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La technique de culture continue: théorie et applications
... In response to these challenges, mathematical models have been developed to predict cellular behaviour and maximise the productivity of target metabolites [2,4]. These models range from simple methodologies, such as the Monod kinetic mass balance [5], to more sophisticated approaches. One example is metabolic flux analysis (FBA), which quantitatively assesses cellular metabolic networks [6]. ...
... Despite these limitations, the model allowed us to adjust parameters and develop optimal strategies to maximise the production of the desired metabolite. Figure 2C presents a comparison between the multiscale model and an unstructured model that applies Monod kinetics [5] (see "Methods" section). We used the objective function F, specified in Eq. 22, to fit the biomass and substrate concentrations for the four Escherichia coli strains previously examined. ...
... where the specific growth rate is given by = max S K s +S , with max as the maximum specific growth rate and K s as the half-velocity constant [5]. ...
Background
Fermentation processes are essential for the production of small molecules, heterologous proteins and other commercially important products. Traditional bioprocess optimisation relies on phenomenological models that focus on macroscale variables like biomass growth and protein yield. However, these models often fail to consider the crucial link between macroscale dynamics and the intracellular activities that drive metabolism and proteins synthesis.
Results
We introduce a multiscale model that not only captures batch bioreactor dynamics but also incorporates a coarse-grained approach to key intracellular processes such as gene expression, ribosome allocation and growth. Our model accurately fits biomass and substrate data across various Escherichia coli strains, effectively predicts acetate dynamics and evaluates the expression of heterologous proteins. By integrating construct-specific parameters like promoter strength and ribosomal binding sites, our model reveals several key interdependencies between gene expression parameters and outputs such as protein yield and acetate secretion.
Conclusions
This study presents a computational model that, with proper parameterisation, allows for the in silico analysis of genetic constructs. The focus is on combinations of ribosomal binding site (RBS) strength and promoters, assessing their impact on production. In this way, the ability to predict bioreactor dynamics from these genetic constructs can pave the way for more efficient design and optimisation of microbial fermentation processes, enhancing the production of valuable bioproducts.
... The concept of "continuous culture" originated in the late 1940s and was widely used and fully developed in the 1950s. The basic design and theory of continuous culture were originally described independently by [1,2]. They called this device as "chemostat". ...
... In industry, chemostats can be used to simulate the decomposition of biological wastes, or to evolve water with microorganisms, etc. Chemostat model has been widely studied by many scholars, because it can not only be used to cultivate microorganisms in laboratory, but also plays a very key role in our daily life. There are many research results on chemostats, which can be viewed in [1,[3][4][5][6][7]. ...
... In chemostat, a single microbial model was first proposed by [1]. Moreover, in [3], Smith and Waltman described a deterministic chemostat model with Monod-type functional re-Y. ...
Stochastic ultimate boundedness has always been a very important property, which plays an important role in the study of stochastic models. Thus, in this paper, we will study a stochastic periodic chemostat system, in which we assume that the nutrient input concentration and noise intensities are periodic. In order to make the stochastic periodic model have mathematical and biological significance, we will study a very important issue: the existence, uniqueness and ultimate boundedness of a global positive solution for a stochastic periodic chemostat system.
... The dynamical model of the CSTR is derived from the mass balance law [8], resulting in a coupled system of Ordinary Differential Equations (ODE). One of the most critical components of the ODEs of the CSTR is the models that describe the microbial specific growth rates, of which the Monod [16] and Haldane [17] types are commonly used in the literature. The first is used to study the CSTR behaviour in a fixed environment where the growth of microorganisms depends only on the limited substrate. ...
... • The first one is the Monod type [16] given by the following expression: ...
... The idea of SINDy-PI is to circumvent the null space of the Implicit-SINDy and create a robust identification framework of implicit dynamics. The concept is that if there is at least one term, represented by ω j (x,ẋ) ∈ Ω(x,ẋ), in the dynamics (15) that is known, then it is feasible to rephrase (16) as ω j (X,Ẋ) = Ω(X,Ẋ|ω j (X,Ẋ))σ j (17) where the term Ω(X,Ẋ|ω j (X,Ẋ)) is the same as the library Ω(X,Ẋ) excluding the ω j column. Specifically, the sparse coefficient vector σ j minimises the following: ...
This paper presents a data-driven approach for identifying the governing equations or ODEs of a continuously stirred tank reactor (CSTR) system. The paper employs the sparse identification of nonlinear dynamics (SINDy) algorithm, a popular and versatile method used for discovering nonlinear dynamical system models from data. The SINDy-PI (parallel, implicit) framework, a robust variation of the SINDy method, is used to find the implicit dynamics and rational nonlinearities of the CSTR for both Monod and Haldane types of specific growth rates. The simulation results demonstrate the accuracy of the method in inferring the ODEs of the CSTRs using limited and noisy data. The proposed method can be used to improve our understanding of complex systems and inform the design of control strategies.
... The chemostat is a laboratory apparatus invented simultaneously by J. Monod and by A. Novick and L. Sziland in the 1950s, in which organisms grow in a controlled way. 1,2 This apparatus consists of an enclosure, containing the reaction volume, connected by input to feed the system with resources, and outlet through which microorganisms are removed. The chemostat model describes several ecological interactions (commensalism, syntrophy, competition, mutualism, etc.) and can capture complex relationships, such as the combination of mutualism and competition. ...
... In this work, we perform the mathematical analysis of model (1), which was proposed in Vet et al 5 and studied only numerically. The model considers the phenomenon of a simultaneous competitive and mutualistic relationship between two microbial species in a chemostat. ...
... In the case 1 2 > 1, the bifurcation diagrams and the operating diagrams show that the system exhibits bistability, once between the washout steady-state E 0 and the steady-state of coexistence E * 1 and once between the extinction of the second species steady-state E 1 and the positive steady-state E * 1 . By our study, we generalize Vet et al 5 by allowing a larger class of growth functions and by giving rigorous proofs for the results on the existence of steady states obtained numerically in Vet et al 5 for system (1). The numerical study of Vet et al 5 corresponds to the case 1 2 > 1, which is equivalent to v 11 v 22 > v 12 v 21 and means that the production of cross-fed nutrients is higher than the consumption. ...
We perform the mathematical analysis of a model describing the interaction of two species in a chemostat, involving competition and mutualism, simultaneously. The model is a five‐dimensional system of differential equations with nonlinear growth functions. We give a comprehensive description of the dynamics of the system by determining analytically the existence and the local stability conditions of all steady‐states, considering a large class of growth rates. We prove that there exists a unique stable coexistence steady‐state and give the conditions under which bistability can occur. We give bifurcation diagrams and operating diagrams showing the rich behavior of the system.
... The chemostat is a basic laboratory instrument for the continuous culture of microorganisms. In the chemostat, the flow rate keeps the medium constant and is easy to be controlled, which is its advantage [1]- [2]. A chemostat consists of three parts: the nutrient container, the culture container and the collection container. ...
... Considering the roles of different response functions in different systems, different response functions are usually considered to express their specific response requirements. Generally, the response function g(·) takes the following forms: (1) The linear response functional [9][10][11][12][13]: g(·) = S. (2) lvlev [9]: g(·) = 1 − e λS where λ > 0 is a constant. (3) [9]: g(·) = arctanS. ...
... (7) The Beddington-DeAngelis functional response [18,19]: g(·) = mS 1+m 1 S+m 2 x where m > 0, m 1 > 0 and m 2 > 0 are constants. Clearly, all of these are nonlinear except for (1), and all but (6) are monotonous.g( ) can take similar response functionals. ...
This paper aims to study the long-time dynamic of a stochastic microorganism flocculation model with Monod response functionals. By introducing the flocculant into the classical chemostat, the studied model is extended to a stochastic one with nonlinear reaction terms. For the corresponding deterministic model, several sufficient conditions are given for the stability of equilibrium points. Considering the influence of noises, a unique threshold for determining persistence or not of the microorganism is derived firstly; and then influences of the flocculation on the output of chemostat is investigated. Results show that the input concentration of flocculants has significant influences on the dynamic and output of the chemostat. Finally, theoretical conclusions are illustrated by numerical method for wastewater treatment.
... The chemostat, which is a simple laboratory apparatus used for the continuous culture of microorganisms, was introduced by Novick and Szilard [1], and Monod [2]. For details on mathematical analysis of models of growth and competition in the chemostat, the reader is referred to [3][4][5]. ...
... with λ s defined by (2). ...
This paper investigates the dynamics of a chemostat model incorporating two populations of one bacterial species: susceptible and virus-infected. Through the two operating parameters of the model, represented by the input concentration of the nutrient and the dilution rate of the chemostat, we analyze the existence and stability conditions of all possible equilibria, and then describe the operating diagram of the model, which is the bifurcation diagram giving its behavior with respect to those operating parameters, that visually depicts the various regions of stability of those equilibria. This diagram gives a better understanding of the complex interplay between bacterial populations growth, viral infection and environmental factors, in a controlled environment.
... To allow comparisons of the study sites in terms of species richness, it is necessary to develop a relation between the contiguous plot area and the number of species in each plot. A number of models have been proposed to describe the species-area relation (de Caprariis et al. 1976;Gitay et al. 1991;Monod 1950;Tjørve 2003;Williams 1995). Asymptotic functions are appropriate in very large plots where all species are likely to be captured by the samples. ...
... The power function is more suitable for small plot sizes where the maximum number of species is unknown. The hitherto almost unknown Monod function (Monod 1950), which represents a flexible compromise that includes the advantages of both, the power function and the asymptotic functions, will be used in our study to estimate the species-area relation: ...
Tropical dry forests (TDF) support the livelihoods of millions of people worldwide, but in contrast with the humid tropical forests, knowledge of their structure and biology is limited. This study aims to fill that gap using observations from the South Indian Deccan Plateau. Based on large, tree-mapped field plots, within or near a densely populated metropolitan area, we present details of 130 woody plant species, including a large number of climbers. The modelling approach includes a new function for developing species–area relations (SAR’s). In addition to the greater flexibility of the function, when compared with traditional power and asymptotic functions, the Monod function not only provides greater flexibility, but also allows reasonable estimates of SAR's if the overall regional species richness is known. This is an important advantage when compared to the standard methods. Another new finding concerns the species abundance distribution (SAD) which explains processes of community assembly and species turnover. The SAD follows the Weibull model which is a significant improvement compared with the traditional use of the Lognormal model because the Weibull parameters seem to be related to species richness. We also present a new cell-based (in addition to the individual neighbourhood-based) approach for analysing structure. The cell-based analysis combines small-grain measures of density and crowding, richness and size variation and can be used to assess the degree of similarity or dissimilarity among forest stands, or between a current and some ideal target structure. The methods of analysis and the proposed framework for pro-active conservation presented in this study may be helpful in regions of the world where complex multi-species forests require advanced methods for sustaining their resilience and functions.
... Le chémostat a été inventé en 1950 concomitamment par le biologiste Monod [43] qui l'appela dans un premier temps bactogène, ainsi que le physicien Novick & le chimiste Szilard [47] qui ont opté par la suite pour l'appellation chémostat. C'est un bioréacteur à alimentation continue conçu sous forme d'un réservoir dont le contenu est un milieu homogène où la température, le potentiel d'hydrogène et le volume total sont constants. ...
... Enfin, en 1942, elle a été utilisée par Monod dans le cas de la croissance de micro-organismes, voir [42]. Pour de plus amples détails, voir aussi [3,37,43]. ...
Cette thèse se situe dans le domaine de la modélisation des systèmes dynamiques et traite des modèles spatialisés, décrivant des configurationsde deux bioréacteurs continûment alimentés et reliés en série que nous retrouvons dans les laboratoires de microbiologie. Les spécialistes en géniedes bio-procédés portent un grand intérêt à ce genre de bioréacteurs appelés depuis les années 1950 chémostats, car ils donnent des résultats trèssatisfaisants notamment dans le secteur de l’écologie microbienne.La théorie du chémostat, considérée comme une théorie mathématique soulève différentes questions auxquelles appartient la problématique decette thèse. Dans ce travail, nous nous intéressons au modèle de deux chémostats interconnectés en série. Tout d’abord, nous faisons une étude mathématique approfondie qui aboutit à une analyse des performances de ce dispositif en série, dans le but de les comparer aux performances d’un seul chémostat de même volume total.L’objectif de cette thèse et la question fondamentale à laquelle nous apportons une réponse est: "Dans quelles circonstances, le fait de spatialiser, c’est-à-dire de considérer deux chémostats reliés en série, donne un meilleur résultat en termes de performances qu’un seul chémostat ?" En effet, tout au long des différents chapitres de cette thèse, nous comparons, à l’état d’équilibre, les performances du dispositif en série à celles d’un seul chémostat en se basant sur trois critères de performances différents: la concentration de substrat en sortie, le débit de biogaz et la productivité de la biomasse. L’originalité de ce travail se situe dans la généricité des fonctions de croissances. En effet, nous considérons toujours une fonction de croissance positive et monotone, ou positive et non-monotone. Chaque chapitre est caractérisé par une particularité dans le modèle mathématique correspondant au dispositif des deux chémostats en série. Après avoir consacré un chapitre à une introduction générale et un second chapitre à des rappels sur un seul chémostat, nous retrouvons dans le troisième chapitre le modèle classique des deux chémostats interconnectés en série où la fonction de croissance est strictement monotone. Ce modèle peut être considéré comme le modèle fondamental sur lequel est basée le travail de cette thèse. Les résultats de ce troisième chapitre étendent certains résultats de la littérature obtenus pour des fonctions de croissances linéaires et de type Monod. Ces résultats donnent des conditions nécessaires et suffisantes sur l’existence d’une configuration de deux chémostats en série plus efficace qu’un seul chémostat sans mortalité, selon les trois différents critères de performances. Ensuite, étendu sur deux chapitres consécutifs, nous considérons le modèle fondamental où nous prenons en compte la mortalité de la biomasse de l’espèce ce qui a été encorefort peu étudié de façon approfondie dans la littérature. L’ajout du paramètre de mortalité induit à de surprenants résultats. Les résultats obtenuspour la concentration de substrat en sortie dans le cas sans mortalité, sont étendus au cas avec mortalité. Certains résultats obtenus pour le débit de biogaz différent du cas sans mortalité et se sont révélés non-intuitifs aux praticiens du génie des bio-procédés. Les résultats obtenus pour la productivité de la biomasse dans le cas sans mortalité ont mérité une analyse plus approfondie lorsque la mortalité est non nulle et de nouveaux phénomènes inexistants dans le cas sans mortalité sont apparus. Ainsi, lorsque la mortalité est prise en considération, le cinquième chapitre est exclusivement dédié à l’étude de la productivité de la biomasse. Dans un sixième chapitre encore en cours d’étude, nous traitons le même modèle initial mais avec une concentration de substrat en entrée répartie sur les deux réacteurs en série. Enfin, dans un septième chapitre aussi encore en cours d’étude, nous considérons le modèle fondamental avec une fonction de croissance non-monotone. De premiers résultats sont fournis pour ces deux derniers chapitres et sont en cours d’évolution. Finalement, pour chaque modèle étudié, nous donnons des conditions dépendant des paramètres biologiques et des paramètres opératoires qui permettent de savoir précisément, selon le critère de performances choisi, quelle structure entre le dispositif en série et un seul chémostat, donne le meilleur résultat recherché par le praticien.
... In the 1950s, Jacques Monod [1], Aaron Novick and Leo Szilard [2] almost simultaneously invented the chemostat used to the continuous culture of microorganisms. A chemostat consists of three vessels: the nutrient medium, the reaction medium and the receiver medium [3]. ...
... Suppose that when S(t) = λ, dS(t) dt = 0. Then there exists a time T 1 > 0 such that either case (1): S(t) ≥ λ for all t > T 1 or case (2): S(t) ≤ λ for all t > T 1 . For case(1), if the assertion holds, then dxi(t) dt ≥ 0, i = 1, 2, · · · , n. According to S(t) + n i=1 xi(t) yi = S 0 , which implies that for any microorganism x i (t) always is ...
In order to investigate the effect of telegraph noise on coexistence of multi-microorganisms, we propose a multi-species chemostat model with Markov switchings. First, we investigate the basic dynamic properties of the model and its subsystem. Then, we establish sufficient criteria for the coexistence of the model. Finally, the coexistence of multi-microorganisms with the same break-even concentrations for every state is verified, which expands the theory of coexistence for the corresponding deterministic model. The results indicate that telegraph noise is conducive to the coexistence of multi-microorganisms.
... The chemostat is a simple laboratory apparatus used for the continuous culture of microorganisms. It was introduced simultaneously by Novick and Szilard [29], and Monod [22]. For details and complements on continuous culture, the reader is referred to [17,23]. ...
We study a mathematical model of bacterial growth on a single limiting nutrient in a chemostat where a virus is present.
The assumption is that the virus can infect the population, resulting in the emergence of two distinct populations: the susceptible and the infected, which are in competition. The model has the structure of an SIS epidemic model. We assume that the growth functions are general and not just linear as in previous works in the literature. We analyze the stability of both disease-free and endemic equilibria. The model can exhibit a multiplicity of endemic equilibria, as well as the appearance of periodic orbits by supercritical or subcritical Hopf bifurcations. Bistability between several equilibrium states or limit cycles is also possible. We present an explicit expression for the basic reproduction number of the epidemic in terms of biologically significant parameters. To better understand the richness of the model's behavior, a few bifurcation diagrams with respect to input nutrient concentration are examined.
... These findings are relevant for understanding how nutrient co-limitation influences the efficiency of biomass formation in a given habitat 13 . While microbial utilization of multiple nutrients has been extensively studiedleading to significant findings such as the diauxic shift [21][22][23] and the complex interplay of factors in multi-nutrient environments 24-26 -these studies often fail to differentiate between different nutrient types and tend to focus on specific growth phases. Consequently, they may not adequately capture biomass gain across the entire growth curve. ...
Microorganisms primarily utilize nutrients to generate biomass and replicate. When a single nutrient source is available, the produced biomass typically increases linearly with the initial amount of that nutrient. This linear trend can be accurately predicted by “black box models”, which conceptualize growth as a single chemical reaction, treating nutrients as substrates and biomass as a product. However, natural environments usually present multiple nutrient sources, prompting us to extend the black box framework to incorporate catabolism, anabolism, and biosynthesis of biomass precursors. This modification allows for the quantification of co-utilization effects among multiple nutrients on microbial biomass production. The extended model differentiates between different types of nutrients: non-degradable nutrients, which can only serve as a biomass precursor, and degradable nutrients, which can also be used as an energy source. We experimentally demonstrated using Escherichia coli that, in contrast to initial model predictions, different nutrients affect each other’s utilization in a mutually dependent manner; i.e., for some combinations, the produced biomass was no longer proportional to the initial amounts of nutrients present. To account for these mutual effects within a black box framework, we phenomenologically introduced an interaction between the metabolic processes involved in utilizing the nutrient sources. This phenomenological model qualitatively captures the experimental observations and, unexpectedly, predicts that the total produced biomass is influenced not only by the combination of nutrient sources but also by their relative initial amounts – a prediction we subsequently validated experimentally. Moreover, the model identifies which metabolic processes – catabolism, anabolism, or precursor biosynthesis—is affected in each specific nutrient combination, offering insights into microbial metabolic coordination.
... The study of interactions within microbial ecological communities is one of the most important issues in microbial ecology [1][2][3][4][5]. An interaction has an impact (neutral, beneficial, or detrimental) on the partner microorganisms involved. ...
The aim of this paper is to show that Tilman's graphical method for the study of competition between two species for two resources can be advantageously used for the study of commensalism or syntrophy models, where a first species produces the substrate necessary for the growth of the second species. The growth functions of the species considered are general and include both inhibition by the other substrate and inhibition by the species' limiting substrate, when it is at a high concentration. Because of their importance in microbial ecology, models of commensalism and syntrophy, with or without self-inhibition, have been the subject of numerous studies in the literature. We obtain a unified presentation of a large number of these results from the literature. The mathematical model considered is a differential system in four dimensions. We give a new result of local stability of the positive equilibrium, which has only been obtained in the literature in the case where the removal rates of the species are identical to the dilution rate and the study of stability can be reduced to that of a system in two dimensions. We describe the operating diagram of the system: this is the bifurcation diagram which gives the asymptotic behavior of the system when the operating parameters are varied, i.e., the dilution rate and the substrate inlet concentrations.
... There are several laws that govern the precise growth rate (S) . The Monod legislation is the most popular (Monod 1950) (1) ...
In this article, we take a look at an Ordinary Differential Equation model that describes the bacteria’s role in anaerobic biodegradation dynamics of domestic garbage in a landfill. A nonlinear Ordinary Differential Equation system is used to describe biological activities. In the current study, the Levenberg–Marquardt Backpropagation Neural Network is used to locate alternate solutions for the model. The Runge–Kutta order four (RK-4) method is employed to produce reference solutions. Different scenarios were looked at to analyse our surrogate solution models. The reliability to verify the equilibrium of the mathematical model, physical quantities such as the half-saturation constant (), the maximum growth rate (), and the inhibition constant (), can be modified. We categorise our potential solutions into training, validation and testing groups in order to assess how well our machine learning strategy works. The advantages of the Levenberg-Marquardt Backpropagation Neural Network scheme have been shown by studies that compare statistical data based on Mean Square Error Function, efficacy, regression plots, and error histograms. From the whole process we conclude that Levenberg–Marquardt Backpropagation Neural Network is accurate and authentic.
... In a controlled continuous microbial cultivation system, such as a chemostat [45,46], it is possible to grow microbial cultures at a steady state with pre-defined growth rates. In chemostat cultures, the dilution rate D is an experimental control parameter and, when the culture has reached a stationary state, the specific growth rate is forced to be equal to the dilution rate. ...
The biotechnological exploitation of microorganisms enables the use of metabolism for the production of economically valuable substances, such as drugs or food. It is, thus, unsurprising that the investigation of microbial metabolism and its regulation has been an active research field for many decades. As a result, several theories and techniques were developed that allow for the prediction of metabolic fluxes and yields as biotechnologically relevant output parameters. One important approach is to derive macrochemical equations that describe the overall metabolic conversion of an organism and basically treat microbial metabolism as a black box. The opposite approach is to include all known metabolic reactions of an organism to assemble a genome-scale metabolic model. Interestingly, both approaches are rather successful at characterizing and predicting the expected product yield. Over the years, macrochemical equations especially have been extensively characterized in terms of their thermodynamic properties. However, a common challenge when characterizing microbial metabolism by a single equation is to split this equation into two, describing the two modes of metabolism, anabolism and catabolism. Here, we present strategies to systematically identify separate equations for anabolism and catabolism. Based on metabolic models, we systematically identify all theoretically possible catabolic routes and determine their thermodynamic efficiency. We then show how anabolic routes can be derived, and we use these to approximate biomass yield. Finally, we challenge the view of metabolism as a linear energy converter, in which the free energy gradient of catabolism drives the anabolic reactions.
... There exists a vast literature devoted to both the modeling of the chemostat and its applications and the study of the arising dynamical issues. We would like to refer the reader to the seminal works [4][5][6] and the monographs [7][8][9] where more information on the subject can be found. ...
This paper revisits a recently introduced chemostat model of one–species with a periodic input of a single nutrient which is described by a system of delay differential equations. Previous results provided sufficient conditions ensuring the existence and uniqueness of a periodic solution for arbitrarily small delays. This paper partially extends these results by proving—with the construction of Lyapunov–like functions—that the evoked periodic solution is globally asymptotically stable when considering Monod uptake functions and a particular family of nutrient inputs.
... where x n and x n-1 are the cell density (cells/mL) at time t n and t n-1 (d), r espectiv el y. Based on μ v alue, the half-satur ation constant ( K s ) of N le v el was computed using Monod function (Equation 2) (Monod 1978, Tan et al. 2019 to compare and assess ov er all Nutilization capacity between MC + and MC − Microcystis at the assigned N le v el r ange: ...
Microcystin (MC)-producing (MC+) and MC-free (MC−) Microcystis always co-exist and interact during Microcystis-dominated cyanobacterial blooms (MCBs), where MC+ Microcystis abundance and extracellular MCs content (EMC) determine the hazard extent of MCBs. The study elucidated intraspecific interaction between MC+ and MC− Microcystis at various nitrogen (N) levels (0.5-50 mg/L) and how such N-mediated interaction impacted algicidal and EMC-inhibiting effect of luteolin, a natural bioalgicide. Conclusively, MC+ and MC− Microcystis were inhibited mutually at N-limitation (0.5 mg/L), which enhanced algicidal and EMC-inhibiting effects of luteolin. However, at N-sufficiency (5-50 mg/L), MC− Microcystis promoted MC+ ecotype growth and dominance, and such intraspecific interaction induced cooperative defense of two ecotypes, causing weakened luteolin's algicidal and EMC-inhibiting effects. Mechanism analyses further revealed that MC+ Microcystis in luteolin-stress co-culture secreted exopolymeric substances (EPSs) for self-protection against luteolin-stress and also released more EMC to induce EPS-production by MC− Microcystis as protectants, thus enhancing their luteolin-resistance and promote their growth. This study provided novel ecological implications of MC− Microcystis toward MC+ ecotype in terms of assisting the dominant establishment of MC+ Microcystis and cooperative defense with MC+ ecotype against luteolin, which guided the application of bioalgicide (i.e. luteolin) for MCBs and MCs pollution mitigation in different eutrophication-degree waters.
... Generally, Monod model is the most used model to describe the relationship between the growth rate of a microorganism (µ) and the concentration of the growth-limiting substrate [26]: ...
This study presents a novel integrated model to describe the fermentation-pervaporation system for bioethanol recovery. The proposed model incorporates the effects of by-product inhibition on the fermentation process and variation in feed concentration during the pervaporation process. The model is based on a modified Monod model to describe the fermentation process and the solution diffusion model for the pervaporation process, which can predict the concentrations of biomass, glucose, and ethanol in the bioreactor, as well as the partial flux of ethanol and water, total flux, the volume of permeate, and concentration of ethanol permeate. Simulation results showed that the proposed modified Monod model outperforms the conventional Monod model in predicting biomass, glucose, and ethanol concentrations due to its ability to consider the inhibitory effect of by-products with low mean absolute prediction error values of 4.1%, 12.6 and 5.2%. Moreover, the inclusion of the byproduct inhibitory effect in the fermentation model improved the accuracy of the integrated model in predicting the bioethanol permeation flux by 7.9%. The proposed integrated model provides a useful tool for designing and optimizing fermentation-pervaporation systems for bioethanol recovery to promote the adoption of continuous fermentation at the industrial scale.
... , (c > 0) [8,9] Sigmoidal G N(t) = N 2 (t) (c 1 + N(t))(c 2 + N(t)) ...
A chemostat is a laboratory device (of the bioreactor type) in which organisms (bacteria, phytoplankton) develop in a controlled manner. This paper studies the asymptotic properties of a chemostat model with generalized interference function and Poisson noise. Due to the complexity of abrupt and erratic fluctuations, we consider the effect of the second order Itô-Lévy processes. The dynamics of our perturbed system are determined by the value of the threshold parameter C ⋆ 0. If C ⋆ 0 is strictly positive, the stationarity and ergodicity properties of our model are verified (practical scenario). If C ⋆ 0 is strictly negative, the considered and modeled microorganism will disappear in an exponential manner. This research provides a comprehensive overview of the chemostat interaction under general assumptions that can be applied to various models in biology and ecology. In order to verify the reliability of our results, we probe the case of industrial waste-water treatment. It is concluded that higher order jumps possess a negative influence on the long-term behavior of microorganisms in the sense that they lead to complete extinction.
... Developed by Novick and Szilard [27], and Monod [26], the chemostat is a laboratory device used by biologists to raise the microorganisms and study their interactions while at the same time regulating the population size and the experimental medium. It consists in a culture in a container of constant volume in which a substrate is con-B Josué Tchouanti josue.tchouanti-fotso@unice.fr 1 Neuromod Institute, Université Côte d'Azur, 2004 Route des Lucioles, 06902 Valbonne, France tinuously injected and extracted at the same rate. ...
We study the existence and uniqueness of the solution of a non-linear coupled system constituted of a degenerate diffusion-growth-fragmentation equation and a differential equation, resulting from the modeling of bacterial growth in a chemostat. This system is derived, in a large population approximation, from a stochastic individual-based model where each individual is characterized by a non-negative trait whose dynamics is described by a diffusion process. Two uniqueness results are highlighted. They differ in their hypotheses related to the influence of the resource on individual trait dynamics, the main difficulty being the non-linearity due to this dependence and the degeneracy of the diffusion coefficient. Further we show by probabilistic arguments that the semi-group of the stochastic trait dynamics admits a density. We deduce that the diffusion-growth-fragmentation equation admits a function solution with a certain Besov regularity.
... Here μ i (R 1 , ⋯, R K ) is the specific growth rate of species i as a function of the resource availability; m i is the specific mortality rate of species i; D is the system's turnover rate; S j is the supply concentration of resource j; and c ij is the content of resource j in species i. It needs to be assumed that the specific growth rates follow the Monod equation (Monod, 1950), and are determined by the resource that is most limiting according to Liebig's 'law of the minimum' (Liebig, 1840): ...
Empirical Dynamic Modeling (EDM) has been a powerful tool for complex ecosystem prediction by providing an equation-free modelling framework. Theoretically, it allows future ecosystem behavior to be predicted by connecting current state to the similar, adjacent and future state on the attractor manifold which is reconstructed by single or multiple time series observed from natural systems. However, the Euclidean distance metric used in these algorithms could bias the true distance on the attractor manifold and consequently decrease the prediction performance. This could become worse if the dimension of the ecosystem is much higher and the system behavior is much complicated so that the reconstructed attractor manifold is more intricate. Therefore, manifold distance metric for both Simplex Projection and S-map was proposed. Our results clearly showed that the prediction accuracy of EDM had a general improvement after manifold distance metric was adopted. Experiments conducted on both synthetic and empirical data proved this advancement. Interestingly, these improvements were unequal for different implementations and the number of variables for embedding. Analysis demonstrated that S-map under multivariate embedding achieved the best prediction performance when manifold distance metric was applied. This suggested that the proposed manifold distance metric can work particularly well for predicting high dimensional ecosystem with complex behaviors. The main contribution of this research is that a new ecological indicator has been developed to more accurately estimate the similarity between ecological states in a reconstructed manifold and therefore provide higher prediction accuracy for EDM framework.
... Case 1 -Monotonic growth -Monod type This first case consists of the Monod relation of growth (Monod, 1950), which assumes that the growth rate is zero when there is no substrate and tends to an upper limit when the substrate is in great excess. ...
This paper concerns the inverse problem of characterising the state of a bioreactor from observations. In laboratory settings, the bioreactor is represented by a device called a chemostat. We consider a differential description of the evolution of the state of the chemostat under environmental fluctuations. First, we model the state evolution as a stochastic process driven by Brownian motion. Under this model, our best knowledge about the state of the chemostat is described by its probability distribution in time, given the distribution of the initial state. The corresponding probability density function solves a deterministic partial differential equation (PDE), the Kolmogorov forward equation. While this provides a probabilistic description, incorporating an observation process allows for a more refined characterisation of the state. More formally, we are interested in obtaining the distribution of the state conditional on an observation process as the solution to a filtering problem, with the corresponding conditional probability density function solving a non-linear stochastic PDE, the Kushner–Stratonovich equation. This paper focuses on the pathwise formulation of this filtering problem in which inferences about the state are obtained conditional on a fixed stream of observations. We establish the existence and uniqueness of solutions to the governing differential equations, ensuring well-posedness before presenting numerical approximations. We then approximate the pathwise solution to the filtering problem by combining the finite difference and splitting methods for solving PDEs, and then compare the approximated solution with results from a linearisation method and a classical sequential Monte Carlo method.
In this paper, we study a well-known two-step anaerobic digestion model in a configuration of two chemostats in series. The model is an eight-dimensional system of ordinary differential equations. Since the reaction system has a cascade structure, the model can be reduced to a four-dimensional one. Using general growth rates, we provide an in-depth mathematical analysis of the asymptotic behavior of the system. First, we determine all the equilibria of the model where there can be fifteen equilibria with a nonmonotonic growth rate. Then, the necessary and sufficient conditions of existence and local stability of all equilibria are established according to the operating parameters: the dilution rate, the input concentrations of the two nutrients, and the distribution of the total process volume considered. The operating diagrams are then theoretically analyzed to describe the asymptotic behavior of the process according to the four control parameters. The system exhibits a rich behavior with bistability, tri-stability, and the possibility of coexistence of the two microbial species in the two bioreactors.
The antagonistic coevolution of microbes and viruses influences fundamentally the diversity of microbial communities. Information on how environmental variables interact with emergent defense-counterdefense strategies and community composition is, however, still scarce. Following biological intuition, diversity should increase with improved growth conditions, which offset evolutionary costs; however, laboratory and regional data suggest that microbial diversity decreases in nutrient-rich conditions. Moreover, global oceanic data show that microbial and viral diversity decline for high latitudes, although the underlying mechanisms are unknown. This article addresses these gaps by introducing an eco-evolutionary model for bacteria-virus antagonistic coevolution. The theory presented here harmonizes the observations above and identifies negative density dependence and viral plasticity (dependence of virus performance on host physiological state) as key drivers: environmental conditions selecting for slow host growth also limit viral performance, facilitating the survival of a diverse host community; host diversity, in turn, enables viral portfolio effects and bet-hedging strategies that sustain viral diversity. From marine microbes to phage therapy against antibiotic-resistant bacteria or cancer cells, the ubiquity of antagonistic coevolution highlights the need to consider eco-evolutionary interactions across a gradient of growth conditions.
This work casts new insights into the model of a chain of chemostats in series, constructed in order to avoid the competitive exclusion in a nonautonomous chemostat. The main contribution of this paper is the idea to consider a periodic input of limiting nutrient, and it also proves that the chain of chemostats it is still a useful tool to ensure the periodical coexistence of the microbial species.
At the core of understanding growth are two fundamental concepts (that apply not only to phytoplankton but to all organisms): first, understanding the rate of growth of an organism is not the same as measuring its biomass, and second, rates of biological processes (including growth) increase with increasing concentration of the limiting substrate up to a point of “saturation” where limitation is overtaken by some other factor such as another substrate or physiological limitations. This chapter uses these fundamental concepts to introduce how growth of phytoplankton is measured, how cell division occurs, and how growth can be manipulated and quantified. Different culture systems, from laboratory to mesocosms, are introduced along with their associated experimental considerations.
The coupling between anaerobic digestion and hydrothermal carbonization (HTC) is a promising alternative for sustainable energy production. This study presents a dynamic model tailored for a lab-scale anaerobic digester operating on HTC products, specifically hydrochar and HTC liquor derived from sewage and agro-industrial digestate. Leveraging a modified version of the Anaerobic Model 2 (AM2), our simplified model of four states integrates pH and biomass decay rates into biomass kinetics. Simulation results of the mode were compared with experimental data collected over 164 days from the digester. The obtained results have proven the ability of the proposed model to predict the trend of the biogas production as well as important measured outputs of the bioreactor. The developed model could be used to control and optimize the performance of the digester, which provides potential for bioenergy production from waste streams such as digestate and digestate treated through the HTC process.
Durante un año se estudió el comportamiento de dos columnas de agua en el embalse El Peñol-Guatapé. La estación uno se ubicó a la entrada del río Nare (principal tributario del embalse) y la dos en la zona denominada isla del Sol (de características predominantemente lénticas). En cada estación se hicieron muestreos mensuales con el propósito de calcular la tasa de velocidad de reacción biológica y se aplicaron algunos modelos de carga de nutrientes y de eutroficación. En la estación uno las tasas de asimilación de nutrientes por parte de los microorganismos y de sedimentación de organismos clorofilados fueron más altas que en la estación dos. En general, la producción primaria en la estación uno estuvo limitada por el fósforo y en la estación dos, se detectaron limitaciones por nitrógeno durante cinco meses y en siete por el fósforo. A partir del modelo de predicción de la eutroficación, en lagos cálidos tropicales, se encontró que el embalse El Peñol-Guatapé es oligoproductivo.
In this paper, we formulate a classical stochastic delayed chemostat model and verify that this model has a unique global positive solution. Furthermore, we investigate the dynamical behavior of this solution. We find that the solution of stochastic delayed system will oscillate around the equilibriums of the corresponding deterministic delayed model, moreover, analytical findings reveal that time delay has very significant effects on the extinction and persistence of the microorganism, that is to say, when the time delay is smaller, microorganism will be persistent; when the time delay is larger, microorganism will be extinct. Finally, computer simulations are carried out to illustrate the obtained results. In addition, we can also find by the computer simulation that larger noise may lead to microorganism become extinct, although microorganism is persistent in the deterministic delayed system when the time delay is smaller.
Most ecosystems are characterized by seasonality, which, through biotic and abiotic changes, influences species biomass dynamics. Recent studies have shown that ecologically important traits can evolve rapidly in response to environmental changes, resulting in eco-evolutionary dynamics with consequences for population and community dynamics. Evidence for seasonal effects on intraspecific variation is still scarce and understanding eco-evolutionary dynamics in the presence of seasonal fluctuations remains a major challenge. Following the phytoplankton spring bloom in Lake Constance, we investigated how seasonal changes influence the intraspecific diversity of Asterionella formosa both at genotypic and phenotypic levels. We found a high degree of genetic and phenotypic differentiation characterizing Asterionella population, explained by a clustering of the isolates into early and late spring according to lake thermal stratification. Yet, most traits related to environmental parameters as well as fitness in different seasonal environments did not show a clear response to seasonality (i.e., temperature and nutrients), indicating that seasonal changes in biotic interactions (i.e., parasitic chytrids) were the major drivers of the observed seasonal shift in Asterionella genotypes. Our results highlight the importance of studying eco-evolutionary processes for understanding variations in population and community dynamics in response to seasonal environmental fluctuations.
How do individuals survive, develop, and grow in their environment? The multitude of challenges falls into two major categories:
Coping with the abiotic environment, i.e., basically the physical (e.g., temperature, pressure, wave energy) and chemical conditions (e.g., salinity) which set limits to survival and may be more or less favorable within those limits.
Beyond survival of abiotic stressors, organisms also have to extract resources from their environment, i.e., energy and substances needed for the production of their own body mass and for supporting their activity. At the same time, also materials are returned to the environment, consisting of “wastes,” i.e., end products of metabolism, and of architectural material, e.g., substances to build shells, reefs, etc. The matter and energy exchange with the environment is coupled to an internal transformation of matter and energy, called metabolism. There are two main categories of metabolic transformations: The processes building up biomass, called anabolism or assimilatory metabolism, and those processes which gain energy for maintenance and activity, called catabolism or dissimilatory metabolism.
A produção de cerveja artesanal tem grande potencial de crescimento no Brasil e pode ser classificada de acordo com o teor de álcool, variedade do extrato de malte, lupulagem ou com o tipo de fermentação. O presente estudo comparou os modelos cinéticos de Monod, Lee e Jin para definir qual é mais consistente com os dados experimentais de uma produção de cerveja artesanal. Através da variável independente densidade do mosto, foi estudado perfil da fermentação através do consumo de substrato e da formação de etanol fazendo a leitura da densidade do mosto, então convertida em g/L de açúcar fermentável da cerveja. Com o auxílio do software de simulação de processos EMSO, os modelos cinéticos resultantes foram ajustados e calculados por parametrização para cada modelo e densidade inicial do mosto. Os resultados mostram que os modelos propostos por Lee e Jin apresentam perfis muito próximos com os dados experimentais, prevendo o comportamento da fermentação, podendo ser utilizado para novas pesquisas na comunidade científica e projetos na área industrial, mostrando que a inclusão de termos de inibição por substrato e produto e a adequação do perfil matemático da curva foram decisivos para este caso, quando comparado com o modelo proposto por Monod, que apresentou perfis de consumo e formação não adequados para este processo.
In this article, we consider the chemostat system with n≥1 species, one limiting substrate, and mutations between species. Our objective is to globally stabilize the corresponding dynamical system around a desired equilibrium point. Doing so, we introduce auxostat feedback controls which are controllers allowing the regulation of the substrate concentration. We prove that such feedback controls globally stabilize the resulting closed‐loop system near the desired equilibrium point. This result is obtained by combining the theory of asymptotically autonomous systems and an explicit computation of solutions to the limit system. The performance of such controllers is illustrated on an optimal control problem of Lagrange type which consists in maximizing the production of species over a given time period w.r.t. the dilution rate chosen as control variable.
In this paper, we study the stochastic periodic behavior of a chemostat model with periodic nutrient input. We first prove the existence of global unique positive solution with any initial value for stochastic non-autonomous periodic chemostat system. After that, the sufficient conditions are established for the existence of nontrivial positive periodic solution. Moreover, we also analyze the conditions for extinction exponentially of microorganism, and we find that there exists a unique boundary periodic solution for stochastic chemostat model, which is globally attractive. At the same time, in the end of this paper, we also give some numerical simulations to illustrate our main conclusions.
This paper investigates the stochastic dynamics of trophic cascade chemostat model perturbed by regime switching, Gaussian white noise and impulsive toxicant input. For the system with only white noise interference, sufficient conditions for stochastically ultimate boundedness and stochastically permanence are obtained, and we demonstrate that the stochastic system has at least one nontrivial positive periodic solution. For the system with Markov regime switching, sufficient conditions for extinction of the microorganisms are established. Then we prove the system is ergodic and has a stationary distribution. The results show that both impulsive toxins input and stochastic noise have great effects on the survival and extinction of the microorganisms. Finally, a series of numerical simulations are presented to illustrate the theoretical analysis.
In this work, we consider a model of two microbial species in a chemostat in which one of the competitors can produce a toxin (allelopathic agent) against the other competitor, and is itself inhibited by the substrate. The existence and stability conditions of all steady states of the reduced model in the plane are determined according to the operating parameters. With Michaelis-Menten or Monod growth functions, it is well known that the model can have a unique positive equilibrium which is unstable as long as it exists. By including both monotone and non-monotone growth functions (which is the case when there is substrate inhibition), it is shown that a new positive equilibrium point exists which can be stable according to the operating parameters of the system. This general model exhibits a rich behavior with the coexistence of two microbial species, the multi-stability, the occurrence of stable limit cycles through super-critical Hopf bifurcations and the saddle-node bifurcation of limit cycles. Moreover, the operating diagram describes some asymptotic behavior of this model by varying the operating parameters and illustrates the effect of the inhibition on the emergence of the coexistence region of the species.
In this article, we are interested in identifying the parameters of an aerobic bioprocess modelused for wastewater treatment. In the field of biotechnology, various computer bugs caused by roundingerrors can induce an error interval that is too wide during data acquisition. For this reason, weare testing a new identification method using a set method based on interval arithmetic. The processstudied is the chemical transformation of ammoniacal nitrogen which takes place in two stages: Reactionof nitrificationdenitrification.The parameters chosen for the identification are the yields andthe maximum growth rates. Initially, the study of observability by a differential algebraic method willsimplify the study of the mathematical model. This nonlinear model is described by six differentialequations. Subsequently, we apply a set method, in particular the propagation of constraints also calledforwardbackward propagation, this technique allowed us to determine intervals containing the variablereturns as well as the maximum specific growth rates defined from the Monod model which describesthe operation of the bioreactor. This method also guarantees the result by rejecting all inconsistentvalues.
Anaerobic co‐digestion is defined as the simultaneous anaerobic digestion of two or more substrates. Mixing several substrates has many advantages. In particular, it can increase biogas production. In our work, we built a mathematical model to describe the anaerobic co‐digestion of two substrates, with preference and mortality of bacteria in closed mode. The existence and boundedness of positive solutions are investigated. We show that trajectories converge toward a global attractor which is composed of an infinite number of non‐hyperbolic equilibria. The choice of growth functions, for both bacteria, can completely change the performance of the process. Numerical tests are presented to illustrate the theoretical analysis. They allow the prediction of the quantities of the initial mixture for a better biogas production.
In this paper, we study the asymptotic dynamics of two chemostat models with random environmental fluctuations modeled by means of real noise, where different consumption functions for the consumer species (Monod and Haldane) are taking into account. For each model, our main goal is to investigate the existence of deterministic attracting sets. This allows us to provide conditions under which the species become extinct or persist, which is the main goal in industrial setup. In addition, we depict different numerical simulations to support the theoretical results. Finally, we present some conclusions to sum up our contribution and we compare how the species behave depending on its consumption function.
In this paper we investigate four non-autonomous chemostat models with non-monotonic consumption function, where wall growth and nutrient recycling are also taken into account. In each case, we prove the existence and uniqueness of non-negative global solution that generates a non-autonomous dynamical system. In addition, we also prove the existence of a unique (global) pullback attractor whose internal structure provides detailed information about the long-time behavior of the state variables, for instance, conditions to ensure the extinction and the persistence of the species. We also display numerical simulations to illustrate the theoretical results.
The resource’s diffusion and the populations’ migration may alter the ecosystem’s structure such that the species’ dynamics are changed. Moreover, the delay phenomenon in population behaviors, such as digestion or maturation, will inevitably affect the species’ dynamics. We, in this work, investigate a chemostat system with delay and spatial diffusion. The existence conditions of the Hopf bifurcation from the time lag and diffusive terms are determined. The concentration of population in the chemostat approaches a positive value when the bifurcation parameter’s value does not cross the critical point. The microorganisms’ concentration will fluctuate periodically as the value of the bifurcation parameter passes through the critical point. By the theory of norm form and center manifold, we further talked about the direction of the Hopf bifurcation and the stability of the periodic solutions. Several numerical examples are provided to support the theoretical results in this work.
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