The promise of dawn: Microalgae photoacclimation as an optimal control problem of resource allocation.

Photosynthetic microorganisms are known to adjust their photosynthetic capacity according to light intensity. This so-called photoacclimation process is thought to maximize growth at equilibrium, but its dynamics under varying conditions remains less understood. To tackle this problem, microalgae growth and photoacclimation are represented by a (coarse-grained) resource allocation model. Using optimal control theory (the Pontryagin maximum principle) and numerical simulations, we determine the optimal strategy of resource allocation to maximize microalgal growth rate over a time horizon. We show that, after a transient, the optimal trajectory approaches the optimal steady state, a behavior known as the turnpike property. Then, a bi-level optimization problem is solved numerically to estimate model parameters from experimental data. The fitted trajectory represents well a Dunaliella tertiolecta culture facing a light down-shift. Finally, we study photoacclimation dynamics under day/night cycle. In the optimal trajectory, the synthesis of the photosynthetic apparatus surprisingly starts a few hours before dawn. This anticipatory behavior has actually been observed both in the laboratory and in the field. This shows the algal predictive capacity and the interest of our method which predicts this phenomenon.

Twelve quick tips for designing sound dynamical models for bioprocesses.

Because of the inherent complexity of bioprocesses, mathematical models are more and more used for process design, control, optimization, etc. These models are generally based on a set of biochemical reactions. Model equations are then derived from mass balance, coupled with empirical kinetics. Biological models are nonlinear and represent processes, which by essence are dynamic and adaptive. The temptation to embed most of the biology is high, with the risk that calibration would not be significant anymore. The most important task for a modeler is thus to ensure a balance between model complexity and ease of use. Since a model should be tailored to the objectives, which will depend on applications and environment, a universal model representing any possible situation is probably not the best option.

Optimal control of bacterial growth for the maximization of metabolite production.

Microorganisms have evolved complex strategies for controlling the distribution of available resources over cellular functions. Biotechnology aims at interfering with these strategies, so as to optimize the production of metabolites and other compounds of interest, by (re)engineering the underlying regulatory networks of the cell. The resulting reallocation of resources can be described by simple so-called self-replicator models and the maximization of the synthesis of a product of interest formulated as a dynamic optimal control problem. Motivated by recent experimental work, we are specifically interested in the maximization of metabolite production in cases where growth can be switched off through an external control signal. We study various optimal control problems for the corresponding self-replicator models by means of a combination of analytical and computational techniques. We show that the optimal solutions for biomass maximization and product maximization are very similar in the case of unlimited nutrient supply, but diverge when nutrients are limited. Moreover, external growth control overrides natural feedback growth control and leads to an optimal scheme consisting of a first phase of growth maximization followed by a second phase of product maximization. This two-phase scheme agrees with strategies that have been proposed in metabolic engineering. More generally, our work shows the potential of optimal control theory for better understanding and improving biotechnological production processes.

Theory of turbid microalgae cultures.

Microalgae can be cultivated in closed or open photobioreactors (PBR). In these systems, light rapidly decreases as it passes through the culture due to the turbidity of the medium. Thus, microalgae experiment different light intensities depending on their position in the medium. In this paper, we study theoretically how the growth rate of microalgae is affected by different factors; incident light intensity, form of the PBR, microalgae population density, turbidity of non-microalgae components, and light path-length of the reactor. We show that for different types of PBR the average growth rate is completely determined by the incident light intensity and the optical depth. In the case of vertical cylindrical PBRs illuminated from above (e.g. race-way or panel-type reactors), we described (and we prove under general assumptions) in details the dependence of the AGR on the aforementioned factors. Finally, we discuss some implications of our analysis; the occurrence of the Allee effect, if light ostensibly limits or inhibits the growth rate in outdoor cultures, and how the geometry of the PBR affects microalgae growth rate and productivity.

Dynamical Allocation of Cellular Resources as an Optimal Control Problem: Novel Insights into Microbial Growth Strategies.

Microbial physiology exhibits growth laws that relate the macromolecular composition of the cell to the growth rate. Recent work has shown that these empirical regularities can be derived from coarse-grained models of resource allocation. While these studies focus on steady-state growth, such conditions are rarely found in natural habitats, where microorganisms are continually challenged by environmental fluctuations. The aim of this paper is to extend the study of microbial growth strategies to dynamical environments, using a self-replicator model. We formulate dynamical growth maximization as an optimal control problem that can be solved using Pontryagin's Maximum Principle. We compare this theoretical gold standard with different possible implementations of growth control in bacterial cells. We find that simple control strategies enabling growth-rate maximization at steady state are suboptimal for transitions from one growth regime to another, for example when shifting bacterial cells to a medium supporting a higher growth rate. A near-optimal control strategy in dynamical conditions is shown to require information on several, rather than a single physiological variable. Interestingly, this strategy has structural analogies with the regulation of ribosomal protein synthesis by ppGpp in the enterobacterium Escherichia coli. It involves sensing a mismatch between precursor and ribosome concentrations, as well as the adjustment of ribosome synthesis in a switch-like manner. Our results show how the capability of regulatory systems to integrate information about several physiological variables is critical for optimizing growth in a changing environment.

Estimation of neutral lipid and carbohydrate quotas in microalgae using adaptive interval observers.

Under stress conditions, microalgae are known to accumulate large amounts of neutral lipids and carbohydrates, which can be used for biofuel production. However, on-line measurement of microalgal biochemical composition is a difficult task which makes the microalgal process rather difficult to manage. In this paper, we propose a so called adaptive interval observer for the on-line estimation of neutral lipid and carbohydrate quotas in microalgae. The observer is based on a change of coordinates that involves a time-varying gain. We introduce dynamics for the gain, whose trajectory converges toward a predefined optimal value (which maximizes the convergence rate of the observer). The observer performance is illustrated with experimental data of Isochrysis sp. cultures under nitrogen limitations and day-night cycle. The proposed observer design appears to be a suitable robust estimation technique.

Minimal time control of fed-batch bioreactor with product inhibition.

This paper is devoted to the minimal time control problem for fed-batch bioreactors, in presence of an inhibitory product, which is released by the biomass proportionally to its growth. We first consider a growth rate with substrate saturation and product inhibition, and we prove that the optimal strategy is fill and wait (bang-bang). We then investigate the case of the Jin growth rate which takes into account substrate and product inhibition. For this type of growth function, we can prove the existence of singular arc paths defining singular strategies. Several configurations are addressed depending on the parameter set. For each case, we provide an optimal feedback control of the problem (of type bang-bang or bang-singular-bang). These results are obtained gathering the initial system into a planar one by using conservation laws. Thanks to Pontryagin maximum principle, Green's theorem, and properties of the switching function, we obtain the optimal synthesis. A methodology is also proposed in order to implement the optimal feeding strategies.

Three-reaction model for the anaerobic digestion of microalgae.

Coupling an anaerobic digester to a microalgal culture has received increasing attention as an alternative process for combined bioenergy production and depollution. In this article, a dynamic model for anaerobic digestion of microalgae is developed with the aim of improving the management of such a coupled system. This model describes the dynamics of inorganic nitrogen and volatile fatty acids since both can lead to inhibition and therefore process instability. Three reactions are considered: Two hydrolysis-acidogenesis steps in parallel for sugars/lipids and for proteins, followed by a methanogenesis step. The proposed model accurately reproduces experimental data for anaerobic digestion of the freshwater microalgae Chlorella vulgaris with an organic loading rate of 1 gCOD L(-1) d(-1). In particular, the three-reaction pathway allows to adequately represent the observed decoupling between biogas production and nitrogen release. The reduced complexity of this model makes it suitable for developing advanced, model-based control and monitoring strategies.

The paradoxes hidden behind the Droop model highlighted by a metabolic approach.

We propose metabolic models for the haptophyte microalgae Tisochrysis lutea with different possible organic carbon excretion mechanisms. These models-based on the DRUM (Dynamic Reduction of Unbalanced Metabolism) methodology-are calibrated with an experiment of nitrogen starvation under day/night cycles, and then validated with nitrogen-limited chemostat culture under continuous light. We show that models including exopolysaccharide excretion offer a better prediction capability. It also gives an alternative mechanistic interpretation to the Droop model for nitrogen limitation, which can be understood as an accumulation of carbon storage during nitrogen stress, rather than the common belief of a nitrogen pool driving growth. Excretion of organic carbon limits its accumulation, which leads to a maximal C/N ratio (corresponding to the minimum Droop N/C quota). Although others phenomena-including metabolic regulations and dissipation of energy-are possibly at stake, excretion appears as a key component in our metabolic model, that we propose to include in the Droop model.

Sharing Vitamin B12 between Bacteria and Microalgae Does Not Systematically Occur: Case Study of the Haptophyte Tisochrysis lutea.

Haptophyte microalgae are key contributors to microbial communities in many environments. It has been proposed recently that members of this group would be virtually all dependent on vitamin B12 (cobalamin), an enzymatic cofactor produced only by some bacteria and archaea. Here, we examined the processes of vitamin B12 acquisition by haptophytes. We tested whether co-cultivating the model species Tisochrysis lutea with B12-producing bacteria in vitamin-deprived conditions would allow the microalga to overcome B12 deprivation. While T. lutea can grow by scavenging vitamin B12 from bacterial extracts, co-culture experiments showed that the algae did not receive B12 from its associated bacteria, despite bacteria/algae ratios supposedly being sufficient to allow enough vitamin production. Since other studies reported mutualistic algae-bacteria interactions for cobalamin, these results question the specificity of such associations. Finally, cultivating T. lutea with a complex bacterial consortium in the absence of the vitamin partially rescued its growth, highlighting the importance of microbial interactions and diversity. This work suggests that direct sharing of vitamin B12 is specific to each species pair and that algae in complex natural communities can acquire it indirectly by other mechanisms (e.g., after bacterial lysis).

Optimal proteome allocation and the temperature dependence of microbial growth laws.

Although the effect of temperature on microbial growth has been widely studied, the role of proteome allocation in bringing about temperature-induced changes remains elusive. To tackle this problem, we propose a coarse-grained model of microbial growth, including the processes of temperature-sensitive protein unfolding and chaperone-assisted (re)folding. We determine the proteome sector allocation that maximizes balanced growth rate as a function of nutrient limitation and temperature. Calibrated with quantitative proteomic data for Escherichia coli, the model allows us to clarify general principles of temperature-dependent proteome allocation and formulate generalized growth laws. The same activation energy for metabolic enzymes and ribosomes leads to an Arrhenius increase in growth rate at constant proteome composition over a large range of temperatures, whereas at extreme temperatures resources are diverted away from growth to chaperone-mediated stress responses. Our approach points at risks and possible remedies for the use of ribosome content to characterize complex ecosystems with temperature variation.

A new approach to define optimized range of medium composition for enhancement of hairy root production in fed-batch process.

This work proposes a new methodology to identify the best medium concentrations for fed-batch production of hairy root using Datura innoxia as a model. Firstly, the role of each component on the growth rate is investigated separately. Then, an experimental design allows refining the optimization studying the interactions between the major species. The result analysis let to define concentration range optimized for fed-batch process. The work novelties lie in two aspects. Firstly, concentrations have been kept constant during each run. Thus, biomass uptakes do not affect the optimization and the growth rate is maintained constant during the exponential phase. Secondly, the effects of salts are generally studied. In this work, the influences of each ion are investigated in order to avoid bias due to the counter-ion effects. Compared to the classical B(5) medium, the optimized medium shows a significant improvement leading to more than 80% increase of final biomass production.

How haptophytes microalgae mitigate vitamin B12 limitation.

Vitamin B12 (cobalamin) can control phytoplankton development and community composition, with around half of microalgal species requiring this vitamin for growth. B12 dependency is determined by the absence of cobalamin-independent methionine synthase and is unrelated across lineages. Despite their important role in carbon and sulphur biogeochemistry, little is known about haptophytes utilization of vitamin B12 and their ability to cope with its limitation. Here we report the first evaluation of B12 auxotrophy among this lineage based on molecular data of 19 species from 9 families. We assume that all species encode only a B12-dependent methionine synthase, suggesting ubiquitous B12 auxotrophy in this phylum. We further address the effect of different B12 limitations on the molecular physiology of the model haptophyte Tisochrysis lutea. By coupling growth assays in batch and chemostat to cobalamin quantification and expression analyses, we propose that haptophytes use three strategies to cope with B12 limitation. Haptophytes may assimilate dissolved methionine, finely regulate genes involved in methionine cycle and B12 transport and/or limit B12 transport to the mitochondrion. Taken together, these results provide better understanding of B12 metabolism in haptophytes and represent valuable data for deciphering how B12-producing bacteria shape the structure and dynamics of this important phytoplankton community.

A biomolecular proportional integral controller based on feedback regulations of protein level and activity.

Homeostasis is the capacity of living organisms to keep internal conditions regulated at a constant level, despite environmental fluctuations. Integral feedback control is known to play a key role in this behaviour. Here, I show that a feedback system involving transcriptional and post-translational regulations of the same executor protein acts as a proportional integral (PI) controller, leading to enhanced transient performances in comparison with a classical integral loop. Such a biomolecular controller-which I call a level and activity-PI controller (LA-PI)-is involved in the regulation of ammonium uptake by Escherichia coli through the transporter AmtB. The P II molecules, which reflect the nitrogen status of the cell, inhibit both the production of AmtB and its activity (via the NtrB-NtrC system and the formation of a complex with GlnK, respectively). Other examples of LA-PI controller include copper and zinc transporters, and the redox regulation in photosynthesis. This scheme has thus emerged through evolution in many biological systems, surely because of the benefits it offers in terms of performances (rapid and perfect adaptation) and economy (protein production according to needs).

How do microalgae perceive light in a high-rate pond? Towards more realistic Lagrangian experiments.

Hydrodynamics in a high-rate production reactor for microalgae cultivation affects the light history perceived by cells. The interplay between cell movement and medium turbidity leads to a complex light pattern, whose forcing effects on photosynthesis and photoacclimation dynamics are non-trivial. Hydrodynamics of high density algal ponds mixed by a paddle wheel has been studied recently, although the focus has never been on describing its impact on photosynthetic growth efficiency. In this multidisciplinary downscaling study, we first reconstructed single cell trajectories in an open raceway using an original hydrodynamical model offering a powerful discretization of the Navier-Stokes equations tailored to systems with free surfaces. The trajectory of a particular cell was selected and the associated high-frequency light pattern was computed. This light pattern was then experimentally reproduced in an Arduino-driven computer controlled cultivation system with a low density Dunaliella salina culture. The effect on growth and pigment content was recorded for various frequencies of the light pattern, by setting different paddle wheel velocities. Results show that the frequency of this realistic signal plays a decisive role in the dynamics of photosynthesis, thus revealing an unexpected photosynthetic response compared to that recorded under the on/off signals usually used in the literature. Indeed, the light received by a single cell contains signals from low to high frequencies that nonlinearly interact with the photosynthesis process and differentially stimulate the various time scales associated with photoacclimation and energy dissipation. This study highlights the need for experiments with more realistic light stimuli to better understand microalgal growth at high cell densities. An experimental protocol is also proposed, with simple, yet more realistic, step functions for light fluctuations.

Modelling of Microalgae Culture Systems with Applications to Control and Optimization.

Mathematical modeling is becoming ever more important to assess the potential, guide the design, and enable the efficient operation and control of industrial-scale microalgae culture systems (MCS). The development of overall, inherently multiphysics, models involves coupling separate submodels of (i) the intrinsic biological properties, including growth, decay, and biosynthesis as well as the effect of light and temperature on these processes, and (ii) the physical properties, such as the hydrodynamics, light attenuation, and temperature in the culture medium. When considering high-density microalgae culture, in particular, the coupling between biology and physics becomes critical. This chapter reviews existing models, with a particular focus on the Droop model, which is a precursor model, and it highlights the structure common to many microalgae growth models. It summarizes the main developments and difficulties towards multiphysics models of MCS as well as applications of these models for monitoring, control, and optimization purposes.

Optimizing microalgal production in raceway systems.

The industrial exploitation of microalgae is characterized by the production of high-value compounds. Optimization of the performance of microalgae culture systems is essential to render the process economically viable. For raceway systems, the optimization based on optimal control theory is rather challenging, because the process is by essence periodically forced and, as a consequence, optimization must be carried out in a periodic framework. In this article, we propose a simple operational criterion for raceway systems that when integrated in a strategy of closed-loop control allows attaining biomass productivities very near to the theoretical maximal productivities. The strategy developed was tested numerically using a mathematical model of microalgae growth in raceways. The model takes into account the temporal variation of the environmental variables temperature and light intensity and their influence on microalgae growth.

Modelling neutral lipid production by the microalga Isochrysis aff. galbana under nitrogen limitation.

This article proposes a dynamical model of microalgal lipid production under nitrogen limitation. In this model, intracellular carbon is divided between a functional pool and two storage pools (sugars and neutral lipids). The various intracellular carbon flows between these pools lead to a complex dynamic with a strong discrepancy between synthesis and mobilization of neutral lipids. The model has been validated with experiments of Isochrysis aff. galbana (clone T-iso) culture under various nitrogen limitation conditions and under nitrogen starvation. The hysteresis behavior of the neutral lipid quota observed experimentally is accurately predicted.

Modeling anaerobic digestion of microalgae using ADM1.

The coupling between a microalgal pond and an anaerobic digester is a promising alternative for sustainable energy production by transforming carbon dioxide into methane using solar energy. In this paper, we demonstrate the ability of the original ADM1 model and a modified version (based on Contois kinetics for the hydrolysis steps) to represent microalgae anaerobic digestion. Simulations were compared to experimental data of an anaerobic digester fed with Chlorella vulgaris. The modified ADM1 fits adequately the data for the considered 140 day experiment encompassing a variety of influent load and flow rates. It turns out to be a reliable predictive tool for optimising the coupling of microalgae with anaerobic digestion processes.