# Mechanistic Modeling of Rotating Algal Biofilms

Yan Gao, Patrick Perré, Ignacio Fierro, Filipa Lopes, Olivier Bernard

► To cite this version:

Yan Gao, Patrick Perré, Ignacio Fierro, Filipa Lopes, Olivier Bernard. Mechanistic Modeling of Rotating Algal Biofilms. Biotechnology and Bioengineering, 2025, 122 (11), pp.2980-3006. 10.1002/bit.70028 . hal-05458109

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Submitted on 15 Jan 2026

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# Mechanistic Modelling of Rotating Algal Biofilms

GAO Yan<fnref n="1" />,<fnref n="2" />, PERRÉ Patrick<fnref n="3" />, FIERRO Ignacio<fnref n="2" />, LOPES Filipa<fnref n="1" />, and BERNARD Olivier<fnref n="2" />,\*

## ABSTRACT

Biofilm-based microalgal cultivation systems have emerged as a promising alternative to conventional suspended growth methods, offering improved light utilization and biomass productivity. Among these, Rotating Algal Biofilm (RAB) systems are particularly advantageous by subjecting cells to short periodic light/dark (L/D) cycles to mitigate photoinhibition. Through experimental validation and modeling, this study demonstrates that optimized L/D cycles enhance photosynthetic efficiency by temporally diluting high-intensity light. To investigate the impact of light regimes, a model was developed based on the Han's photosynthesis framework, incorporating respiration dynamics for broad ranges of cycle times and L/D ratios. Calibrated with experimental data, it accurately predicts biofilm behavior under varying light conditions. A key innovation is the integration of respiration variations during intermittent illumination, providing insights into growth dynamics across frequencies and duty cycles. Key findings show that high light frequencies reduce photoinhibition and enhance growth at given intensities. Increasing the light fraction improves growth rates by reducing peak intensity and shortening dark periods. The model elucidates biofilm responses to fluctuating light and offers strategies for reactor optimization. This study advances algal biofilm photophysiology understanding and provides a predictive tool for optimization and scaling up biofilm-based cultivation systems.

**Keywords:** microalgae; biofilm; growth modelling, respiration; light/dark cycles

## 1 Introduction

Meeting the escalating global food demand without exacerbating climate change stands as one of the most pressing challenges of our time. In this context, microalgae have emerged as a transformative resource for sustainable production of food, feed, and high-value bioactive compounds for the pharmaceutical and cosmetic industries<fnref n="1" />. Beyond their immediate applications, microalgae hold long-term potential as a feedstock for biofuel production, though economic viability remains a critical hurdle<fnref n="2" />. As photosynthetic microorganisms, microalgae harness solar energy to fix CO₂ while assimilating inorganic nutrients (e.g., nitrates and phosphates) into biomass, releasing O₂ as a byproduct<fnref n="3" />.

Currently, microalgae are predominantly cultivated in suspension-based systems, but these face significant economic

<footnote n="1">¹LGPM, CentraleSupélec, Université Paris-Saclay, Gif-sur-Yvette, 91190, France</footnote>

<footnote n="2">²Biocore, Inria Sophia Antipolis Méditerranée, Université Nice Côte d'Azur, Valbonne, 06902, France</footnote>

<footnote n="3">³LGPM, CEBB, CentraleSupélec, Université Paris-Saclay, Pomacle, 51110, France</footnote>

<footnote>\*olivier.bernard@inria.fr</footnote>
challenges due to high harvesting costs. Conventional harvesting methods such as centrifugation, filtration, and flotation processes<sup>4,5</sup> are both energy-intensive and time-consuming, driving recent interest in biofilm-based microalgae technology<sup>6</sup>. Unlike planktonic cells that require constant mixing for light access, microalgae biofilms form adherent microbial layers on solid surfaces<sup>6</sup>, offering two key advantages: simplified harvesting through mechanical scraping potential for automated processing<sup>7,8</sup>, as well as sustainable regrowth with reduced contamination risks between harvest cycles<sup>9</sup>.

Biofilm systems also appear to demonstrate superior light transmittance efficiency compared to suspended cultures<sup>10,11</sup>. Light attenuation in biofilms occurs primarily through algal cells and their extracellular matrix, while suspended systems experience additional photon loss through media absorption - particularly pronounced in colored substrates like digestate. Remarkably, even in clear water, approximately 20% of red photons are absorbed within just 0.3 m path length. This light transmittance advantage, combined with enhanced CO<sub>2</sub> transfer efficiency when biofilms are exposed to air<sup>11</sup>, contributes to significantly improved productivity in biofilm systems<sup>12</sup>.

Biofilm reactor technology has evolved into diverse configurations, as comprehensively reviewed by Wang et al.<sup>13</sup>. These systems critically influence light regimes for attached microalgae, with revolving designs specifically engineered to provide periodic illumination. Among these, Rotating Algal Biofilm (RAB) reactors have emerged in three principal configurations: drum-type<sup>14</sup>, rotating disk<sup>15,16</sup>, and revolving conveyor systems<sup>12</sup>. These designs share a common operational principle - subjecting cells to controlled light/dark (L/D) cycles characterized by two key parameters: the cycle time (T) representing complete L/D period duration, and the duty cycle (ε) defining the light-exposed fraction of each cycle<sup>17</sup>. The resulting growth rate (μ) becomes a function of T, ε, and peak light intensity (I<sub>peak</sub>). Unlike suspended cultures, RAB systems offer unique design-controlled optimization of these photoperiod parameters.

The physiological basis of these systems traces back to foundational work by Brown and Escombe<sup>18</sup> on plant photobiology, later extended to algae by Emerson<sup>19</sup>. While most subsequent research focused on suspended cultures<sup>20-23</sup>, critical insights emerged: (1) photoinhibition mitigation through reduced cycle times, and (2) the full light integration phenomenon<sup>22,24</sup>, where high-frequency flashing (millisecond-scale<sup>25</sup>) enables photosynthetic efficiency equivalent to continuous illumination at the same average intensity (I<sub>avg</sub> = I<sub>peak</sub> · ε). Under optimal conditions, this yields maximal growth rates (μ<sub>S</sub>(I<sub>peak</sub> · ε)), where μ<sub>S</sub> represents steady-state gross growth. Conversely, suboptimal L/D cycles incur growth penalties<sup>26</sup>, particularly with prolonged periods where photoinhibition peaks during light phase while carbon fixation in Calvin's cycle ceases rapidly in dark phases<sup>22</sup>, reducing average growth rate to μ<sub>S</sub>(I<sub>peak</sub>) · ε.

While the effects of fluctuating light on planktonic cultures have been extensively studied, research on biofilm systems remains limited, with only a few studies addressing this critical aspect<sup>27,28</sup>. Unlike suspended cultures, biofilm reactors offer precise control over cellular light regimes—including duty cycle (ε) and cycle duration (T)—through engineered design and operational parameters. All the cells at the same depth experience the same light pattern, and the average light intensity varies with biofilm depth due to light attenuation across the microbial layer. Even if the biological response when cells are under biofilm lifestyle is not fully understood<sup>27</sup>, the effect of light pattern on microalgal growth in a RAB was reported in key works:
* Gross et al. demonstrated the superiority of smaller light fractions in a yearlong study of *Chlorella vulgaris* biofilm productivity across various reactor designs¹².

* Reducing the cycle time from 180 s to 15 s (at $\epsilon \approx 1/3$) boosted daily areal productivity by 75% in a rotating algal biofilm system⁸.

* Full light integration was achieved for *Scenedesmus dimorphus* biofilms under $\epsilon = 1/5$ when cycle times decreased from hours to seconds²⁷, highlighting the regime where flashing light mimics continuous illumination.

These observations on net growth rate (or productivity) result from the combined effect of photosynthesis and respiration. A significant fraction (typically 10%²⁹,³⁰) of the energy provided by photosynthesis is consumed through dark respiration to produce cofactors (ATP, NADPH,...) and biomolecules for growth and maintenance³¹,³². Respiration has been shown to vary as a response to the light regime, with an increase under light phases³³⁻³⁵.

To advance the predictive understanding and optimization of algal biofilm productivity, we propose here a comprehensive model that integrates both photosynthesis and respiration dynamics under intermittent light regimes. This is a critical limitation in existing models²⁵,³⁶⁻³⁸, which assume constant respiration rates and consequently yield biased productivity estimates.

Our framework explicitly accounts for light attenuation through biofilm thickness and respiration variations during light/dark cycles, enabling accurate computation of average growth rates across diverse periodic regimes. The model was calibrated and validated using *Chlorella vulgaris* biofilms subjected to four continuous light and six light/dark regimes (detailed in Table 1), bridging the gap between theoretical predictions and experimental observations. Beyond fundamental insights into photophysiological responses, this work provides actionable criteria for designing rotating biofilm reactors, offering a pathway to optimize large-scale productivity by strategically controlling light/dark cycle parameters.

# 2 Model development

## 2.1 Recall of Han's model

The model relies on the mechanistic model of Han³⁹ representing the dynamics of photoinhibition in photosynthesis. It is based on the assumption that the reaction centres can take three states⁴⁰: open and ready for catching photons (A), closed while processing photons (B), or inhibited (C) when some key proteins (such as the D1 protein) get damaged due to an excess of energy resulting from too many harvested photons. The variables A, B and C represent the proportion of the photosynthetic units in each of these states, and therefore $A+B+C=1$. The dynamics of the three states of the model are given by:

$$ egin{cases} 
\dot{A} = -\sigma_H I A + \frac{B}{\tau}, \\ 
\dot{B} = \sigma_H I A - \frac{B}{\tau} + k_r C - k_d \sigma_H I B, \\ 
\dot{C} = -k_r C + k_d \sigma_H I B. 
\end{cases} 
\quad (1) $$
Here $\sigma_H$ ($\mu\text{mol}^{-1}\cdot\text{m}^2$) is the cross-section of PSII; $I$ is the light intensity ($\mu\text{mol}\cdot\text{m}^{-2}\cdot\text{s}^{-1}$); $\tau$ (s) represents the turnover time of the electron transfer chain. $k_d$ (dimensionless) is the damage rate and $k_r$ ($\text{s}^{-1}$) is the repair rate. System (1) can be reduced into a two-equation system, considering that $B = 1 - A - C$ $^{41}$:

$$ \dot{A} = -(\sigma_H I + \frac{1}{\tau})A + \frac{1-C}{\tau}, \quad (2a) $$

$$ \dot{C} = -(k_r + k_d \sigma_H I)C + k_d \sigma_H I(1-A). \quad (2b) $$

The gross growth rate (noted as $\mu_g$), which is the rate of CO$_2$ fixation from the Calvin cycle, can then be written as:

$$ \mu_g = k\sigma_H I A. \quad (3) $$

Where $k$ is the growth yield for dark-adapted cells. This parameter links the rate of absorbed photons to the microalgae growth rate. Indeed, the gross growth yield is $Y = \frac{\mu_g}{\sigma_H I} = kA$. For dark-adapted cultures, $A = 1$.

Simulating Han's model is computationally intensive due to the wide range of time scales involved. The model requires tracking fast light capture dynamics (on the order of milliseconds) over extended periods (e.g., days), leading to a high numerical stiffness that demands fine temporal resolution. This multi-scale nature poses significant challenges for both simulation efficiency and numerical stability, making the calibration of Han's model particularly demanding.

To address these computational constraints, the next section develops two approximations of the model, each tailored to distinct regimes of light fluctuation timescales. These simplified formulations, based on former works$^{42,43}$ enable tractable analysis while preserving the key dynamical features of the full system.

## 2.2 Considered light regimes

In this paper, we consider piece-wise constant light/dark cycles of period $T$, as experienced by biofilms in a rotating process. We denote by $t_D$ the moment when the dark period finishes; by $t_L$ the moment when the light period finishes. The moment $t_L + T$ is when the cycle finishes. The biofilm is exposed to a photosynthetic flux density (PPFD), $I_{peak}$, for a duration of $T_L = \varepsilon T = t_L + T - t_D$ and then the biofilm remains in darkness for a duration $T_D = (1 - \varepsilon)T = t_D - t_L$. This light pattern is presented in Figure 1.

## 2.3 Continuous light regime

Under continuous light regime, $A$, and $C$ reach their steady state ($\dot{A} = \dot{C} = 0$). Considering (2a), the equilibrium is given by:

$$ \left\{ \begin{aligned} A &= \frac{1}{1 + \tau\sigma_H I + \left(\frac{k_d}{k_r}\right)\tau(\sigma_H I)^2}, \\ C &= \frac{\left(\frac{k_d}{k_r}\right)\tau(\sigma_H I)^2}{1 + \tau\sigma_H I + \left(\frac{k_d}{k_r}\right)\tau(\sigma_H I)^2}. \end{aligned} \right. \quad (4) $$

From this expression of $A$, and using the definition of $\mu_g$ expressed by equation (3), it results that:

$$ \mu_g = p_{max} \frac{I}{I + \frac{p_{max}}{\alpha} (1 - \frac{I}{I_r})^2} \quad (5) $$
<figure><img src="image_2.png" /><figcaption>**Figure 1.** (A) A revolving microalgal biofilm reactor design ($\epsilon = 1/3$) similar to the one proposed by Gross et al.⁸. 1: Drive shaft; 2: Rotating conveyor as cell attachment material; 3: Growth medium with planktonic microalgal cells sheared off from the conveyor; 4: Medium container. 5: Light source. (B) Graphic description of the light pattern that biofilm cells are exposed to in RAB as (A).</figcaption></figure>

Where the maximum growth rate $p_{max}$, at optimal light $I^*$, and the initial slope $\alpha$ are related to the Han parameters as follows:

$$ \left\{ \begin{array}{l} \alpha = k\sigma_H, \\ I^* = \frac{1}{\sigma_H} \sqrt{\frac{k_r}{k_d \tau}}, \\ p_{max} = \frac{k}{\tau + 2\sqrt{\frac{k_d \tau}{k_r}}} \end{array} \right. \qquad (6) $$

## 2.4 Short light/dark periods

When light changes rapidly (e.g., on sub-second timescales), the dynamics of C in the Han model evolve too slowly to respond. Applying classical results from the theory of dynamical systems with multiple time scales, C approaches an approximately constant value⁴² ($\bar{C}$), while A and B continue to evolve rapidly. We reduce the dynamics (2a) and (2b) into one equation as follows:

$$ \dot{A} = \frac{1 - A - \bar{C}}{\tau} - \sigma_H IA. \qquad (7) $$

The average value of A in the light phase ($t \in [t_D, t_L + T]$) (light period integrated, noted as $\bar{A}_l$) can be computed as follows (see details in Appendix Section A.1):

$$ \bar{A}_l = \frac{1}{T\epsilon} \int_{t_D}^{t_L+T} A(t) dt = (1 - \bar{C})\lambda, \qquad (8) $$
where

$$ 
\begin{aligned}
\lambda &= \frac{1}{\sigma_H I \tau + 1} + \left( \frac{1}{\sigma_H I \tau + 1} - g \right) \cdot \frac{1}{T\varepsilon(\sigma_H I + \frac{1}{\tau})} \cdot (y-1), \\
\bar{C} &= \frac{k_d \sigma_H I - k_d \sigma_H I \lambda}{\frac{k_r}{\varepsilon} - k_d \sigma_H I \lambda + k_d \sigma_H I}.
\end{aligned}
\quad (9) $$

As the rates of decrease (dark phase) and increase (light phase function of I) of C are different, the average value of C over an entire period, $\bar{C}$, is different from the value obtained under continuous light at the average light intensity.

Finally, the average gross growth rate (light period integrated, noted as $\bar{\mu}_l$) in light phase is then:

$$ \bar{\mu}_l = \frac{1}{T\varepsilon} \cdot \int_{t_D}^{t_L+T} k\sigma_H I A(t) dt = k\sigma_H I \bar{A}_l. \quad (10) $$

This computation, derived through both simplification of Han's model for rapid light variations and analytical determination of the periodic solution, forms the foundation of Simplified Model 1 (SM1) that will be detailed later on.

## 2.5 Long light/dark periods

For light/dark periods a range of order longer than the time constant characterizing its dynamics (on timescales of seconds), A will reach its steady state quickly in the light phase and then in the dark phase$^{41}$. However, during each light phase, A will evolve with C (progressive inhibition during the light phase and recovery during the dark phase). The value of A is simply obtained from equation (2a) where $\dot{A}$ is assumed to be zero:

$$ A = \frac{1-C}{1+\tau\sigma_H I}. \quad (11) $$

Then in light period ($t \in [t_D, t_L + T]$), the dynamics of C in Equation (2b) can be rewritten in:

$$ \dot{C} = -\alpha_H C + \beta, \quad (12) $$

Where

$$ \alpha_H = \frac{k_d \tau (\sigma_H I)^2}{1 + \tau \sigma_H I} + k_r, \quad \beta = \frac{k_d \tau (\sigma_H I)^2}{1 + \tau \sigma_H I}. \quad (13) $$

The average value of C during light phase ($t \in [t_D, t_L + T]$) (noted as $\bar{C}_l$) can then be computed as follows:

$$ \bar{C}_l = \frac{1}{T\varepsilon} \cdot \int_{t_D}^{t_L+T} C(t) dt = \frac{\beta}{\alpha_H} + \left( \frac{\beta}{\alpha_H^2 T\varepsilon} - \frac{C(t_L)}{\alpha_H T\varepsilon} \right) \cdot (e^{-\alpha_H T\varepsilon} - 1). \quad (14) $$

where $C(t_L)$ is determined by the periodicity (see Appendix Section A.2).

The growth rate in the light phase (noted as $\mu_l$) is deduced from Equations (3) and (11):

$$ \mu_l = \frac{k\sigma_H I}{1 + \tau\sigma_H I} (1 - C), \quad (15) $$

Once the periodic regime has been reached ($C(t) = C(t+T)$), the average gross growth rate in light phase ($t \in [t_D, t_L + T]$) (light period integrated, noted as $\bar{\mu}_l$) is given by:

$$ \bar{\mu}_l = \frac{1}{T\varepsilon} \cdot \int_{t_D}^{t_L+T} \frac{k\sigma_H I}{1 + \tau\sigma_H I} (1 - C) dt = \frac{k\sigma_H I}{1 + \tau\sigma_H I} (1 - \bar{C}_l). \quad (16) $$

This computation, derived through both simplification of Han's model for slow light variations and analytical determination of the periodic solution, forms the foundation of Simplified Model 2 (SM2).
## 2.6 Respiration dynamics

The main originality of our model is to account for the respiration variations along the light dark cycles. As our first attempts to accurately reproduce the experimental results with a constant respiration rate failed, we derived the following respiration model which considerably improved the prediction capability of the growth model in situations where frequency and duty cycles vary. The main hypothesis, in line with Ippoliti et al.⁴⁴, is that respiration is triggered by the light sensed by the cells. The detailed mechanism of how light controls respiration rate is not represented, but it is assumed that a by-product of photosynthesis (such as carbohydrates or oxygen) is accumulated in the cell³⁵, favouring the respiration rate.

Two very different mechanisms can support this hypothesis of respiration enhancement: mitochondrial respiration and photorespiration. Accumulation of oxygen can trigger photorespiration, which happens when RuBisCo in the Calvin cycle fixes O₂ instead of CO₂, releasing glycolate. Photorespiration results both in a loss of carbon and enhanced O₂ consumption. On the other hand, high light stimulates photosynthesis, resulting in increased storage of carbon compounds. In turns, it promotes further metabolic processes that ultimately support protein building³⁵,⁴⁵,⁴⁶ for which mitochondrial respiration is ultimately increased by (slower) regulation mechanisms. This pathway still results in a higher O₂ consumption, and a release of carbon, but under the form of CO₂. Therefore, in our model, respiration refers to the sum of dark respiration and photorespiration, both mechanisms being associated to an increase in oxygen consumption and a release of carbon. The major difference with the model of Ippoliti et al.⁴⁴, is the notion of average sensed light which can be related to an internal pool of molecules accumulated from the use of the received light. The concept of average sensed light (denoted $\bar{I}$) in our respiration model considers the history of PPFD to which the cell was exposed to. Later, it will be demonstrated that the parameters of this respiration model derived from data analysis are associated to fast time scales, which support the hypothesis that the changes in respiration rate are more likely to be due to enhanced photorespiration.

The dynamics of $\bar{I}$ across light/dark cycles are governed by the following equation, capturing the cumulative time delays associated with cellular regulatory mechanisms (e.g., delays spanning transcription to translation).

$$ \dot{\bar{I}} = k_{R_I}(I - \bar{I}), \quad (17) $$

where $1/k_{R_I}$ is the time constant of the average sensed light $\bar{I}$. At low light frequency $\bar{I}$ has time to tend towards the actual light, $\bar{I} \approx I$, while for very high frequency, the average sensed light $\bar{I}$ remains close to the average light. The value of $\bar{I}$ further determines the respiration rate, following the model of Ippoliti et al. (2016)⁴⁴:

$$ R(\bar{I}) = R_D + (R_L - R_D) \frac{\bar{I}^n}{\bar{I}^n + I_k^n} \quad (18) $$

$R_L$ and $R_D$ are two constants, representing the highest respiration enhanced by light ($R_L$, s⁻¹), and lowest respiration rate relieved in dark phase ($R_D$, s⁻¹), respectively. $I_k$ is the half saturating light of R. The constant n determines the switching rate between $R_D$ and $R_L$ as a function of the average sensed light.

The dynamics of $\bar{I}$ given by Eq. (17) provides a fluctuating respiration rate R. Depending on where is $\bar{I}$ compared to the half saturation constant $I_k$, the respiration rate at depth z will experiment cycles between $R_D$ and $R_L$ (see Appendix Section A.3, Figure A.1). The fraction of time during which R is larger than $R_D/R_L$ is denoted $\tilde{\epsilon}$. Note that in the deepest part of the biofilm
where $I(z) < I_k$, then $\bar{I}(z) < I_k$ and $R(z)$ is at its lowest value and ultimately tends towards $R_D$.

The time average respiration rate $\bar{R}(z)$ can then be expressed by the fraction of time

$$ \bar{R}(z) = \frac{1}{T} \int_{I_L}^{t_L+T} R(\bar{I}(z,t)) dt = R_D + \frac{R_L - R_D}{T} \left( \int_{I_L}^{t_D} \frac{\bar{I}(z,t)^n}{\bar{I}(z,t)^n + I_k^n} dt + \int_{I_D}^{t_L+T} \frac{\bar{I}(z,t)^n}{\bar{I}(z,t)^n + I_k^n} dt \right) \quad (19) $$

Note that when $k_{Rl}$ is faster than the light fluctuation, $\bar{I} \approx I$ and $\bar{\epsilon} \approx \epsilon$, while for low frequencies $\bar{\epsilon}$ is 1 if the average light value is larger than $I_k$, and 0 otherwise.

## 2.7 The net growth rate in biofilm

When the light penetrates the biofilm, its intensity is attenuated according to the Lambert-Beer law, meaning that, at each depth $z$, the PPFD is:

$$ I(z) = I \cdot e^{-k_z z}. \quad (20) $$

where $k_z$ is the light extinction coefficient. Note that $k_z = \frac{1}{z} \ln \frac{I_{peak}}{I(z)}$. The Beer Lambert equation can be rewritten as $dI = -k_z Idz$.

The gross growth rate $\mu_g$ averaged over time and depth is obtained by integrating the local growth, given by the local PPFD, along the film depth:

$$ \mu_g = \frac{1}{h} \int_0^h \bar{\mu}_l(I) \cdot \epsilon dz = \frac{1}{\ln \frac{I_{peak}}{I(h)}} \int_{I(h)}^{I_{peak}} dI. \quad (21) $$

Likewise, the net growth rate (photosynthesis minus respiration) (noted as $\mu_n$) of the film averaged over time is:

$$ \mu_n = \frac{1}{h} \int_0^h \bar{\mu}_l(I(z)) \epsilon - \bar{R}(\bar{I}(z)) dz = \frac{1}{\ln \frac{I_{peak}}{I(h)}} \int_{I(h)}^{I_{peak}} \frac{\bar{\mu}_l(I) \epsilon - \bar{R}(\bar{I})}{I} dI. \quad (22) $$

## 2.8 Biomass productivity

The net surface productivity ($P_s$) of a RAB, taking into account both the light/dark alternation due to biofilm rotation and the daily PPFD fluctuations is computed assuming that the model solutions remain periodic, since the surface light slowly evolves along the day. This relies on the fact that the light period in the system is much smaller than the daily period of light.

RAB's net surface productivity over the whole day is therefore deduced from the study of productivity within a light/dark cycle, considering that the peak PPFD is varying along the day.

The gross productivity per footprint area ($P_{gf}$ in $g \cdot m^{-2} \cdot d^{-1}$) is then simulated for various designs of biofilm reactor (affecting $\epsilon$) and rotating speed (referring $T = 60/v$, $v$ is the speed in rpm) under a certain PPFD:

$$ P_{gf} = \mu_g(I, \epsilon, T, h) \cdot \rho \cdot h / \epsilon. \quad (23) $$

where $\rho$ is the biofilm density and $h$ the biofilm thickness. The respiratory biomass loss per footprint area ($R_f$ in $g \cdot m^{-2} \cdot d^{-1}$) is:

$$ R_f = \frac{1}{\ln \frac{I_{peak}}{I(h)}} \int_{I(h)}^{I_{peak}} \bar{R}(I, \epsilon, T, h) dI \cdot \rho \cdot h / \epsilon. \quad (24) $$

Finally, the net productivity per footprint area ($P_f$ in $g \cdot m^{-2} \cdot d^{-1}$) is:

$$ P_f = P_{gf} - R_f = \mu_n(I, \epsilon, T, h) \cdot \rho \cdot h / \epsilon. \quad (25) $$
The footprint light yield ($Y_f$ in $g \cdot mol^{-1}$) can be computed based on $P_f$:

$$Y_f = P_f/I. \qquad (26)$$

# 3 Model calibration and validation

## 3.1 Used dataset

The model was calibrated with data from experiments using a flow-cell system$^{47,48}$, where biofilm biomass was estimated via light transmittance, following our previously developed and validated protocols$^{47,48}$. Briefly, the experimental protocol consisted in submitting biofilms of *Chlorella vulgaris* to various light/dark patterns and to follow the biofilm development through light transmittance among other parameters. A set of LED was used to generate continuous light with PPFD of 100, 200, 300, 500 $\mu mol \cdot m^{-2} \cdot s^{-1}$, and four intermittent light with peak PPFD of 300 $\mu mol \cdot m^{-2} \cdot s^{-1}$ (L/D = 5s/10s; 30s/60s; 1min/2min), and 500 $\mu mol \cdot m^{-2} \cdot s^{-1}$ (L/D = 5s/20s) (L/D represents the duration of the light and dark phases). The average growth rate was then computed from these measurements. Microalgae cells were previously photoacclimated in suspension and then submitted to the light patterns.

## 3.2 Calibration strategy

<table><caption>**Table 1.** Light regimes applied to the biofilms. In the first column, the number in brackets indicates the number of replicates of each light regime.</caption><thead><tr><th>Noted light regimes</th><th>Peak light intensity</th><th>Average light intensity</th><th>Light fraction ε</th><th>T<sub>L</sub></th><th>T<sub>D</sub></th><th>T</th></tr></thead><tbody><tr><td>100 cont (14)</td><td>106 ± 2 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>106 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1</td><td>/</td><td>/</td><td>/</td></tr><tr><td>5s@300-10s@0 (11)</td><td>310 ± 5 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>103 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1/3</td><td>5 s</td><td>10 s</td><td>15 s</td></tr><tr><td>30s@300-60s@0 (10)</td><td>310 ± 5 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>103 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1/3</td><td>30 s</td><td>60 s</td><td>90 s</td></tr><tr><td>1min@300-2min@0 (6)</td><td>310 ± 5 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>103 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1/3</td><td>1 min</td><td>2 min</td><td>3 min</td></tr><tr><td>5s@500-20s@0 (10)</td><td>496 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>99 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1/5</td><td>5 s</td><td>20 s</td><td>25 s</td></tr><tr><td>200 cont (8)</td><td>200 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>200 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1</td><td>/</td><td>/</td><td>/</td></tr><tr><td>8s@300-4s@0 (7)</td><td>310 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>207 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>2/3</td><td>8 s</td><td>4 s</td><td>12 s</td></tr><tr><td>5s@400-5s@0 (8)</td><td>400 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>200 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1/2</td><td>5 s</td><td>/ 5 s</td><td>/ 10 s</td></tr><tr><td>300 cont (10)</td><td>310 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>310 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1</td><td>/</td><td>/</td><td>/</td></tr><tr><td>500 cont (15)</td><td>500 ± 3 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>500 μmol·m<sup>-2</sup>·s<sup>-1</sup></td><td>1</td><td>/</td><td>/</td><td>/</td></tr></tbody></table>

The model parameters vector $p = (k, k_r, k_d, \tau, \sigma, R_D, R_L, k_{RI}, n, I_k)$ was determined by adjustment so that the model predictions fit the experimental records. It was chosen to minimize the error between the simulated model output $\mu_{sim}$ and the measured one $\mu_m$. For this purpose, the likelihood function of the error was maximized with respect to $p$. Assuming that the error is distributed according to a Gaussian white noise with a diagonal covariance matrix, this is equivalent to minimize the optimization criterion $\Psi_{MLE}(p)^{49}$:

$$\Psi_{MLE}(p) = \frac{1}{N} \sum_{k=1}^{K} \sum_{j=1}^{J} \left( \frac{\mu_{m_{kj}} - \mu_{sim_{kj}}}{\sigma_k} \right)^2, \qquad (27)$$

Where $\sigma_k$ is the standard deviation of measured growth rates of each light regimes. K is the number of light regimes tested,
and *J* is the number of the replicates of each tested light regime (see Table 1). The Matlab Nelder-Mead simplex direct search minimization algorithm (fminsearch) was used and initialized with the first parameter guess (Han model parameters values from Wu et al.⁵⁰ and respiration model parameters values from Grenier et al.⁵¹ and Ippoliti et al.⁴⁴), as shown in Table 2. Root Mean Square Error (RMSE), and Mean Absolute Error (MAE) are calculated as follows:

$$ \Psi_{\text{RMSE}}(p) = \frac{1}{N} \sum_{k=1}^{K} \sum_{j=1}^{J} \left( \mu_{m_{kj}} - \mu_{\text{sim}_{kj}} \right)^2 $$

$$ \Psi_{\text{MAE}}(p) = \frac{1}{N} \sum_{k=1}^{K} \sum_{j=1}^{J} \left| \mu_{m_{kj}} - \mu_{\text{sim}_{kj}} \right| $$

The model simplification for large *T* (Equations (14) to (16)) was applied to light/dark periods larger than tens of seconds. A trick to considerably speed up the simulations and improve the accuracy of the parameter identification consisted in using the simplified model according to Section 2.5. The model validity was assessed with two intermittent light regimes: 300 µmol·m⁻²·s⁻¹(L/D = 4s/8s) and 400 µmol·m⁻²·s⁻¹(L/D = 5s/5s) while the other data were used for calibration. This partitioning of data into calibration (80%) and validation (20%) subsets follows standard modeling practices, where this ratio ensures robust model training while retaining independent data for unbiased evaluation. To rigorously test generalizability, we selected validation data with distinct light/dark (L/D) ratios (1/2 and 2/3) that differed from the primary training conditions (L/D = 1/3, 1/5).

# 4 Results

## 4.1 Model fitting

The fitted parameter values are shown in Table 2, together with the initial values used to initialize the identification algorithm⁵⁰. From Figure 2A, model growth rate predictions show good correlation with experimental outputs (p = 0.01, RMSE = 0.0413, and MAE = 0.0344). The maximum net growth rate is 0.41 d⁻¹ for the optimal light intensity (*I*\*) estimated at 171 µmol·m⁻²·s⁻¹ in agreement with the reported values of *Chlorella* sp. biofilm cultures¹¹,⁵², while growth inhibition is significant when PPFD exceeds *I*\* (Figure 2B).

<table><caption>**Table 2.** Model parameters obtained by minimization of the prediction error and first guess values.</caption><thead><tr><th>Parameters</th><th>First guess</th><th>Calibrated value</th><th>Unit</th></tr></thead><tbody><tr><td><i>k</i></td><td>8.70 · 10<sup>-6</sup> <sup>50</sup></td><td>4.90 · 10<sup>-6</sup></td><td>s · s<sup>-1</sup></td></tr><tr><td><i>k</i><sub>r</sub></td><td>6.80 · 10<sup>-3</sup> <sup>50</sup></td><td>1.16 · 10<sup>-2</sup></td><td>s<sup>-1</sup></td></tr><tr><td><i>k</i><sub>d</sub></td><td>2.99 · 10<sup>-4</sup> <sup>50</sup></td><td>0.15</td><td>/</td></tr><tr><td>σ<sub>H</sub></td><td>4.70 · 10<sup>-2</sup> <sup>50</sup></td><td>1.59 · 10<sup>-2</sup></td><td>μmol<sup>-1</sup> · m<sup>2</sup></td></tr><tr><td>τ</td><td>0.25 <sup>50</sup></td><td>1.06 · 10<sup>-3</sup></td><td>s</td></tr><tr><td><i>R</i><sub>L</sub></td><td>1.5 · 10<sup>-6</sup> <sup>51</sup></td><td>1.80 · 10<sup>-6</sup></td><td>s<sup>-1</sup></td></tr><tr><td><i>R</i><sub>D</sub></td><td>5.79 · 10<sup>-7</sup></td><td>1.51 · 10<sup>-10</sup></td><td>s<sup>-1</sup></td></tr><tr><td><i>k</i><sub>RI</sub></td><td>0.1</td><td>18.97</td><td>s<sup>-1</sup></td></tr><tr><td><i>n</i></td><td>1.9 <sup>44</sup></td><td>3</td><td>/</td></tr><tr><td><i>I</i><sub>k</sub></td><td>1152 <sup>44</sup></td><td>27.25</td><td>μmol · m<sup>-2</sup> · s<sup>-1</sup></td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>**Figure 2.** (A): Model net growth rate prediction *vs* experimental measurements. (B): With constant illumination, growth rate predicted by Han model as a function of light intensity. Gray cross, blue cross, purple triangle, brown dot represent continuous light with intensity of 100, 200, 300, 500 µmol·m⁻²·s⁻¹; Black dot: 5s@300-10s@0; black diamond: 30s@300-60s@0; black circle: 1min@300-2min@0; light blue triangle: 5s@500-20s@0; dark orange: 8s@300-4s@0; yellow triangle: 5s@400-5s@0. Data are shown as mean ± SD of measurements and predictions. The Magenta star represents the maximum net growth rate at the predicted optimal light intensity.</figcaption></figure>

## 4.2 Efficient simulation of the light-respiration multiscale model via timescale-separated approximations

Simulating the coupled photon capture and respiration dynamics presents a significant computational challenge due to its inherent multiscale nature. To compare model outputs with experimentally measured growth rates over multi-day periods, the system must resolve dynamics spanning milliseconds to days. This demanding requirement leads to numerical stiffness.

To address this, we leverage the theory of slow-fast dynamical systems, exploiting timescale separation to derive tractable approximations. By further assuming convergence to a periodic regime, we obtain two simplified models (SM1 and SM2), each valid for distinct light fluctuation timescales:

* SM1 accurately approximates the full model for light periods below 5 s, with a maximum error of 0.5% (observed at $T = 5$ s), see Appendix Section A.2, Figure A.10.

* SM2 is valid for light periods exceeding 5 s, maintaining errors below 1% (peak error at $T = 5$ s), see Appendix Section A.2, Figure A.10.

As demonstrated in Figure A.10, both approximations introduce negligible error while drastically reducing computational cost. This justifies their use for efficient model calibration and exploration of optimization scenarios.

## 4.3 Effect of light pattern on the respiration rate

For continuous light pattern, the respiration model proposed in this work is equivalent to Ipolliti's model$^{44}$. The respiration rate is increasing with PPFD until saturating (see Figure 3A). For intermittent light patterns, the respiration is totally different as governed by the sensed light ($\bar{I}$), which integrates instantaneous irradiance modulated by the cycle period and duty cycle. This
<figure><img src="image_4.png" /><figcaption>**Figure 3.** Light pattern effect on the respiration rate. (A) Respiration rate under constant illumination (black solid line) and time-averaged respiration for low-frequency L/D cycles (black dash line). (B) Light intensity sensed in light/dark regimes, cycle times of 0.1 s (yellow dash line), 0.5 s (dark blue dash line), and 10 s (orange dash line), respectively, compared with constant light of 50 µmol·m⁻²·s⁻¹ (Gray solid line) and 500 µmol·m⁻²·s⁻¹ (Purple solid line). (C) Respiration rate variation in one cycle according to the light intensity sensed in (B) with different cycle times, respectively; (D) Average respiration in dark phase (R̄D, dark blue solid line), light phase (R̄L, light blue solid line), and in one cycle (R̄, black dot line) with respect to cycle time, respectively; (E) Cycle time average respiration rate with respect to light fraction with different cycle time magnitudes (10 ms: dark blue line, 0.1 s: light blue line, 0.2 s: orange line, 0.5 ms: magenta line, 10 s: black line). The simulations for L/D cycles were done for the incident light intensity of 500 µmol·m⁻²·s⁻¹ and for a light fraction of 1/10 (B to D).</figcaption></figure>
approach aligns with findings from Xue et al. (1996)⁵³, who demonstrated that light-enhanced respiration in *Chlamydomonas reinhardtii* depends on both irradiance and exposure duration, reflecting the interplay between photosynthesis (which supplies respiratory substrates) and respiration (sum of dark respiration and photorespiration). Further discussion of these physiological mechanisms and their implications for the model is provided in Section 5.3. The first order dynamics (Eq. 17) driving $\bar{I}$ tracks the step-wise constant light/dark cycle with a delay. For $T = 10$ s, as presented in Figure 3B, $\bar{I}$ follows a square-wave pattern similar to the light pattern. For cycle time below one second, $\bar{I}$ increases towards $I_{peak}$ in light and decreases towards zero in darkness. For faster cycles, $\bar{I}$ does not follow $I$ anymore and stays closer to $I_{ave}$. For cycle times lower than milliseconds, $\bar{I}$ averages the light pattern and stays at $I_{ave}$. The respiration pattern directly follows the sensed light (see Figure 3C). The cycle time impacts the average respiration (see Figure 3D) only in the range from tens of milliseconds to ten seconds. The ability of $\bar{I}$ to track the light signal of period $T$ depends on the time constant $1/k_{Rl}$ of Equation (17). Average respiration ($\bar{R}$) increases with $\varepsilon$ (Figure 3E), but this effect fades out for faster light cycles.

## 4.4 Effect of the light pattern on the photosynthesis rate (gross growth rate)

Figure 4A displays the gross growth rate response to the cycle period for the same average PPFD ($I_{ave}$). The shorter cycle time leads to higher gross growth rate. For the fastest cycle times, it reaches the maximum, corresponding to continuous light. This is in agreement with the work of Hartmann et al.⁴², presenting three plateaus with time cycle, except the time orders are different. Full light integration appears when $T$ is below 10 ms, then gross growth rate declines with increasing $T$ because the level of photoinhibition (variable $C$) increases (see Figure 4D). For cycle time larger than 15 min photoinhibition is maximum, hence growth rate is minimum. An intermediate regime is reached in the range of seconds to tens of seconds.

Figure 4B details the dynamics of the inhibited state along the cycle. With continuous illumination, the inhibition level remains constant with time, and increases with PPFD. It is interesting to note that, even for a limiting light intensity of $100 \ \mu\text{mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}$, there is an inhibition level of 25%. This is consistent with Eq. (12) that stipulates that, under continuous light, a constant value of $C$ is obtained by a balance between the rate of photo-inhibition and the rate of recovery⁵⁴. For intermittent light, $C$ follows a periodic pattern: decreasing in the dark phase due to the reparation of the damaged proteins and increasing again at light. Comparing various light regimes with a duty cycle of 1/3 and same average PPFD of $300 \ \mu\text{mol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}$ (L/D = 5s/10s; 30s/60s; 1min/2min), light regime with cycle time of 3 min induces higher maximum inhibition level than 90 s and 15 s. This is also true for the average inhibition level (Figure 4D). The level of inhibited state $C$ at the shift between dark and light phases is represented in Figure 4C for a light fraction of 1/3. For fast flashing light ($T$ below 10 ms), $C$ stays almost constant in a cycle, which corresponds to *Full light integration*. The population $B$, from which photoinhibition is triggered (Eq. 1) is that of the averaged light. A second plateau appears between 0.1 s and 5 s. This time period is still fast enough to assume a constant value of $C$ but is large enough for the fast dynamic between $A$ and $B$ to be established. During the light phase, the population $B$ therefore increases to the value of the peak light, which explains the higher value of this second plateau. For $T > 5$ s, $C$ evolves between $C_L$ and $C_D$, where $C_L$ increases and $C_D$ decreases with the period. Remark that only the value of $C$ in the light phase matters for growth. The average $C$ in the light phase ($\bar{C}_l$) increases
with the cycle period (Figure 4D).

For larger duty cycles, with constant average PPFD, light is better distributed along the interval, with lower peak light. It follows that the average value of C in the light period decreases with the duty cycle (Figure 5B). Duty cycle influence fades out with faster light frequencies. The evolution of C in the light period shown in Figures 4D and 5B explains why growth rate increases for higher frequencies (Figure 4A) and for higher duty cycles $\epsilon$ (Figure 5A).

**4.5 Resulting effect on net growth rate**

The net growth rate results from the difference between photosynthesis and respiration rates. The simulated dynamics of net growth rate, as a function of T is shown in Figure 6A, keeping $I_{ave}$ constant at 100 µmol·m⁻²·s⁻¹ (simulations for $I_{ave}$ at

<figure><img src="image_5.png" /></figure>

**Figure 4.** Theoretical and experimental effect of cycle time on gross growth and inhibition state C for the average light intensity of 100 µmol·m⁻²·s⁻¹. (A): Gross growth rate with respect to cycle time T with different light fractions ($\epsilon$). (B): C evolution in one photoperiod of each tested light regime. (C): minimum ($C_D$) and maximum ($C_L$) inhibited level C at the end of the light ($t_L$) and dark ($t_D$) periods, respectively, with a light fraction of 1/3. (D) Average C in light phase ($\bar{C}_l$) as a function of T for different light fractions, sharing the same legend as (A).
<figure><img src="image_6.png" /><figcaption>**Figure 5.** Theoretical and experimental effect of light fraction on gross growth rate (A) and average C in light phase (B) at the average light intensity of 100 µmol·m⁻²·s⁻¹ with cycle times ranging from millisecond to 30 min. 30s@300-60s@0 (pink diamond), 1min@300-2min@0 (black circle).</figcaption></figure>

200 µmol·m⁻²·s⁻¹, together with experimental records are presented in Appendix Figure A.4). Globally, growth rate is higher for faster light/dark alternation. Surprisingly, the model predicts a dip in growth for a period around 0.1 s, where growth starts to decrease while respiration is still high, in line with some experimental observations²⁶,²⁷.

The impact of light fraction on the net growth rate is presented in Figure 6B and compared with experimental data. Growth rate increases with the light fraction. In Figure 6, $\mu_n$ behavior with cycle times of 15 s overlaps the one with cycle time of 25 s. It demonstrates that the higher growth rate at 300 µmol·m⁻²·s⁻¹, L/D = 5s/10s, compared with 500 µmol·m⁻²·s⁻¹, L/D = 5s/20s, mostly results from the higher light fraction not from the cycle time difference.

<figure><img src="image_7.png" /><figcaption>**Figure 6.** Net growth rate with respect to cycle time (A) and light fraction (B) for average PPFD of 100 µmol·m⁻²·s⁻¹.</figcaption></figure>
<figure><img src="image_8.png" /></figure>

<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /></figure>

<figure><img src="image_13.png" /><figcaption>**Figure 7.** Growth in RAB. (A) Net footprint productivity in RAB with respect to cycle time at incident light intensity of 2000 µmol·m⁻²·s⁻¹; (B) maximum cycle time ($T_{max}$) with respect to PPFD to keep $P_f$ in the second plateau in (A); (C) average C in light phase ($\bar{C}_l$); (D) footprint respiratory loss ($R_f$); (E) footprint productivity ($P_f$) and (F) footprint light yield ($Y_f$) in function of PPFD in RAB. (C) to (F) were computed with the lowest $T_{max}$ ($T = 45$ s) found in (B). All sub-figures share the same legend.</figcaption></figure>

## 4.6 Productivity and yield in RABs with different designs

The net productivity per footprint area ($P_f$) is computed for a biofilm thickness of 100 µm and a temperature of 24 °C in Figure 7. For a high PPFD of 2000 µmol·m⁻²·s⁻¹, the net footprint productivity in a horizontal flat plate biofilm system
($\varepsilon = 1$) is close to $0 \text{ g} \cdot \text{m}^{-2} \cdot \text{d}^{-1}$ (see Figure 7A). This is due to (1) severe photoinhibition under continuous exposure to such intense light, (2) respiration losses in a biofilm permanently submitted to light, and (3) the low total growth area.

The footprint productivity $P_f$ in the RABs decreases for longer cycle times in general (Figure 7A), due to elevated inhibition. When $T$ is in the order of milliseconds, gross productivity takes advantage of *full light integration* to achieve higher values by diluting the inhibiting PPFD by flashing light effect, in comparison to the productivity that would be reached by a horizontal flat plate reactor submitted to the same average PPFD (see Appendix Figure A.6.D). When considering $\varepsilon$ of 1/80, $P_f$ can be up to $80 \text{ g} \cdot \text{m}^{-2} \cdot \text{d}^{-1}$. This theoretical case provides the maximum potential to be approached by technological developments. However, even if this short $T$ is favorable to get extremely high $P_f$ at high light, they cannot be applied in current actual RABs (Figure 7A). Indeed, it would mean that the cells have to do a full rotation in 1 ms, involving huge biofilm linear velocities. Moderate shear stress can enhance a more compact biofilm⁵⁵, but elevated shear stress leads to a high detachment rate⁸ while being energy-demanding. Too long $T$ (diurnal cycle) are not interesting for industrial exploitation since they lead to biofilm respiration larger than growth. A tradeoff has thus to be found within the operational plateau (Figure 7A) between cell detachment and growth⁸,¹⁷.

The inflexion plateau presented in Figure 7A with relatively high $P_f$ is obtained in a $T$ range of seconds to tens of seconds, which is industrially achievable. Figure 7B shows $T_{max}$, the upper limit of $T$ in RABs to achieve such operational plateau, under different peak light intensities (only represented for PPFD larger than $300 \text{ µmol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}$ for which $P_f$ is significantly high). With $T$ in this range, high productivities are expected in conditions that are realistic. This critic $T_{max}$ decreases with PPFD, for all designs: a shorter period is necessary to mitigate stronger photoinhibition at higher PPFD. Figure 7B also indicates that with the same PPFD, RABs have narrower operational plateaus with larger $\varepsilon$. The minimum $T$ required is 45 s (rotation speed corresponds to 1.3 rpm, as predicted in Figure 7B) which assures that the operational plateau $P_f$ can be reached for all designs under all PPFD assumed. In the following, the highest $P_f$ under different designs refers to the value obtained for $T = 45s$. In this case, $\bar{C}_l$ in RABs with smaller $\varepsilon$ are significantly lower than in flat plate (Figure 7C), remarkably enhancing growth. It is worth remarking that the gross growth rate in RABs never exceeds the one in a flat plate reactor (Appendix Figure A.5.A). However, due to the reduced average respiration in darkness (Appendix Figure A.5.B), net growth rate ($\mu_n$) in RABs can be higher for intense PPFD illumination (Appendix Figure A.5.C). The higher total biomass in the RAB by a factor $1/\varepsilon$ explains the observed gain in gross footprint productivity (Appendix Figure A.5.D). With a higher biomass, losses due to respiration are enhanced (Figure 7D), but this effect stays marginal, so that net footprint productivity (Figure 7E) stays high. For PPFD of $2000 \text{ µmol} \cdot \text{m}^{-2} \cdot \text{s}^{-1}$, the highest $P_f$ (over $50 \text{ g} \cdot \text{m}^{-2} \cdot \text{d}^{-1}$) is obtained with $\varepsilon$ of 1/80. Smaller values of $\varepsilon$ do not contribute to further increase $P_f$ due to reduced average light and increased respiration. It is important to understand that the respiration of the surface exposed to light is similar for the horizontal flat plate systems and for RAB (unless unrealistic very fast rotations are considered). As a consequence, increasing the surface on which the biofilm is growing will always increase the losses due to respiration. The expected gain in gross footprint productivity (Appendix Figure A.5.D) in RABs is therefore reduced by respiration (Figure 7D). The folded surface amplifies footprint-based respiration, but fortunately, the respiration rate is lower in
the dark. Therefore, the gain in footprint productivity results from the larger biofilm area with mitigated average respiratory activity due to the dark period. In practice, this would mean that the dark periods would contribute to reducing the oxygen level by respiration⁵⁶, thus reducing the photorespiration.

Footprint light yield ($Y_f$) with respect to PPFD is shown in Figure 7F. Clearly, RAB with high dilution rate is significantly more efficient in terms of light utilization under high light. The light yield of a RAB with $\varepsilon$ of 1/80 is 173 times higher than for a flat plate for a PPFD of 2000 $\mu$mol·m$^{-2}$·s$^{-1}$. It is worth remarking that $Y_f$ is even increasing for low light intensity before being dominated by photoinhibition at higher PPFD, and then decreasing. The optimal PPFDs associated to $Y_f$ peak are lower than 500 $\mu$mol·m$^{-2}$·s$^{-1}$. The highest $Y_f$ is 0.53, reached at 315 $\mu$mol·m$^{-2}$·s$^{-1}$ with a design of 1/80. It is 50% above the maximum yield for flat plate reactor permanently exposed to light.

# 5 Discussion

## 5.1 The role played by the light dependent respiration

The originality of our model is to explicitly consider respiration dynamics for modelling the microalgae growth rate under intermittent light regimes. It extends the Han model³⁹ by representing the carbon loss with light, which was most of the time assumed constant in the pre-existing models. This new model highlights the role played by respiration, especially for a process inherently submitted to permanent light and dark fluctuations. Respiration has been described for decades as being driven by the metabolism⁴⁵,⁵⁷, showing a light-history dependent pattern³³,³⁵,⁵⁸. This coupling with photosynthetic activity has been reported in planktonic cultures³³,³⁵,⁵⁷, where respiration rate at light can be enhanced two to four times compared with the level in darkness at steady-state.

It is likely that assuming a constant respiration in the classical model for planktonic microalgae growth was not introducing a large bias, however, for biofilms this phenomenon is much more marked⁵⁹,⁶⁰. Our model shows that ignoring the difference between respiration in light and dark biases the estimations in microalgal growth and biomass productivity in revolving biofilm systems. For systems with intermediate duty cycles (e.g., $\varepsilon$ = 1/2 to 1/80) ignoring the respiration dynamics leads to significant overestimations in net productivity. At $\varepsilon$ = 1/2, the overestimation ranges from 26% to 91% across light intensities of 100–2000 $\mu$mol·m$^{-2}$·s$^{-1}$, while at $\varepsilon$ = 1/80, it varies between 5% and 50% (see Figure A.9).

## 5.2 Is photorespiration dominating respiration at light in biofilms?

Light-dependent respiration has been explained by the accumulation of byproducts from the photosynthesis process³³,³⁵ (including dioxygen) which eventually stimulates both dark respiration and photorespiration. Most of the studies are based on oxygen consumption records, for which it is not possible to distinguish dark respiration and photorespiration. Photorespiration, when RuBisCo (ribulose-1,5-bisphosphate) fixes O₂ instead of CO₂, yields only one phosphoglycerate (PGA) molecule plus P-glycolate (vs. two PGA molecules in carboxylation)⁶¹. This ATP/NADPH-intensive process can reduce carbon fixation efficiency by up to 28% in dense cultures⁶². Photorespiration is likely to occur in phototrophic biofilms, where O₂ concentration can locally be over-saturated⁶³, up to fourfold saturation (at 1000 $\mu$mol·m$^{-2}$·s$^{-1}$) and even eleven-fold at 5% CO₂⁵⁹,⁶⁴.
Oxygen accumulation in biofilms is probably the main reason explaining a strong increase in (photo)respiration⁶⁵. Producing glycolate from biofilms has even been considered as a promising solution to produce organic carbon which can be further valorized in biotechnological applications⁶⁶,⁶⁷.

The hypothesis that photorespiration in biofilm dominates dark respiration is also supported by our estimations of $K_{Rl}$ in the order of 10 s⁻¹, so that the simulated respiration reaches a steady state within 10 ms (Figure 3D). This is relevant only for photorespiration whose timescale is associated with the production of oxygen⁵⁹. Only a few studies estimated photorespiration dynamics from the kinetics of RuBisCo in leaves⁶⁸ demonstrating a fast dynamics.

Photorespiration process is often thought less important in phytoplankton³⁰,⁶⁹ where the level of oxygen stays limited. This is a major difference with biofilms, where oxygen at high levels, related to high cell concentration, are accumulated in the matrix.

## 5.3 Prediction of respiration dynamics from other studies

Light respiration in intermittent light regimes was rarely studied and only a few works on planktonic cultures have been published³³,³⁵,⁴⁵. The trends observed by other authors for various L/D patterns seem to follow some of the key features highlighted by our model, which are not intuitive.

In the study of Beardall et al.³³, the dynamics of respiration were tracked after a 10-minute illumination under 1000 µmol·m⁻²·s⁻¹. The respiration model (see Equation 17) will predict an increase in $\bar{I}$ during the illumination period, at a rate depending on the time constant $1/k_{Rl}$, and thus an increase of the respiration rate towards $R_L$. The extend of the respiration enhancement depends on how far $\bar{I}$ can get above $I_k$, which results from the light exposure time and intensity. This theory is coherent with the observations after light exposure by Beardall et al.³³, for *Isochrysis galbana* who recorded an enhanced respiration with longer exposition periods or with higher PPFD.

When recording dark respiration at different moments after the light exposition, Shimakawa et al.³⁵ observed a decreasing trend of respiration with time in darkness for the cyanobacterium *Synechocystis* sp. PCC 6803 p, in line with the model prediction and similarly to the one observed by Beardall et al.³³.

Grobbelaar⁴⁵ studied planktonic cultures of *Scenedesmus* submitted to periodic sinusoidal light, without dark phase. Light period varied from 5.1 s to 260 s, with peak light at 750 µmol·m⁻²·s⁻¹. The respiration model (see Eq (17) to (19)) predicts, for a sinusoidal light, that the sensed light $\bar{I}$ also follows a sinusoidal pattern with similar average ($I_{ave}$ = 375 µmol·m⁻²·s⁻¹) but lower amplitude (see Appendix Figure A.2.A, B, C). It results that respiration will oscillate between $R_D$ and $R_L$, depending on how long $\bar{I}$ stays above the threshold $I_k$. *Scenedesmus* respiration behavior under continuous light indicates a higher $I_k$ value⁴⁵. Assuming 500 µmol·m⁻²·s⁻¹ for $I_k$ and 0.1897 s⁻¹ for $k_{Rl}$ and values for $R_D$ and $R_L$ identical to the ones of *Chlorella*, the predicted increased average respiration fits the experimental records of Grobbelaar et al.⁴⁵ (see Appendix Figure A.2.D). Figure A.2.(A, B, C) show how $\bar{I}$ sinusoidally oscillates around $I_{ave}$ = 375 µmol·m⁻²·s⁻¹ < $I_k$, and passes more time above $I_k$ for the slowest light signal. It follows that the average respiration predicted by the model increases with the light period (see Appendix Figure A.2.D), since more time is spent above $I_k$, in the region where respiration is closer to $R_L$. For
higher average PPFD ($I_{ave}$), $\bar{I}$ spends more time above $I_k$ leading to an enhanced respiration, also in line with the observations of Grobbelaar et al.<sup>45</sup>.

It is worth remarking that the influence of the illumination period on $\bar{R}$ depends on the position of the average PPFD compared to $I_k$ (comparison between Figure 3 and Figure A.3). In Appendix Figure A.2.D, Figure A.3.D for $I_{ave} < I_k$, $\bar{R}$ increases with larger $T$, in agreement with the observation of Grobbelaar (1991)<sup>45</sup>. Now, if $I_{ave} > I_k$ the reverse phenomenon happens as shown in Figure 3D, since at slower period $\bar{I}$ will go further towards zero, and thus $R$ will stay longer close to $R_D$. With fast flashing light ($T < 10$ ms), the respiration process is too slow to follow the light signal, thus $\bar{R}$ is the same as respiration with equivalent dose supplied with continuous light $R(I_{peak} \cdot \varepsilon)$.

## 5.4 When dynamics should be taken into account?

With constant illumination or with too long illumination periods (above 15 min), the dynamics of A, B, C reach an equilibrium and the gross growth rate can be efficiently represented by the Haldane model (Eq. 5) with only three parameters<sup>70</sup>, with a simplified respiration model, being $R_D$ during the dark periods and $R_L$ during the light periods. A dynamical approach becomes necessary for lower illumination time<sup>41,42,50</sup>. The gross growth rate computation was so far determined only for duty cycles $\varepsilon$ of 0.5<sup>42</sup>, here we propose a simple approach to estimate it for any $\varepsilon$. On top of this, the respiration pattern complicates the relationship between gross and net growth rate in periodic light regimes. The distortion of Han's theory is likely to be more marked for biofilms, due to the higher oxygen level, definitely making their behaviour more tricky to understand.

## 5.5 Can flashing light enhance growth rate?

Net growth rate ($\mu_n$) at high flashing light does not (Figure 6) overpass the value obtained for constant light, at identical light dose, as it has been already stated by several authors<sup>22,42</sup>. This aspect is often unclear in literature, where growth rate enhancement is sometimes due to an increase of the light dose, typically with the duty cycle. The increase of $\mu_n$ with the frequency of alternating L/D regimes is in agreement with literature<sup>22,24,71</sup>. The magnitude of full light integration (tens of milliseconds) is confirmed by the studies of suspended cultures<sup>22,72-74</sup>, which is in line with electron turnover time scale<sup>75,76</sup>. Flashing light is, above all, a matter of diluting light peaks in time, distributing them on a longer period. The respiration pattern does not modify this behaviour since respiration at high frequency is also governed by average light.

Predictions show that the *Full light integration* requires very short time cycle, of the order of a few milliseconds, which cannot be achieved in RABs due to energy cost and cell detachment produced by the high shear stress. Instead, the goal is to work at intermediate time cycles, of the order of ten seconds, that remain short enough for photoinhibition to be diluted over the entire light/dark cycle. The notion of operational plateau (see Figure 6) for which a growth enhancement is expected in realistic condition is the cornerstone for further process optimization, and more experiments are needed to more accurately identify the time range of this plateau.

Now, when considering the net footprint productivity, this effect is slightly reduced by a larger biomass respiration.
## 5.6 About non-represented effects

There is a need for further studies to understand better how respiration fluctuates in intermittent light regimes³⁵,⁷⁷. Our findings show that, for biofilm, respiration is probably mainly influenced by the high level of oxygen. This seems to make photorespiration a major issue in biofilm. This point must still be confirmed by more experimental campaigns where oxygen in biofilm is measured along depth and for various periods and duty cycles. Such experimental approaches will be necessary to further calibrate and validate the proposed model. Consistently, the respiration expression might be improved by considering the effect of light intensity on the O₂ profile, hence on photorespiration, in line with (Li et al. 2016)⁵⁹ which demonstrate that the O₂ concentration varies in the film thickness. Finally, it should be mentioned that the value of R<sub>D</sub> found with our set of experiments is very low. The respiration during the dark phase is then negligible with the conditions tested here. Further experiments, with longer dark periods, would be required, for example to predict the mass loss during a daytime.

## 6 Conclusion

Light/dark cycles have long been studied in microalgae research, primarily in planktonic systems, to optimize growth conditions. We proposed a new model for predicting biofilm productivity under fluctuating light. Crucially, we incorporated respiration dynamics into this framework, offering a more comprehensive representation of microalgal growth under fluctuating light conditions. In this work, we extended the analytical framework of the Han model to derive general solutions for arbitrary light regimes, encompassing a broad range of cycle durations and light duty cycles. Previously, analytical solutions to the Han model were limited to specific cases (e.g., duty cycles of 0.5 or 1). Our results fill this gap, enabling model-based exploration of growth performance under more realistic and variable light regimes. Light induced respiration—is systematically integrated into the growth modeling of photosynthetic biofilms. This dynamics significantly impacts growth predictions, especially in biofilms where photorespiration may reroute fixed CO₂ toward glycolate excretion, effectively reducing net carbon fixation. Rotating biofilm systems offer a promising platform to enhance areal productivity by reducing photoinhibition, but key design parameters such as rotation speed and surface exposure ratio require further optimization. Future developments will involve incorporating temperature dependencies, integrating detachment processes due to shear stress, accounting for species-specific growth responses. Additionally, disentangling the contributions of dark respiration and photorespiration will be critical to advancing both the accuracy of mechanistic models and the engineering of more efficient biofilm-based systems.

## References

1. Bhattacharya, M. & Goswami, S. Microalgae-a green multi-product biorefinery for future industrial prospects. *Biocatal. Agric. Biotechnol.* **25**, 101580 (2020).

2. Hoang, A. T. et al. Biofuel production from microalgae: challenges and chances. *Phytochem. Rev.* **22**, 1089–1126 (2023).

3. Markou, G. & Nerantzis, E. Microalgae for high-value compounds and biofuels production: a review with focus on cultivation under stress conditions. *Biotechnol. advances* **31**, 1532–1542 (2013).
4. Costa, J. A. V., Freitas, B. C. B., Santos, T. D., Mitchell, B. G. & Morais, M. G. Open pond systems for microalgal culture. In *Biofuels from Algae*, 199–223 (Elsevier, 2019).

5. Milledge, J. J. & Heaven, S. A review of the harvesting of micro-algae for biofuel production. *Rev. Environ. Sci. Bio/Technology* **12**, 165–178 (2013).

6. Mantzorou, A. & Ververidis, F. Microalgal biofilms: A further step over current microalgal cultivation techniques. *Sci. Total. Environ.* **651**, 3187–3201 (2019).

7. Zhao, Z., Muylaert, K., Szymczyk, A. & Vankelecom, I. F. Enhanced microalgal biofilm formation and facilitated microalgae harvesting using a novel ph-responsive, crosslinked patterned and vibrating membrane. *Chem. Eng. J.* **410**, 127390 (2021).

8. Gross, M., Henry, W., Michael, C. & Wen, Z. Development of a rotating algal biofilm growth system for attached microalgae growth with in situ biomass harvest. *Bioresour. technology* **150**, 195–201 (2013).

9. Singh, G. & Patidar, S. Development and applications of attached growth system for microalgae biomass production. *BioEnergy Res.* **14**, 709–722 (2021).

10. Wang, J., Liu, J. & Liu, T. The difference in effective light penetration may explain the superiority in photosynthetic efficiency of attached cultivation over the conventional open pond for microalgae. *Biotechnol. for biofuels* **8**, 1–12 (2015).

11. Huang, Y. *et al*. Comparison of chlorella vulgaris biomass productivity cultivated in biofilm and suspension from the aspect of light transmission and microalgae affinity to carbon dioxide. *Bioresour. technology* **222**, 367–373 (2016).

12. Gross, M. & Wen, Z. Yearlong evaluation of performance and durability of a pilot-scale revolving algal biofilm (rab) cultivation system. *Bioresour. technology* **171**, 50–58 (2014).

13. Wang, J., Liu, W. & Liu, T. Biofilm based attached cultivation technology for microalgal biorefineries-a review. *Bioresour. Technol.* **244**, DOI: 10.1016/j.biortech.2017.05.136 (2017).

14. Christenson, L. B. & Sims, R. C. Rotating algal biofilm reactor and spool harvester for wastewater treatment with biofuels by-products. *Biotechnol. bioengineering* **109**, 1674–1684 (2012).

15. Sebestyén, P. *et al*. Upscale of a laboratory rotating disk biofilm reactor and evaluation of its performance over a half-year operation period in outdoor conditions. *Algal research* **18**, 266–272 (2016).

16. Blanken, W. *et al*. Biofilm growth of chlorella sorokiniana in a rotating biological contactor based photobioreactor. *Biotechnol. bioengineering* **111**, 2436–2445 (2014).

17. Gross, M., Mascarenhas, V. & Wen, Z. Evaluating algal growth performance and water use efficiency of pilot-scale revolving algal biofilm (rab) culture systems. *Biotechnol. bioengineering* **112**, 2040–2050 (2015).
18. Brown, H. T. & Escombe, F. Researches on some of the physiological processes of green leaves, with special reference to the interchange of energy between the leaf and its surroundings. *Proc. Royal Soc. London. Ser. B, Containing Pap. a Biol. Character* **76**, 29–111 (1905).

19. Emerson, R. & Arnold, W. A separation of the reactions in photosynthesis by means of intermittent light. *The J. general physiology* **15**, 391 (1932).

20. Grobbelaar, J. U., Nedbal, L. & Tichỳ, V. Influence of high frequency light/dark fluctuations on photosynthetic characteristics of microalgae photoacclimated to different light intensities and implications for mass algal cultivation. *J. applied phycology* **8**, 335–343 (1996).

21. Combe, C. et al. Long-term adaptive response to high-frequency light signals in the unicellular photosynthetic eukaryote *dunaliella salina*. *Biotechnol. bioengineering* **112**, 1111–1121 (2015).

22. Schulze, P. S. et al. Flashing light does not improve photosynthetic performance and growth of green microalgae. *Bioresour. Technol. Reports* **9**, 100367 (2020).

23. Levasseur, W., Pozzobon, V. & Perré, P. Green microalgae in intermittent light: a meta-analysis assisted by machine learning. *J. Appl. Phycol.* **1**–24 (2021).

24. Sforza, E., Simionato, D., Giacometti, G. M., Bertucco, A. & Morosinotto, T. Adjusted light and dark cycles can optimize photosynthetic efficiency in algae growing in photobioreactors. *PloS one* **7**, e38975 (2012).

25. Vejrazka, C. *Microalgal photosynthesis under flashing light* (Wageningen University and Research, 2012).

26. Graham, P. J., Nguyen, B., Burdyny, T. & Sinton, D. A penalty on photosynthetic growth in fluctuating light. *Sci. reports* **7**, 1–11 (2017).

27. Toninelli, A. E., Junfeng Liu, M., Wu, H. & Liu, T. Scenedesmus dimorphus biofilm: Photoefficiency and biomass production under intermittent lighting. *Sci. Reports* **6**, 1–10., DOI: 10.1038/srep32305 (2016).

28. Martín-Girela, I., Curt, M. D. & Fernández, J. Flashing light effects on co2 absorption by microalgae grown on a biofilm photobioreactor. *Algal research* **25**, 421–430 (2017).

29. J. GEIDER, R. & Osborne, B. A. Respiration and microalgal growth: a review of the quantitative relationship between dark respiration and growth. *New phytologist* **112**, 327–341 (1989).

30. Suggett, D. et al. Assessment of photosynthesis in a spring cyanobacterial bloom by use of a fast repetition rate fluorometer. *Limnol. Oceanogr.* **46**, 802–810 (2001).

31. Decostere, B. Assessment of microalgal growth kinetics through respirometry and titrimetry: data collection and model analysis. Ph.D. thesis, Ghent University (2016).

32. Wolf, G., Picioreanu, C. & van Loosdrecht, M. C. Kinetic modeling of phototrophic biofilms: the phobia model. *Biotechnol. bioengineering* **97**, 1064–1079 (2007).
33. Beardall, J. et al. Studies on enhanced post-illumination respiration in microalgae. *J. Plankton Res.* **16**, 1401–1410 (1994).

34. Kliphuis, A. M., Janssen, M., van den End, E. J., Martens, D. E. & Wijffels, R. H. Light respiration in chlorella sorokiniana.
*J. applied phycology* **23**, 935–947 (2011).

35. Shimakawa, G., Kohara, A. & Miyake, C. Characterization of light-enhanced respiration in cyanobacteria. *Int. J. Mol. Sci.* **22**, 342 (2021).

36. Pozzobon, V. & Perre, P. Han's model parameters for microalgae grown under intermittent illumination: Determined using particle swarm optimization. *J. Theor. Biol.* **437**, 29–35 (2018).

37. Jones, G. B., Sims, R. C. & Zhao, J. Experimental and theoretical investigations of rotating algae biofilm reactors (rabrs): Areal productivity, nutrient recovery, and energy efficiency. *Biotechnol. Bioeng.* **120**, 2865–2879 (2023).

38. Jones, G. B., Sims, R. C. & Zhao, J. Computational modeling of microalgal biofilm growth in heterogeneous rotating algal biofilm reactors (rabrs) for wastewater treatment. *Appl. Math. Model.* **131**, 487–504 (2024).

39. Han, B.-P. A mechanistic model of algal photoinhibition induced by photodamage to photosystem-ii. *J. theoretical biology* **214**, 519–527 (2002).

40. Megard, R., Tonkyn, D. & Senft, I., W.H. Kinetics of oxygenic photosynthesis in planktonic algae *. *J. Plankton Res.* **6**, 325–337, DOI: 10.1093/plankt/6.2.325 (1984). https://academic.oup.com/plankt/article-pdf/6/2/325/7496768/6-2-325.pdf.

41. Bara, O., Bonnefond, H. & Bernard, O. Model development and light effect on a rotating algal biofilm. *IFAC-PapersOnLine* **52**, 376–381 (2019).

42. Hartmann, P., Béchet, Q. & Bernard, O. The effect of photosynthesis time scales on microalgae productivity. *Bioprocess biosystems engineering* **37**, 17–25 (2014).

43. Lamare, P.-O. et al. Gradient-based optimization of a rotating algal biofilm process. *Automatica* **105**, 80–88 (2019).

44. Ippoliti, D. et al. Modeling of photosynthesis and respiration rate for isochrysis galbana (t-iso) and its influence on the production of this strain. *Bioresour. technology* **203**, 71–79 (2016).

45. Grobbelaar, J. U. The influence of light/dark cycles in mixed algal cultures on their productivity. *Bioresour. Technol.* **38**, 189–194, DOI: 10.1016/j.automatica.2013.11.014 (2014).

46. Falkowski, P. G., Dubinsky, Z. & Santostefano, G. Light-enhanced dark respiration in phytoplankton: With 1 figure and 1 table in the text. *Int. Vereinigung für theoretische und angewandte Limnol. Verhandlungen* **22**, 2830–2833 (1985).

47. Gao, Y., Bernard, O., Fanesi, A., Perré, P. & Lopes, F. The impact of light/dark regimes on structure and physiology of chlorella vulgaris biofilms. *Front. Microbiol.* **14** (2023).

48. Gao, Y., Bernard, O., Fanesi, A., Perré, P. & Lopes, F. The effect of light intensity on microalgae biofilm structures and physiology under continuous illumination. *Sci. Reports* **14**, 1151 (2024).
49. Grenier, J., Lopes, F., Bonnefond, H. & Bernard, O. Worldwide perspectives of rotating algal biofilm up-scaling (2020). Submitted.

50. Wu, X. & Merchuk, J. C. A model integrating fluid dynamics in photosynthesis and photoinhibition processes. *Chem. engineering science* **56**, 3527–3538 (2001).

51. Grenier, J., Bonnefond, H., Lopes, F. & Bernard, O. The impact of light supply to moving photosynthetic biofilms. *Algal Res.* **44**, 101674 (2019).

52. Wang, Y. et al. The self-adaption capability of microalgal biofilm under different light intensities: Photosynthetic parameters and biofilm microstructures. *Algal Res.* **58**, 102383 (2021).

53. Xue, X., Gauthier, D. A., Turpin, D. H. & Weger, H. G. Interactions between photosynthesis and respiration in the green alga chlamydomonas reinhardtii (characterization of light-enhanced dark respiration). *Plant Physiol.* **112**, 1005–1014 (1996).

54. Nikolaou, A., Hartmann, P., Sciandra, A., Chachuat, B. & Bernard, O. Dynamic coupling of photoacclimation and photoinhibition in a model of microalgae growth. *J. theoretical biology* **390**, 61–72 (2016).

55. Fanesi, A. et al. Shear stress affects the architecture and cohesion of chlorella vulgaris biofilms. *Sci. Reports* **11**, 1–11 (2021).

56. Huesemann, M. H. et al. A screening model to predict microalgae biomass growth in photobioreactors and raceway ponds. *Biotechnol. bioengineering* **110**, 1583–1594 (2013).

57. Weger, H. G., Herzig, R., Falkowski, P. G. & Turpin, D. H. Respiratory losses in the light in a marine diatom: Measurements by short-term mass spectrometry. *Limnol. Oceanogr.* **34**, 1153–1161 (1989).

58. Grobbelaar, J. U. & Soeder, C. J. Respiration losses in planktonic green algae cultivated in raceway ponds. *J. Plankton Res.* **7**, 497–506 (1985).

59. Li, T. et al. Microscale profiling of photosynthesis-related variables in a highly productive biofilm photobioreactor. *Biotechnol. bioengineering* **113**, 1046–1055 (2016).

60. Garbayo, I. et al. Enhancement of photorespiration in immobilized chlamydomonas reinhardtii cells. *Biotechnol. letters* **27**, 265–267 (2005).

61. Foyer, C. H., Bloom, A. J., Queval, G. & Noctor, G. Photorespiratory metabolism: genes, mutants, energetics, and redox signaling. *Annu. review plant biology* **60**, 455–484 (2009).

62. Birmingham, B. C., Coleman, J. R. & Colman, B. Measurement of photorespiration in algae. *Plant Physiol.* **69**, 259–262 (1982).

63. Wieland, A. & Kühl, M. Regulation of photosynthesis and oxygen consumption in a hypersaline cyanobacterial mat (camargue, france) by irradiance, temperature and salinity. *FEMS microbiology ecology* **55**, 195–210 (2006).
64. Pringault, O. & Garcia-Pichel, F. Monitoring of oxygenic and anoxygenic photosynthesis in a unicyanobacterial biofilm, grown in benthic gradient chamber. *FEMS microbiology ecology* **33**, 251–258 (2000).

65. Kühl, M., Glud, R. N., Ploug, H. & Ramsing, N. B. Microenvironmental control of photosynthesis and photosynthesis-coupled respiration in an epilithic cyanobacterial biofilm 1. *J. Phycol.* **32**, 799–812 (1996).

66. Günther, A. et al. Methane production from glycolate excreting algae as a new concept in the production of biofuels. *Bioresour. Technol.* **121**, 454–457 (2012).

67. Taubert, A., Jakob, T. & Wilhelm, C. Glycolate from microalgae: an efficient carbon source for biotechnological applications. *Plant Biotechnol. J.* **17**, 1538–1546 (2019).

68. Fu, X. & Walker, B. J. Dynamic response of photorespiration in fluctuating light environments. *J. Exp. Bot.* **74**, 600–611 (2023).

69. Blanken, W., Cuaresma, M., Wijffels, R. H. & Janssen, M. Cultivation of microalgae on artificial light comes at a cost. *Algal Res.* **2**, 333–340 (2013).

70. Bernard, O. & Rémond, B. Validation of a simple model accounting for light and temperature effect on microalgal growth. *Bioresource Technology* **123**, 520–527., DOI: 10.1016/j.biortech.2012.07.022 (2012).

71. Nedbal, L., Tichỳ, V., Xiong, F. & Grobbelaar, J. U. Microscopic green algae and cyanobacteria in high-frequency intermittent light. *J. Appl. Phycol.* **8**, 325–333 (1996).

72. Kok, B. Experiments on photosynthesis by chlorella in flashing light. *Algal culture from laboratory to pilot plant* 63–75 (1953).

73. Vejrazka, C., Janssen, M., Benvenuti, G., Streefland, M. & Wijffels, R. H. Photosynthetic efficiency and oxygen evolution of chlamydomonas reinhardtii under continuous and flashing light. *Appl. Microbiol. Biotechnol.* **97**, 1523–1532 (2013).

74. Levasseur, W., Perré, P. & Pozzobon, V. Chlorella vulgaris acclimated cultivation under flashing light: An in-depth investigation under iso-actinic conditions. *Algal Res.* **70**, 102976 (2023).

75. Falkowski, P. G. & Raven, J. A. *Aquatic photosynthesis* (Princeton University Press, 2013).

76. Malcata, F. X. Microalgae and biofuels: a promising partnership? *Trends biotechnology* **29**, 542–549 (2011).

77. Ruiz-Martinez, A., Serralta, J., Seco, A. & Ferrer, J. Behavior of mixed chlorophyceae cultures under prolonged dark exposure. *respiration rate modeling*. *Ecol. Eng.* **91**, 265–269 (2016).

7 Acknowledgements

We thank the financial support of China Scholarship Council (NO.201806120042) and ANR PhotoBiofilm Explorer (Project-ANR-20-CE43-0008)/LaSIPS Greenbelt and Fondation des Treilles. We thank also the Département de la Marne, Grand Reims, Région Grand Est and the European Union along with the European Regional Development Fund (ERDF Champagne Ardenne
2014-2020) for their financial support of the Chair of Biotechnology of CentraleSupélec. This work benefited from the ITN
Digitalgaesation project funded by the European Union's Horizon 2020 research, and innovation program under the Marie
Skłodowska-Curie grant agreement No. 955520.

**8 Author contributions statement**

Yan GAO and Ignacio FIERRO developed the model. Yan GAO wrote the first manuscript version. Olivier BERNARD supervised the modelling. Olivier BERNARD, Patrick PERRÉ and Filipa LOPES contributed to the results analysis and reviewed the manuscript.

**9 Conflict of Interest Statement**

We declare no conflict of interest.