Dynamic Modeling of Microalgae Growth and Lipid Production under

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Energy and the Environment

Dynamic Modeling of Microalgae Growth and Lipid Production under Transient Light and Nitrogen Conditions Diyuan Wang, Yi-Chun Lai, Amanda Louise Karam, Francis Lajara de los Reyes III, and Joel Ducoste Environ. Sci. Technol., Just Accepted Manuscript • DOI: 10.1021/acs.est.9b02908 • Publication Date (Web): 26 Aug 2019 Downloaded from pubs.acs.org on August 27, 2019

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Environmental Science & Technology

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Dynamic Modeling of Microalgae Growth and Lipid Production

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under Transient Light and Nitrogen Conditions

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Diyuan Wang, Yi-Chun Lai, Amanda L. Karam, Francis L. de los Reyes, III, Joel J. Ducoste*

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Department of Civil, Construction, and Environmental Engineering

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North Carolina State University

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Raleigh, NC 27695, United States

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*Corresponding author e-mail: [email protected]

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ABSTRACT

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We developed a new dynamic model to

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characterize how light and nitrogen regulate

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the cellular processes of photosynthetic

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microalgae leading to transient changes in

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the production of neutral lipids,

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carbohydrates, and biomass. Our model recapitulated the versatile neutral lipid synthesis

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pathways via (i) carbon reuse from carbohydrate metabolism under nitrogen sufficiency and (ii)

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fixed carbon redirection under nitrogen depletion. We also characterized the effects of light

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adaptation, light inhibition hysteresis, and nitrogen limitation on photosynthetic carbon fixation.

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The formulated model was calibrated and validated with experimental data of Dunaliella viridis

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cultivated in a lab-scale photobioreactor (PBR) under various light (low/moderate/high) and

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nitrogen (sufficient/limited) conditions. We conducted the identifiability, uncertainty, and

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sensitivity analyses to verify the model reliability using the profile likelihood method, the

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Markov chain Monte Carlo (MCMC) technique, and the extended Fourier Amplitude Sensitivity

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Test (eFAST). Our model predictions agreed well with experimental observations and suggested

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potential model improvement by incorporating a lipid degradation mechanism. The insights from

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our model-driven analysis helped improve the mechanistic understanding of transient algae

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growth and bioproducts formation under environmental variations and could be applied to

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optimize biofuel and biomass production.

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INTRODUCTION

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To make microalgae-derived biofuel commercially competitive with fossil fuel, recent research

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has suggested different strategies such as producing high-profit coproducts from microalgae

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proteins and nutritional components,1 integrating algae production with wastewater treatment

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and industrial CO2 uptake,2 or designing algal polycultures for multifunctional and stable

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cultivation.3 However, applying these strategies in large-scale cultivation is difficult, because

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environmental variations such as light and nutrient affect not only the biomass yield but also the

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pattern and activity of microalgae metabolic pathways4 that lead to dynamic changes in neutral

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lipids, carbohydrates, and other functional biomass contents. Therefore, the design, control, and

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optimization of the cultivation process continue to be a fundamental challenge in advancing

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microalgae-based biofuel technologies.

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To better understand, explain, and predict dynamic algal behaviors in response to critical

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environmental variations, different mathematical models have been developed. With detailed

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representation of the underlying pathways, metabolic models could reveal the metabolic network

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for production of targeted compounds in microalgae. However, the obtained metabolic carbon


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fluxes are constrained to a specific condition due to the steady-state and balanced-growth

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assumption for solving the stoichiometric reactions.5 To predict dynamic carbon fluxes during

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culture condition transitions such as light-dark cycles6 or switching between nitrogen repletion

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and depletion,7 metabolic models require multiple pseudo-steady-state time intervals, thus

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becoming highly complex and computationally expensive to solve. Guest et al.8 developed a

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lumped-pathway metabolic model that simplified the dynamic metabolic modeling by integrating

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metabolic reactions into the kinetic model. However, the model was still subject to the steady-

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state and balanced-growth assumption and required two sets of reactions to separately determine

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the stoichiometric yields for nutrient-available and nutrient-depleted conditions. Consequently,

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the transient culture behaviors under varying light and nitrogen conditions were not predicted

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accurately.

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Kinetic models allow the incorporation of multiple environmental factors that

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dynamically modulate algae growth and biofuel production. However, kinetic models based

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solely on empirical relationships9,10 are usually black-box representations of the algal system and

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do not reveal how environmental factors regulate different bioprocesses. Consequently, these

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models provide limited understanding of the tradeoffs between algal growth and lipid

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production. Some kinetic models11–13 simulated the carbon fixation and distribution processes to

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describe storage molecule production. However, these models were only valid for nitrogen-

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limited conditions or required a new set of parameters to fit the nitrogen-sufficient conditions,

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indicating that the underlying mechanism of neutral lipid synthesis was not fully characterized.

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Moreover, these prior works did not clearly study the potential for the regulatory effects of light.

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For example, most prior models determined the photosynthesis rate based solely on the

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instantaneous light intensity but ignored the hysteresis effect of the pre-exposure to high or low


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light intensities.14 Without comprehensively accounting for the light variation effect on

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photosynthesis rate, prior photosynthesis models may not be applicable to wide-ranging light

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conditions and outdoor cultivation systems.14 In addition, light could also affect the lipid

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production through influences on the carbon distribution processes. A simple representation of

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the carbon distribution process may not properly capture the detailed regulatory and

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combinatorial effects of light and nitrogen on lipid accumulation.

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In this study, we establish a new kinetic model to improve the dynamic modeling of

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photosynthetic carbon fixation and carbon partitioning processes regulated by both light and

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nitrogen factors. The proposed carbon partitioning mechanism recapitulates the versatile neutral

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lipid synthesis pathways. Moreover, the formulated photosynthesis process incorporates not only

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light adaptation and inhibition effects but also the nitrogen-limited effect on carbon growth. Our

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model could characterize and forecast transient culture behaviors including the production of

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neutral lipids, carbohydrates, and functional biomass under a wide range of light and nitrogen

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conditions. The model reliability is verified via the identifiability, uncertainty, and sensitivity

82

analyses. The proposed modeling framework facilitates rational study of the microalgae dynamic

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cellular processes in response to environmental variations and provides useful insights on

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cultivation process control for achieving optimal biomass growth and biofuel production.

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METHODS

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We established our model (Figure 1) based on the major cellular processes of most green algae

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species to describe the culture behaviors of Dunaliella viridis cultivated under different light and

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nitrogen conditions. The experiments were performed in a bench-scale 3.2L flat-panel

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photobioreactor (Figure S1) with a light sensor mounted inside and the pH and temperature


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controlled at 7.6 and 25 C, respectively. We used data from eleven experiments (Table S1)

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including four light conditions (categorized into low/moderate/high levels) and ten initial nitrate

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concentrations (categorized into high/low levels). Concentrations of the biomass, carbohydrate,

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neutral lipid, Chlorophyll a (Chl-a), and nitrate were measured daily in triplicates for

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approximately ten days. The experiments were fully discussed by Lai et al.15 Our model contains

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six state variables, and Table 1 summarizes the entire set of ordinary differential equations

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(ODEs) for the model formulation (see Nomenclature for detailed parameter definitions).

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Figure 1. Diagrammatic representation of carbon flows under (a) nitrogen-sufficient conditions and (b) nitrogendeficient conditions. Dashed arrows indicate small or decreasing carbon flow rates under corresponding nitrogen conditions. Dark gray arrows denote carbon flows related to photosynthetic carbon fixation rates. Yellow arrows indicate additional carbon flows regulated by light.

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Table 1. Full set of ODEs for the proposed model formulation

Carbon Flow

State Variable Rate Equation 𝑑𝑋𝑐𝑑 Functional biomass 𝑋𝑐𝑑

𝑑𝑡

(

= 1―

𝑞𝑛

)

𝑄𝑛

∙ 𝑃 ∙ 𝑋𝑐𝑑

Equation Number

R1

(1)

(-) R2

―𝑚 ∙ 𝑋𝑐𝑑 See 𝑃 defined in Eqn (12), 𝑄𝑛 in Eqn (18) 𝑑𝑋𝑐𝑎𝑟𝑏 𝑑𝑡

Carbohydrate 𝑋𝑐𝑎𝑟𝑏

=

𝑞𝑛 𝑄𝑛


R3

― 𝑚𝑐 ∙ exp( ― 𝑘𝑚𝑐 ∙ 𝐼𝑎𝑣𝑔_𝑅) ∙ 𝑋𝑐𝑎𝑟𝑏

(-) R4

―𝑚𝑐𝑁 ∙ 𝑣𝑁 ∙ 𝑋𝑐𝑑

(-) R5

― 𝛽 ∙ exp(𝑘𝑛𝑙 ∙ 𝐼𝑎𝑣𝑔) ∙ 𝑣𝑁 ∙ 𝑋𝑐𝑑 𝑞𝑛 ― 𝑞𝑛𝑙 ∙ ∙ 𝑃 ∙ 𝑋𝑐𝑑 𝑄𝑛

(-)* R6

(2)

(-)* R7

See 𝐼𝑎𝑣𝑔 defined in Eqn (10), 𝐼𝑎𝑣𝑔_𝑅 in Eqn (11), 𝑞𝑛𝑙 in Eqn (15), 𝑣𝑁 in Eqn (17) 𝑑𝑋𝑛𝑙 Neutral lipid 𝑋𝑛𝑙

Chlorophyll a 𝑋𝑐ℎ𝑙

𝑑𝑡

= 𝛽 ∙ exp(𝑘𝑛𝑙 ∙ 𝐼𝑎𝑣𝑔) ∙ 𝑣𝑁 ∙ 𝑋𝑐𝑑 + 𝑞𝑛𝑙 ∙

𝑑𝑋𝑐ℎ𝑙 𝑑𝑡

𝑞𝑛 𝑄𝑛

R6 (3)


R7

= 𝜃 ∙ exp( ― 𝑘𝑐ℎ𝑙 ∙ 𝐼𝑎𝑣𝑔) ∙ 𝑣𝑁 ∙ 𝑋𝑐𝑑 ― 𝑚𝑐ℎ𝑙 ∙ 𝑋𝑐ℎ𝑙

(4)

See 𝑚𝑐ℎ𝑙 defined in Eqn (16)

Extracellular nitrogen 𝑆𝑁𝑂3

𝑑𝑆𝑁𝑂3

Intracellular nitrogen 𝑋𝑁

𝑑𝑋𝑁

𝑑𝑡

𝑑𝑡

= ― 𝑣𝑁 ∙ 𝑋𝑐𝑑

= 𝑣𝑁 ∙ 𝑋𝑐𝑑

(5)

(6)

Note: (-) denotes the carbon loss. (-)* denotes the carbon conversion from one molecule to another.

104 105

Light Estimation. The light condition in our experiment was controlled by adjusting

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the light source position (i.e., the distance 𝑑 in Figure S1). As source light passes through the

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photobioreactor, some light attenuation will occur. The level of attenuation is affected by the


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amount of chlorophyll, biomass, and background absorbing material.16 Eqn (7) is formulated to

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express the local light intensity 𝐼𝑙 along the optical path 𝑙 inside the reactor. The light attenuation

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due to background absorbing material is modeled in 𝐼𝑅,𝑙, which is estimated by an empirical

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equation (Eqn. 8) obtained from experimental measurements (Figure S2). The light attenuation

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due to Chl-a and the remaining biomass (𝑋𝐶 ― 𝑋𝑐ℎ𝑙) is modeled by formulating the light

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absorbance 𝐴 (Eqn. 9) based on the Beer-Lambert Law. We calibrated the values of 𝑎 (the

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optical cross section of Chl-a) and 𝑏 (the optical cross section of non-Chl-a biomass) in Eqn (9)

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by minimizing the discrepancy between the experimentally measured light profile 𝐼𝑝 and the

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calculated light profile 𝐼𝑧𝑙. Specifically, the light intensity 𝐼𝑝 at location 𝑙 = 𝑧𝑙 was continuously

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monitored by the light sensor during cultivation, and this intensity could be calculated as 𝐼𝑧𝑙 by

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plugging the experimental values of Chl-a and biomass concentration into Eqn (9) and specifying

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𝑙 = 𝑧𝑙 in Eqn (7). We assumed that 𝑎 and 𝑏 were independent of the cultivation conditions and

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estimated their values (𝑎 = 80.5 m2 mol-1 𝑋𝑐ℎ𝑙-C and 𝑏 = 1.4718 m2 mol-1 C) using the 𝐼𝑝 of

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different light and nitrogen conditions (calibration results shown in Figure S3). Our value of 𝑎 is

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equivalent to 4.96 m2 g-1 Chl-a, which is close to the value of 4.82 m2 g-1 Chl-a estimated by

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Packer et al.12

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𝐼𝑙 = 𝐼𝑅,𝑙 ∙ exp ( ―𝐴 ∙ 𝑙)

(7)

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𝐼𝑅,𝑙 = 1.7324(𝑙 + 𝑑) ―1.993

(8)

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𝐴 = 𝑎 ∙ 𝑋𝑐ℎ𝑙 + 𝑏 ∙ (𝑋𝐶 ― 𝑋𝑐ℎ𝑙)

(9)

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The average light intensity 𝐼𝑎𝑣𝑔 (Eqn. 10) inside the reactor is calculated by averaging the

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local light intensities 𝐼𝑙 over the optical path from 𝑙 = 0 to 𝑙 = 𝑧𝑡. In addition, the average

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background light intensity 𝐼𝑎𝑣𝑔_𝑅 is expressed in Eqn (11). The instantaneous light intensity 𝐼𝑎𝑣𝑔


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regulates the rates of cellular processes that can be dynamically affected by light variations such

131

as photosynthesis, carbon partitioning, and Chl-a synthesis. In contrast, 𝐼𝑎𝑣𝑔_𝑅 does not vary in

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time. We used 𝐼𝑎𝑣𝑔_𝑅 to adjust the parameter values that only change when the algae are

133

cultivated under different light conditions. 1

𝑧

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𝐼𝑎𝑣𝑔 = 𝑧𝑡 ∙ ∫0𝑡1.7324(𝑙 + 𝑑) ―1.993 ∙ exp ( ―𝐴 ∙ 𝑙)ⅆ𝑙

135

𝐼𝑎𝑣𝑔_𝑅 = 𝑧𝑡 ∙ ∫0𝑡1.7324(𝑙 + 𝑑) ―1.993ⅆ𝑙

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Photosynthetic Carbon Fixation. The photosynthesis rate equation (Eqn. 12) was

1

𝑧

(10) (11)

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formulated by modifying the simplified Steele’s equation,17,18 which expresses the typical

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relationship between photosynthesis rate and light intensity (i.e., P-I relationship) during the

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stages of light limitation, saturation, and inhibition. However, the P-I relationship varies in terms

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of the light saturation point and light-saturated photosynthetic rate because algae cells can adapt

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to light variations by adjusting their cellular pigment contents.19 We thus modified the saturated

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light intensity 𝐼𝑠𝑎𝑡 as a function of the Chl-a content (Eqn. 13) inspired by the Poisson model.20

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In addition, the simplified Steele’s equation overlooks the hysteresis of light inhibition caused by

144

the pre-exposure to high light intensity.14 We thus modified the carbon yield on light energy 𝑌𝑐𝐸

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as a function of 𝐼𝑎𝑣𝑔_𝑅 (Eqn. 14). Consequently, the carbon yield is smaller when cells are

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initially exposed to high background light intensity. This reduced carbon yield is in line with the

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prior finding that certain intracellular inhibitors could cause a decrease in the light utilization

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efficiency during photoinhibition.21 Moreover, the higher carbon yield at low background light

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intensity reflects that cells could utilize low-level light more efficiently during low-light

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adaptation.22 The final expression of the photosynthesis rate in Eqn (12) is normalized by the

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functional biomass. Accordingly, the product of 𝑃 and 𝑋𝑐𝑑 gives the carbon fixation rate (i.e.


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total carbon growth rate). This carbon inflow can then be distributed into the synthesis of

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functional biomass and storage molecules.

( )∙𝑃 𝑋𝐶

𝐼𝑎𝑣𝑔

𝐼𝑎𝑣𝑔

154

𝑃=

155

𝐼𝑠𝑎𝑡 =

156

𝑌𝑐𝐸 = 𝑌𝐸 ∙ exp ( ― 𝑘𝑦𝑒 ∙ 𝐼𝑎𝑣𝑔_𝑅)

157

Functional Biomass (Functional Pool). Functional biomass is the algal biomass

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excluding carbohydrates and neutral lipids. The partitioning of the fixed carbon between the

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functional pool and the storage pool is regulated by the nitrogen cell quota 𝑄𝑛. We modeled the

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carbon fixation to the functional biomass (R1) in Eqn (1) based on the Droop model.23 For

161

simplicity, we modeled the degradation of functional biomass (R2) that contributes to respiration

162

as a first-order relationship. The composition of functional biomass is assumed as

163

CH1.8O0.5N0.2.8,24

164

𝑋𝑐𝑑

𝑚

∙ 𝐼𝑠𝑎𝑡 ∙ exp (1 ― 𝐼𝑠𝑎𝑡 )

𝑃 𝑚 ∙ 𝑋𝐶

(12) (13)

𝑎 ∙ 𝑌𝑐𝐸 ∙ 𝑋𝑐ℎ𝑙

(14)

Carbohydrates and Neutral Lipids (Storage Pool). The assimilated carbon that

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has not been used to produce functional biomass is first stored in carbohydrates as shown in R3

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of Eqn (2). The degradation of carbohydrates is categorized into three types based on the carbon

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flow direction: 1) the carbon loss during respiration (R4, R5), 2) the carbon reuse for lipid

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synthesis during nitrogen-available conditions (R6), and 3) the carbon redirection to lipid

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synthesis under nitrogen deficiency (R7). The carbohydrate respiration rate includes two

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components: a basal-level rate (R4) and a growth-related rate (R5). R4 is based on the first-order

171

relationship with the respiration rate constant 𝑚𝑐 adjusted by 𝐼𝑎𝑣𝑔_𝑅 to account for the

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photoinhibition of respiration. As reported in prior studies, moderate to high light intensities

173

could damage the respiratory system of cells and impact respiration activities.25 Consequently,


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cells under the light-inhibition condition could maintain a lower respiration rate than cells that

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have recovered from high-light stress.26 R5 is coupled with the nitrate uptake rate via coefficient

176

𝑚𝑐𝑁 to reflect the growth cost27 as the prior observation showed that a faster growth rate required

177

a higher respiration rate for more energy supplies.28 R6 is also coupled with the nitrate uptake

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rate via the coefficient 𝛽 to describe that some carbon from carbohydrate metabolism could be

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reutilized to accumulate neutral lipids during normal algal growth. However, the coefficient 𝛽 is

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further regulated by 𝐼𝑎𝑣𝑔 because neutral lipids production could increase with increasing light

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intensities as shown in prior studies.29–31 We further formulated R7 to direct a portion 𝑞𝑛𝑙 of the

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previously fixed carbon from carbohydrates to neutral lipids when nitrogen is depleted. 𝑞𝑛𝑙 is

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expressed as an error function (Eqn. 15) triggered by deficient nitrogen concentrations. Some

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detailed carbon partitioning processes between functional biomass components and storage

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molecules were not specifically simulated in this work. For example, functional biomass

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components such as proteins and polar lipids could also contribute to lipid synthesis, and these

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processes were assumed to be covered by R7 as R7 is directly related to the carbon partitioning

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between the functional pool and the storage pool. Additionally, some degraded carbon from

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carbohydrates may also be reused to accumulate functional biomass, and such a process was

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assumed to be covered by the Droop model. We modeled carbohydrates in the form of glucose

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and neutral lipids in the form of triacylglycerols.32

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𝑞𝑛𝑙 = 𝜙 ∙ (1 ― erf (𝑘𝑁 ∙ 𝑆𝑁𝑂3))

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Chlorophyll a. The synthesis of Chl-a is dependent on the assimilated nitrogen.

(15)

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Following prior modeling work,12,13,27,33 we linked the Chl-a synthesis rate to the nitrogen

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uptake rate via the maximum Chl-a:N ratio 𝜃. To account for the light adaptation effect,34 𝜃 is

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regulated by 𝐼𝑎𝑣𝑔 via an exponential function. We further applied the error function in Eqn (16)


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to model the degradation of Chl-a under stress conditions because our experiments showed that

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the Chl-a concentration decreased under nitrogen starvation and high-light conditions, and such

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phenomenon was also observed in previous studies.29,35

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𝑚𝑐ℎ𝑙 = 𝑚 ∙ (1 ― erf (𝑘𝑁 ∙ 𝑆𝑁𝑂3)) ∙ erf (𝑘𝐼 ∙ 𝐼𝑎𝑣𝑔_𝑅)

201

Extracellular Nitrate. In prior models, the nitrate uptake rates were related to the

(16)

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external nitrate concentrations following the Michaelis–Menten kinetics V = Vmax [S] / (KS,N +

203

[S]). However, Fachet et al.33 found that the half-saturation coefficient KS,N was non-identifiable.

204

Moreover, the value of the profile likelihood (which measures the discrepancy between the

205

model prediction and the experimental result) decreases as KS,N increases, indicating that the

206

optimal solution may prefer a larger KS,N. If KS,N >> [S], the Michaelis–Menten equation is

207

essentially reduced to V = Vmax [S] / KS,N, which becomes a first-order relationship. This linear

208

relationship was also observed in some microalgae species when the nitrate concentrations were

209

above 60 𝜇mol-N L-1.36 On the other hand, Jiménez and Niell37 suggested that nitrate uptake rate

210

may be independent of the initial nitrate concentration since they noticed that the initial nitrate

211

concentrations (between 0.5 to 10 mmol-N L-1) had negligible effect on the specific growth rate

212

of Dunaliella viridis. As a result, the nitrate uptake rate 𝑣𝑁 in Eqn (17) is linearly related to the

213

extracellular nitrate concentration 𝑆𝑁𝑂3 and is divided by the initial concentration 𝑆𝑁𝑂3_0 to offset

214

the effect of initial nitrate level. Additionally, the maximum nitrate uptake rate 𝑣𝑛𝑚 is regulated

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by the nitrogen quota to potentially limit the maximum amount of nitrogen that cells can

216

assimilate.12,13,27

(

)∙

𝑄𝑛𝑚 ― 𝑄𝑛

𝑣𝑛𝑚 ∙ 𝑆𝑁𝑂3

𝑄𝑛𝑚 ― 𝑞𝑛

𝑆𝑁𝑂3_0

217

𝑣𝑁 =

218

Intracellular Nitrogen and Nitrogen Quota. We assumed that algae cells

219

assimilated the nitrate removed from the medium. The resulting rate of intracellular nitrogen

(17)


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accumulation (Eqn. 6) was obtained by negating the sign of the nitrate uptake rate. We assumed

221

the initial concentration of intracellular nitrogen 𝑋𝑁_0 to be 0.2𝑋𝑐𝑑_0. In prior models, the

222

nitrogen quota 𝑄𝑛 was usually defined as the ratio of the cellular nitrogen to either the total

223

cellular carbon (or biomass)11,27,33 or the functional carbon (or biomass).8,12,13 However, based on

224

prior experimental results suggesting that the storage molecule production could be enhanced

225

with elevated carbon fixation via excess CO2 supply or mixotrophic cultivation,38–40 it seemed

226

more reasonable to regulate the carbon distribution between the functional pool and the storage

227

pool using the total N:C ratio (Eqn. 18). 𝑋𝑁

228

𝑄 𝑛 = 𝑋𝐶

229

Model Calibration and Validation. The experimental data was split into two subsets

(18)

230

(Table S1): the calibration set (~ 65% of total data) and the validation set (~ 35% of total data).

231

The calibration set was expected to cover most conditions in terms of the wide-ranging light

232

intensities and the transition of nitrogen level from sufficiency to deficiency. Our model was

233

systematically calibrated via identifiability analysis, uncertainty quantification, and sensitivity

234

analysis. We chose the profile likelihood method to perform identifiability analysis owing to its

235

ability to detect both structurally and practically non-identifiable parameters.41 After balancing

236

the model complexity with the available data, we estimated the parameter posterior distributions

237

and quantified the model output uncertainties using the Markov chain Monte Carlo (MCMC)

238

technique with the Delayed Rejection Adaptive Metropolis (DRAM) algorithm.42 We further

239

investigated the sensitivity of each parameter to the model responses using a variance-based

240

global sensitivity analysis technique - the extended Fourier Amplitude Sensitivity Test (eFAST) -

241

well-known for its flexibility and adaptability in dealing with nonlinear monotonic and non-

242

monotonic relationships.43 Details regarding the use of above methods for our model assessment


12

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243

are discussed in the Supporting Information (SI). Finally, we compared the light estimation with

244

experimental data and validated the model by predicting new experiments in the validation set.

245

RESULTS AND DISCUSSION Model Calibration Evaluation. Identifiability Analysis. We detected one non-

246 247

identifiable parameter 𝑄𝑛𝑚 as shown in the profile likelihood plots (Figure S4). 𝑄𝑛𝑚 was only

248

used in Eqn (17) to regulate the nitrate uptake rate. As we further increased the value of 𝑄𝑛𝑚, the

249

discrepancy between the model prediction and the experimental data decreased. The discrepancy

250

reached the minimum when 𝑄𝑛𝑚 ― 𝑞𝑛 was set to one (Figure S5). Therefore, we excluded the term

251


252

Consequently, the nitrogen quota had no regulatory effect on the nitrate uptake rate in our model.

253

It is possible that the nitrate supply in our experiments did not satisfy the cells’ maximum

254

nitrogen uptake capacity so the cells could still assimilate nitrate for additional growth. As

255

shown in previous studies, the saturated cell density of Dunaliella viridis did not increase after

256

increasing the nitrate concentration from 5 mM to 10 mM.37 In contrast, Dunaliella tertiolecta

257

did not further grow after increasing the NO3- concentration from 23 mM to 46 mM. The

258

maximum nitrate concentration we provided was around 4.95 mM.44 Therefore, additional

259

experiments with higher nitrate concentrations may be necessary to test the regulatory

260

effectiveness of 𝑄𝑛𝑚 on nitrate uptake rate.

261


𝑄𝑛𝑚 ― 𝑞𝑛

from the nitrate uptake equation to resolve the non-identifiability associated with 𝑄𝑛𝑚.

Uncertainty Quantification. We obtained two MCMC sample chains that converged to

262

similar steady ranges after 250000 iterations (Figure S6). We chose the one with higher stability

263

over the iterations (Table S5) and used the samples that reached the steady range to generate the


13


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264

parameter posterior distributions and pairwise correlation plots (Figure S7 and S8). Table 2 lists

265

the parameter mean values, standard deviations, and two sets of optimal values obtained by

266

ranking the posterior samples based on either the maximum likelihood function (MLF) or the

267

minimum sum squared error (Min SSE).

268

Table 2. Parameter posterior chain statistics

Mean

SD

𝑃𝑚 𝑌𝐸

4.3054 0.2784

0.7197 0.0876

4.2599 0.3075

3.2084 0.1713

𝑘𝑦𝑒

0.0290

0.0015

0.0296

0.0306

𝑞𝑛 𝜈𝑛𝑚

0.0175 0.1154 3.1142 0.1317 0.0279 0.0684 2.1824 2.7548 0.0208 0.2185 0.0248 0.0022 3.3370

0.0035 0.0039 0.8468 0.0108 0.0033 0.0199 0.7697 0.7898 0.0031 0.0068 0.0021 0.0007 0.8794

0.0170 0.1167 3.2883 0.1355 0.0281 0.0584 2.3150 2.9097 0.0208 0.2169 0.0246 0.0019 2.7503

0.0213 0.1135 1.9066 0.1249 0.0268 0.0748 1.6029 2.1583 0.0205 0.2049 0.0190 0.0030 2.7775

𝑚 𝛽 𝑘𝑛𝑙 𝜙 𝑚𝑐𝑁 𝑚𝑐 𝑘𝑚𝑐 𝜃 𝑘𝑐ℎ𝑙 𝑘𝐼 𝑘𝑁

269 270

Optimal values MLF Min SSE

Parameter

To obtain the model output distributions, we randomly selected 2000 parameter sets from

271

the MCMC posterior samples to generate the output medians, 95% credible intervals, and 95%

272

prediction intervals as illustrated in Figure 2. The model output medians, which approximate the

273

predictions of the model best fit (Figure S9), agreed well with the experimental data. The

274

credible intervals reflected the model output distribution due to parameter uncertainties, and the

275

prediction intervals revealed the overall uncertainties of the model outputs resulted from the

276

parameter uncertainties, measurement noises, and prediction inaccuracies inherent to model


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277

formulation.45 Our model predictions were able to capture the variations in the experimental data

278

and account for the measurement errors.

279 280 281 282 283 284 285 286

Figure 2. Experimental data and calibrated model output with uncertainty quantification. Solid lines correspond to the medians of model responses bounded by 95% credible intervals (dark-color shades) and 95% prediction intervals (light-color shades). Symbols with error bars represent experimental data and measurement noises. Blue and red denote high and low nitrate conditions, respectively. Light levels refer to different light conditions (with corresponding initial nitrate concentration in mM as N): low light (100PPFD and 4.95N, 100PPFD and 0.62N), moderate light (400PPFD and 4.89N, 300PPFD and 0.66N), and high light (600PPFD and 4.88N, 600PPFD and 0.71N). (Note: 1 PPFD = 1 µmol photons m-2s-1)

287

Sensitivity Analysis. We evaluated the sensitivity of each parameter on five state variables

288

at three representative time points (day 2, day 4, and day 10) for four conditions (no stress, single

289

high-light stress, single low-N stress, and dual stress). The sensitivity results (Table S6 and S7,

290

Figure S10 and S11) demonstrated the effectiveness of most parameters. Some parameters were


15


291

mainly sensitive to certain variables under certain conditions, indicating potential space for

292

model reduction that could be tailored to specific purposes. In addition, the variations of

293

parameter sensitivity under different time points and conditions could reflect the relative

294

activities of the underlying cellular processes, which is also helpful in understanding the

295

mechanism of cellular metabolism. See SI for detailed discussion.

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296

Several parameters maintained high sensitivity universally. The processes regulated by

297

these parameters may play a central role in characterizing the algae culture behaviors and thus

298

need to be carefully incorporated into the model. 1) Parameters related to the photosynthesis (𝑃𝑚,

299

𝑌𝐸, 𝑘𝑦𝑒), carbon partitioning (𝑞𝑛), nitrate uptake (𝑣𝑛𝑚), and functional biomass degradation (𝑚):

300

Photosynthesis, nitrate uptake, maintenance and respiration are fundamental processes that

301

provide the sources of carbon, nitrogen, and energy for cellular metabolism. In addition, the

302

nitrogen-quota-regulated carbon partitioning between the functional biomass and the storage

303

molecules appears to be the basic process for modeling the intracellular carbon partitioning. 2)

304

R6-related parameters (𝑘𝑛𝑙 and 𝛽): Carbon reuse from carbohydrate metabolism for neutral lipid

305

synthesis could be a necessary carbon partitioning process due to its high impact on neutral

306

lipids, carbohydrates, and functional biomass especially under high-light conditions. R6 has not

307

been incorporated in prior kinetic models. 3) Parameters related to the Chl-a synthesis (𝑘𝑐ℎ𝑙 and

308

𝜃): Chl-a accumulation could affect other variables due to its close correlation with

309

photosynthesis. Such effect has not been studied in prior models.

310

Light Estimation Performance. The model-predicted 𝐼𝑎𝑣𝑔 matched the estimation from

311

experimental data and the model-predicted light profiles 𝐼𝑧𝑙 were in good agreement with both

312

the experimental estimation and the light sensor measurements (Figure S12 and S13). Therefore,

313

our model was able to properly predict the light variations as a function of time and location.


16

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314


Lipid Accumulation and Carbohydrate Metabolism. Neutral lipid production

315

increases rapidly under nitrogen depletion, which is usually accompanied by a decline in

316

carbohydrate accumulation.35,46–49 To characterize this transient behavior, we established a

317

hierarchical carbon partitioning mechanism to quantify the dynamic carbon distributions among

318

functional biomass, carbohydrates, and neutral lipids. First, the intracellular nitrogen quota

319

regulates the carbon distribution between the functional pool and the storage pool. As 𝑄𝑛

320

decreases, the fixed carbon could be shifted from the synthesis of functional biomass to the

321

production of storage molecules (Figure 3a). Next, the extracellular nitrogen level regulates the

322

carbon partitioning between carbohydrates and neutral lipids inside the storage pool. When

323

nitrogen is sufficient, the stored carbon is first accumulated in carbohydrates, and then a small

324

amount of carbon degraded from carbohydrates could be slowly reused for lipid synthesis via R6.

325

As nitrogen is depleted, the rate of R6 decreases and the process of lipid accumulation shifts to

326

R7, rapidly redirecting the previously stored carbon to neutral lipids (Figure 3b). This shift from

327

R6 to R7 properly captures the transient change of lipid production during the transition from

328

nitrogen sufficiency to depletion.

329

R6 and R7 recapitulate the versatile neutral lipid synthesis pathways for most green algae

330

species40 and demonstrate the metabolic relationship between carbohydrate metabolism and lipid

331

accumulation. Carbohydrates are the primary storage molecules during nitrogen-sufficient

332

conditions because they are highly efficient in energy and carbon storage.4 Carbohydrate

333

metabolism is mainly used for respiration to produce energy for cell growth and maintenance (R5

334

and R4). Lipid accumulation is not favored due to its low energy efficiency (i.e., higher energy

335

consumption for synthesis and slower energy release from degradation). R6 represents the minor

336

carbon contribution to lipid synthesis from acetyl-CoA via the carbohydrate metabolism


17


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337

pathway.40 When nitrogen becomes deficient, lipid accumulation is more valuable in the long

338

term because the large energy reserve from lipids could help cells survive during nitrogen

339

starvation.35 R7 represents the enhanced neutral lipid production due to the redirection of

340

assimilated carbon through multiple pathways including the starch degradation, the de novo

341

synthesis of acetyl-CoA, and the recycling of carbon from other byproducts of the primary

342

metabolism.40,49 Therefore, the carbon partitioning strategy proposed in our model may also be

343

applicable to other green algae species and mixed-culture systems. Moreover, the model could

344

also be tailored for mixotrophic conditions by incorporating an additional carbon fixation process

345

from organic carbon sources.

346

The model-predicted neutral lipid accumulation rates are shown in Figure 3c. R6 and R7

347

were both regulated by light intensities but led to different effects on lipid accumulation. When

348

nitrogen was sufficient, lipid accumulation could be stimulated by high light intensity, albeit the

349

carbon fixation was inhibited (see the carbon fixation rate in Figure 4b). However, as in line with

350

a prior study,35 lipid accumulation was not favored by nitrogen-starved cells under high-light

351

conditions. Photoinhibition impaired carbon fixation and thus less carbon could be channeled to

352

neutral lipid after nitrogen depletion. Therefore, introducing the combinatorial stresses of high

353

light and low nitrogen may not enhance lipid production as shown in Figure 2. It is also worth

354

noting that the lipid accumulation rates decreased after prolonged nitrogen depletion due to the

355

decrease in carbon fixation rate (Figure 4b), which could further increase the 𝑄𝑛 value (N:C

356

ratio, Figure 3a) and exacerbate the decline of the rate of R7 (Figure 3b). Accordingly, to

357

improve neutral lipid production for photoautotrophic microalgae culture, we could first enhance

358

the photosynthetic carbon fixation before nitrogen depletion, which would help further reduce

359

the 𝑄𝑛value and direct more carbon to neutral lipid after N depletion. Second, we could prevent


18

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360

or slow down the decrease of the photosynthetic carbon fixation rate after nitrogen depletion.

361

Strategies regarding the photosynthetic carbon fixation improvement are discussed in the next

362

section. Moreover, since the maximum neutral lipid accumulation rate occurs after nitrogen

363

depletion, we may improve the lipid production efficiency by manipulating nitrogen availability

364

to reduce the time needed for reaching the maximum lipid accumulation rate. Our model could

365

thus be used to optimize important control process parameters such as light intensities, nitrate

366

concentrations and time durations during the two-stage cultivation.50 Further, it is worth noting

367

that R7 is triggered by nitrate depletion. Thus, adjusting the N:C ratio by enhancing carbon

368

assimilation without nitrogen deprivation would only increase the rate of R6, which is probably

369

not high enough to mimic lipid overproduction under nitrogen starvation.

370 371 372 373 374

Figure 3. Model-predicted dynamic carbon partitioning and neutral lipid accumulation behaviors. (a) Nitrogenquota-regulated carbon partitioning between function pool and storage pool. (b) Extracellular-nitrogen-regulated carbon partitioning from carbohydrates to neutral lipids. (a) and (b) illustrate the carbon flow rates for the moderatelight conditions. (c) Neutral lipid accumulation rates under different light and nitrogen conditions.


19


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375

Photosynthesis and Chl-a Accumulation. Similar to the carbon fixation process,

376

photosynthesis was formulated to account for both light and nitrogen effects in our model. To

377

illustrate the dynamics of the photosynthesis rate in response to light variations and compare

378

with prior studies, we plotted the photosynthesis rate expressed per Chl-a versus instantaneous

379

light intensity (𝐼𝑎𝑣𝑔) for different light conditions in Figure 4a. When light intensities dropped

380

below 10 mol photons m-2d-1, the algae grown under the low-light condition seemed to have a

381

higher photosynthesis rate than the moderate-light condition. However, this enhanced

382

photosynthetic ability to utilize the low-level light did not guarantee a higher total carbon growth

383

rate under the low-light condition as shown in Figure 4b. This result is consistent with the low-

384

light adaptation phenomenon observed in prior work.22 From the modeling perspective, we

385

consider that although the carbon yield on light energy is higher under the low-light condition,

386

the light energy gained by the cells is still low thus leading to a lower total carbon growth rate.

387

When algae were grown under the high-light condition (Figure 4a), initial exposure to the high

388

background light intensity inhibited the cell photosynthesis rate. The photosynthesis rate

389

recovered as the light intensity was attenuated but was still lower than that of the cells grown

390

under the moderate-light condition, which is in line with the hysteresis feature of light inhibition

391

shown in previous studies.51,52 As the pre-exposure to high light intensity decreases the carbon

392

yield on light energy, this inhibitory effect could continue to modulate the photosynthesis rate

393

despite the attenuation of the ambient light. Therefore, a potential way to enhance the

394

photosynthetic carbon fixation rate is to cultivate algae first under a low-light condition to obtain

395

an optimal light utilization efficiency and then increase the light intensity to a moderate-light

396

condition to provide an optimal light energy environment.


20

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397


Nitrogen availability could also limit the carbon growth. This nitrogen-regulated effect is

398

modeled in the photosynthesis by incorporating the nitrogen effect on Chl-a synthesis and

399

degradation. As shown in Figure 4b, photosynthetic carbon fixation rates stopped increasing and

400

then declined after nitrogen depletion. Chl-a degradation under nitrogen depletion impaired the

401

photosynthesis. Therefore, finding ways to block the Chl-a degradation pathway may help

402

prevent further decrease of the photosynthetic carbon fixation. In our model, Chl-a degrades at

403

the maximum rate when the high-light and low-nitrogen stresses occur simultaneously. Future

404

models could modify the Chl-a degradation rate (Eqn.16) as 𝑚𝑐ℎ𝑙 = 𝑚 ∙ max {

405

(1 ― erf (𝑘𝑁 ∙ 𝑆𝑁𝑂3)), erf (𝑘𝐼 ∙ 𝐼𝑎𝑣g_R)} so that Chl-a would degrade at the maximum rate if either

406

the high-light damage or nitrogen starvation occurs. This integrated nitrogen is significant

407

because it partly explains why prior models could not use a single set of parameters to predict

408

both nitrogen-sufficient and nitrogen-limited conditions.13

409 410 411 412 413

Figure 4. Model-predicted dynamic photosynthetic carbon fixation behaviors. (a) Photosynthesis-Light Intensity (PI) relationship under high-nitrogen conditions demonstrates the light adaptation and inhibition effects. (b) Photosynthetic carbon fixation rates under different light and nitrogen conditions further demonstrate the nitrogenlimited effect on carbon growth.

414

Model Validation and Future Work. The proposed model was validated through the

415

predictions of new data (Figure S14-S16). Overall, our model predictions were in good


21


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416

agreement with the experimental data except that the model 1) overestimated the neutral lipid

417

production after prolonged nitrogen depletion for the 300PPFD and 0.43N condition and 2) did

418

not predict the decrease in neutral lipid concentration toward the end of the 300PPFD and 3.80N

419

experiment. It is possible that neutral lipids started to degrade after prolonged nitrogen starvation

420

or under low light intensities as light was attenuated during algae growth. However, our model

421

did not include such a neutral lipid degradation process because the lipid concentration decline

422

was not observed in the calibration experiments. This hypothesis needs further experimental and

423

modeling work to test. Other aspects such as the variations of the initial states may also

424

contribute to the bias in model predictions (see discussion in SI).

425

Future mechanistic studies also need to investigate how light affects the neutral lipid

426

accumulation during nitrogen-sufficient conditions. As indicated by our lipid accumulation

427

process R6, the underlying lipid biosynthetic pathway is likely sensitive to light variations.

428

However, light could also affect the lipid degradation pathway if lipid degradation exists. For

429

instance, as light intensity declines, more lipids would degrade and lead to an increase in the net

430

lipid accumulation under higher-light conditions. This warrants further investigation to clarify

431

whether the light intensity regulates the neutral lipid synthesis pathway or the degradation

432

pathway. Such knowledge would benefit future work including the modeling of light-dark

433

cycling conditions. Future models could also extend our current framework to incorporate other

434

important environmental factors such as temperature and offer mechanistic insights on how cells

435

regulate the different cellular processes in response to environmental variations. With the ability

436

to characterize transient algae behaviors, these models could help optimize the cultivation

437

process to enhance biofuel and biomass production.


22

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438

ASSOCIATED CONTENT

439

Supporting Information. Detailed information regarding light estimation, model calibration

440

and validation as noted in the text.

441

ACKNOWLEDGEMENTS

442

This work was funded by NSF Emerging Frontiers in Research and Innovation Grant No.

443

1332341. The authors would like to thank Zisu Hao for helpful discussions on model

444

formulation, Samiul Haque for assistance in model calibration, and Dr. Ranji Ranjithan, Dr.

445

James Levis, Dr. Amy Grunden and Dr. Heike Sederoff for their critical comments.

446

NOMENCLATURE Parameter

Unit

(a) Fitting parameters 𝑃𝑚 d-1 𝑌𝐸 mol C (mol photons)-1 𝑘𝑦𝑒 m2 d (mol photons)-1 𝑞𝑛 𝑣𝑛𝑚 𝑚 𝛽

mol N (mol C)-1 mol N (mol 𝑋𝑐𝑑-C)-1 d-1 d-1 mol C (mol N)-1

𝑘𝑛𝑙 𝜙

m2 d (mol photons)-1 -

𝑚𝑐𝑁

mol C (mol N)-1

𝑚𝑐 𝑘𝑚𝑐

d-1 m2 d (mol photons)-1

𝜃 𝑘𝑐ℎ𝑙

mol 𝑋𝑐ℎ𝑙-C (mol N)-1 m2 d (mol photons)-1

Description Maximum photosynthesis rate Carbon yield on light energy Coefficient of background light intensity for calculating carbon yield on light energy Minimum nitrogen quota Maximum nitrogen uptake rate Functional biomass degradation rate Coefficient of carbon reuse for lipid accumulation from carbohydrate metabolism Coefficient of light for lipid synthesis Rate portion of carbon redirection from carbohydrate to lipid accumulation Coefficient of carbohydrate respiration for describing the growth cost Carbohydrate maintenance rate Coefficient of background light intensity for calculating carbohydrate maintenance rate Maximum ratio of Chl-a to nitrogen Coefficient of light for Chl-a synthesis


23


𝑘𝐼 𝑘𝑁 𝑄𝑛𝑚

m2 d (mol photons)-1 m3 (mol N)-1 mol N (mol C)-1

(b) State variables 𝑋𝑐𝑑 mol 𝑋𝑐𝑑-C m-3 𝑋𝑐𝑎𝑟𝑏 𝑋𝑛𝑙 𝑋𝑐ℎ𝑙 𝑆𝑁𝑂3 𝑋𝑁

mol 𝑋𝑐𝑎𝑟𝑏-C m-3 mol 𝑋𝑛𝑙-C m-3 mol 𝑋𝑐ℎ𝑙-C m-3 mol N m-3 mol N m-3

(c) Other parameters m-1 𝐴 m2 (mol 𝑋𝑐ℎ𝑙-C)-1 𝑎 m2 (mol C)-1 𝑏 m 𝑑 m 𝑙 𝑚𝑐ℎ𝑙 d-1 𝐼𝑎𝑣𝑔 mol photons m-2 d-1 𝐼𝑎𝑣𝑔_𝑅

mol photons m-2 d-1

𝐼𝑙

mol photons m-2 d-1

𝐼𝑝 𝐼𝑅,𝑙

mol photons m-2 d-1 mol photons m-2 d-1

𝐼𝑠𝑎𝑡 𝐼𝑧𝑙 𝑃 𝑄𝑛 𝑞𝑛𝑙

mol photons m-2 d-1 mol photons m-2 d-1 mol C (mol 𝑋𝑐𝑑-C)-1 d-1 mol N (mol C)-1 -

𝑣𝑁 𝑋𝐶

mol N (mol 𝑋𝑐𝑑-C)-1 d-1 mol C m-3

𝑌𝑐𝐸 𝑧𝑙 𝑧𝑡

mol C (mol photons)-1 m m

447

REFERENCES

448 449

(1)

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Coefficient of background light intensity for Chl-a degradation Nitrogen deficiency coefficient Maximum nitrogen quota Concentration of functional biomass (algal biomass excluding carbohydrates and neutral lipids) as C Concentration of carbohydrates as C Concentration of neutral lipids as C Concentration of Chl-a as C Concentration of nitrate as N in the reactor Concentration of cellular nitrogen as N Light absorbance Optical cross section of Chl-a Optical cross section of biomass except Chl-a content Distance from the light source to the front of the reactor Optical path coordinate 𝑙 inside the reactor Chl-a degradation rate Instantaneous light intensity averaged over the optical path of the reactor Background light intensity averaged over the optical path of the reactor Light intensity at the location 𝑙 along the optical path inside the reactor Light profile monitored by the light sensor at location 𝑙 = 𝑧𝑙 Background light intensity at the location 𝑙 along the optical path inside the reactor Saturated light intensity Estimated light profile at location 𝑙 = 𝑧𝑙 Photosynthesis rate normalized to functional biomass Time-dependent nitrogen quota Rate portion of carbon redirection from carbohydrate to lipid accumulation Nitrogen uptake rate Concentration of total fixed organic carbon (𝑋𝐶 = 𝑋𝑐𝑑 + 𝑋𝑐𝑎𝑟𝑏 + 𝑋𝑛𝑙) Carbon yield on light energy Distance from the light sensor to the front of the reactor Thickness of the photobioreactor along the optical path

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Dynamic Modeling of Microalgae Growth and Lipid Production under

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