hence assessed. The fitness of the model for biomass dry-weight concentration, lipid global concentration, carbohydrate global concentration, and

microalgae cell count based on all sampling of all reactors are shown in Fig. 7. Good fitness was obtained for these metrics ($R2 = 0.940, 0.842, 0.618, \text{and} 0.414$, respectively, $RMSE = 0.241g\text{dryweight} \cdot L^{-1}, 0.166 \text{glipids} \cdot L^{-1}, 0.118g\text{carbohydrates} \cdot L^{-1}, \text{and} 2.91 \cdot 10^{10}\text{cell} \cdot L^{-1}$ respectively). It proved challenging to obtain accurate prediction for carbohydrate and lipid quotas (both mass and cellular quotas) for all tested conditions, which we foremost linked to imprecision of the cell count prediction and maximum sugar cell quota. We also acknowledge the results presented here are by no means an independent validation of the model but merely a verification that the independently developed model blocks are coherent altogether. Nevertheless, we insist that fair prediction of global carbon sequestration was derived from independent photo-respirometry experiments, and the global repartition of predicted absorbed carbon between lipid and carbohydrates could be verified with fair accuracy in a remarkable variety of illumination conditions based on simple coefficients (maximum cell carbohydrate quota, maximum rate of carbohydrate conversion to lipids, rate of endogenous carbohydrate consumption) thus apparently characterizing the algae strain.

### 4.3.2. Future work needed

Despite satisfying model performance, shortcomings of the present model have appeared and need finer work in the future to verify if accuracy can be increased:

* Predicting cell division proved to be challenging though some clear conclusions have emerged: a Droop-inspired model did not permit to successfully predict cell count evolution in the reactors, but the modified exponential quota model proposed (Eq. (4)) performed well on individual reactors. The parameters appearing in Eq. (4) were clearly depending on the illumination conditions (SI-14), but the experimental set-up in the present work did not provide enough control of light conditions to accurately link kinetic parameter values evolution with light conditions. To refine our approach, we propose

<figure><img src="image_11.png" /><figcaption>Fig. 7. Measured against predicted by the proposed global model for biomass dry-weight concentration (a), algae cell count (b), global carbohydrates concentration in the reactor (c), and global lipids concentration in the reactor (d).</figcaption></figure>
that this equation should instead be calibrated in luminostat culti-
vation conditions.

* Algae respiration rate prediction could not be achieved accurately in the present experiment although the value derived from a global average enabled to satisfyingly predict carbon sequestration. Again, tests in luminostat conditions and specific respiration test of the biomass under N starvation may enable to refine our findings. Our experiments clearly indicated that endogenous decay measured in nutrient replete conditions is not relevant to N-starvation conditions. The sensitivity of the model fitness to the rate of endogenous carbon consumption show that this parameter is in critical need of better assessment.

* The model performance was poorer under highest light densities (particularly visible for carbohydrate prediction, Fig. 7-c)). We suggest that this is due to a decrease in the protein content of the microalgae, noticeable in such conditions in our experiment but that we could not quantify, leading to overestimating the carbon assimilated. Specific study of this phenomenon is therefore needed. Again, experiments in luminostat conditions may help investigate this effect.

* As photo-respirometry experiments are time-consuming to set-up and complex to interpret due to challenges in assessing the amount of light effectively absorbed by the micro-algae in the testing chamber, it would be interesting to replace these experiments by high-throughput methods. The advancement of fluorometry tests enabled to determine Rapid Light Curve (Kruskopf and Flynn, 2006; Levasseur et al., 2023) which may replace photo-respirometry experiments to evaluate the carbon assimilation potential of the microalgae. But the equivalence between the two methodologies has yet to be demonstrated.

* The model was developed at 25 °C. As temperature will influence the photosynthetic activity of the microalgae for any metabolic status, such variations must also be investigated at varying temperature for a fully scalable model.

* Model calibration involved a range of hypotheses that would need to be confirmed in dedicated studies. *a*<sub>PI</sub> and *K*<sub>2</sub> were both assumed to be constant in the present study despite limited evidence. Sensitivity analysis (see SI-16) indicated that moderate variations of these parameters had little impact on the final fit of the model indicating that these hypotheses do not critically impact model performance in the conditions tested. Nevertheless, they may still limit the capacity of the model to predict biomass, lipid, and carbohydrate productivity in photobioreactors presenting significantly higher or lower light density than tested in the present study.

* The model investigated the evolution of photosynthetic activity and carbon partition in *N*-deplete conditions. If microalgae cultivation can be modelled with constant photosynthetic activity under nutrient replete conditions (Béchet et al., 2015, 2014), and no evolution of their composition is expected in such conditions, the present model does not address transient conditions of *N*-limitation. While studies are needed to bridge both sets of conditions, industrial applications for microalgae biofuel production have been proposed in the form of two-step process: a first step biomass production in nutrient replete conditions, followed by lipid accumulation after sudden resuspension in *N*-deplete media (Benasla and Hausler, 2021; Nagappan et al., 2019; Narala et al., 2016). In such applications, the present model could be directly used to design and size the second step of lipid accumulation.

**5. Conclusion**

The present model was effectively applied in a diversity of light and
cell concentration conditions giving strength to the idea that the model

can then be transferred to a higher dimension with success. Viewing the
present model as an extension of the work of Béchet et al., (2015, 2014),
we are hopeful that this model can be successfully upscaled and used to
predict or design full scale photo-bioreactors dedicated to lipid pro-
duction for *Chlorella vulgaris* NIES 227 as the authors had achieved. In
addition, similar dynamics of lipid accumulation with an initial increase
of carbohydrate quota followed by an increase of lipid quota with
concomitant stagnation or decrease of carbohydrate quota was reported
for a variety of microalgae e.g. *Chlorella vulgaris* CCALA924 (Six et al.,
2024b), *Haematococcus pluvialis* and *Nannochloropsis* sp. (Recht et al.,
2012) although exhibiting significantly different quota and thresholds.
We hypothesize from this observation that the model of the general
structure as described in Fig. 1, formalized in the equations presented in
Box 1 can be applied to most microalgae strains, all model parameters
needing to be experimentally calibrated for each new species or strain.

Cultivation conditions at industrial scale can be optimized to
improve lipid yields in microalgae-based biofuel production by
leveraging the model developed. This may be achieved applying the
present calibration for *Chlorella vulgaris* NIES 227, or transferring the
model structure and methodology to other microalgae strain. The pre-
diction of lipid and carbohydrate productivity in full-scale photo-
bioreactors will then enable design and operations optimization (e.g.
light distribution, starvation duration) for maximization.

Funding sources

This project has received internal funding from the Carbon Circular
Economy program of Comissariat à l'Energie Atomique et aux Energies
Alternatives, Paris, France.

CRediT authorship contribution statement

**Paul Chambonniere.** *Evita Dollon:* Writing – review & editing, Visualization, Validation, Investigation, Data curation. **Alexandra Dimitriades-Lemaire:** Resources, Methodology. **Jean-François Sassi:** Resources, Project administration, Funding acquisition. **Florian Delrue:** Writing – review & editing, Supervision, Resources, Project administration, Methodology, Funding acquisition, Conceptualization.

Declaration of competing interest

The authors declare that they have no known competing financial
interests or personal relationships that could have appeared to influence
the work reported in this paper.

Acknowledgements

The authors wish to thank Gatien Fleury (CEA) for his work in the
development of the lipid analysis method used during this study, and
Alexandre Six (CEA) for fruitful discussions around the conceptualiza-
tion of this work.

**Appendix A. Supplementary data**

Supplementary data to this article can be found online at https://doi.
org/10.1016/j.biortech.2025.133585.

Data availability

All data and code has been made fully available with links to re-
pository in the manuscript.
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