Growth models for cells in the chemostat

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1 Growth models for cells in the chemostat V. Lemesle, J-L. Gouzé COMORE Project, INRIA Sophia Antipolis BP93, Sophia Antipolis, FRANCE Valerie.Lemesle, Abstract A chemostat is a continuous device where microorganisms consume a nutrient to grow. To describe microorganisms growth in the chemostat, Monod [12] introduced a very simple model. Droop [5] then presented another model considering the possible nutrient storage in a plankton cell. However, this work was mainly based on experimental observations and was not completely justified by biochemical arguments. In this paper, a new model of phytoplankton growth is proposed. Though our model is finally similar to the Droop one, we base our modelling on biochemical arguments. We study two versions of our model considering or not cell mortality. We give also some indications on the observation of such models, and related observers. Keywords biochemical mechanisms, structured model, ordinary differential equations, chemostat, plankton growth models, observers. 1 Introduction Unstructured mathematical models are often employed to describe cell growth in continuous cultures devices, so called chemostats. The most representative one has been proposed by Monod [12]. Its approach is based on very strong assumptions on the interaction between the two main components in the culture: the microorganisms, more precisely bacteria, and the environment. It has been assumed that the biomass concentration alone is able to characterize the physiological state of the whole cell population and that the biomass specific growth rate is controlled only by the concentration of a unique growth limiting nutrient. Although 1

2 this model could be successfully used to fit steady state data, its prediction, when phytoplankton is considered, is far from being satisfactory. Then, a new modelling approach is therefore required to adequately used this continuous culture model when phytoplankton is studied. In fact, structured models seems very efficient for this purpose. One of the most classical one is proposed by Droop [5]. Its approach is based on two steps: the storage and the metabolization of the limiting nutrient for phytoplankton cell. It is assumed that the intracellular limiting component quantity and the total biomass are able to characterize the cell population and that the biomass specific growth rate is given by a function depending on a minimum of internal nutrient quantity. All this assumptions are based on experimental observations. In this paper, we propose another structured model, more based on mechanistic biochemical explanations. Our approach is explained by biochemical mechanisms and not by biological data only. It is assumed that two components are limiting for the growth and that the metabolized and un-metabolized limiting intracellular components characterize the cell population (phytoplankton). This paper is organized as follows. First, we present the chemostat apparatus, recalling the two classical models and their main properties. Then, we explain the basis of our modelling approach and we compare this model to the Droop one. We give finally some indications on building observers for these models. 2 Growth models 2.1 The Monod model The most classical chemostat model has been introduced by Monod to give a synthetic picture of what happens when a microorganism culture (bacteria) is grown in a reactor. Then, only two variables are chosen to describe the various and complex reactions occurring in the reactor vessel: the biomass concentration and the limiting substrate concentration. The mathematical modelling is divided in two parts: the previously mentioned physical part due to the flow and the biological part, which describes the biological reactions in the vessel. This description leads to the well known mass balanced equations: (1) Biological part Physical part Based on biological data,, the specific growth rate, is often defined by a classical function, so-called Michaelis-Menten function.! 2

3 This model can be formulated as a biochemical reaction defined such that: Reactants and product are considered; the rate of reaction depends on biological mechanisms. Indeed, it is assumed some qualitative hypotheses about this specific rate considering some biological characteristics. Hypothesis H 2.1 and is, increasing and bounded. Hypothesis H 2.2 Moreover, the mass conservation principle is verified. Indeed, take the variable (the total mass in the chemostat) then the dynamical equation of follows equation: "! Then, the asymptotic behavior of the system (1) is given by the following proposition. Proposition 2.3 Under the hypotheses (H2.1) and (H2.2), the washout steady state is unstable and the non trivial steady state is globally asymptotically stable (wrt the positive orthant). This simple unstructured model is very intuitive and gives a good idea of the bacteria dynamics. The main problem of this model is that all the physiological characterization is contained in only one variable which is not enough to describe all the different microorganisms. Moreover, this model predicts, for example, that there is no growth if there is no more substrate ( % ). This contradicts experimental facts for phytoplankton. 2.2 The Droop model The Droop model has been built to describe the phytoplankton growth submitted to limiting vitamin B '& in the chemostat [5]. This model is more descriptive than the Monod one: indeed, it distinguishes the evolution of the biomass concentration and of the intracellular limiting nutrient concentration (. The variable ( is socalled the intracellular quota of limiting nutrient. Thus, considering the biological $# 3

4 phenomena and the physical part due to the through passing flow, the following variable yield model is obtained: ( ( ( Biological part Physical part The equation of ( is often replaced by the equation of ( (deduced of (2)): ( ( ( (3) Let us remark that the quantity ( does not makes a difference between the stored and the metabolized nutrient. This model can be illustrated by the following biochemical mechanisms: ( ( storage metabolization The phenomenon of metabolization cannot be described very precisely using the variables (,, : this is the main problem of this model. Indeed, the variable ( does not make the difference between the storage of nutrient and how this storage is used for metabolization. As in the Monod model, Droop assumed some qualitative hypotheses about the specific growth rate ( and storage rate functions based on biological characteristics. Hypotheses H 2.4 ' and is, increasing and bounded. ( is, non negative, increasing and bounded, such that there exists ( defined as follows (. Hypothesis H 2.5 The mass conservation principle is verified; indeed, take the variable ( (the total mass in the chemostat), the dynamical equation of verifies equation: "! The asymptotic behavior is given by the following proposition proved by Lange et al. in [9, 13]. Proposition 2.6 Under all the hypotheses above, the washout equilibrium is unstable and the non trivial steady state is globally asymptotically stable (wrt. the positive orthant). We propose now a structured model described by variables which have a more biological sense and where biochemical phenomena are more taken into account. 4 $# (2)

5 3 A structured phytoplankton growth model In this section, we consider phytoplankton growth limited by the two major components of photo-synthetic cells: nitrogen and carbon. 3.1 Model formulation It is well known that the phytoplanktonic cells grow via absorption of extracellular nitrogen and carbon. We simplify the complex absorption of extracellular nitrogen in two steps: storage and metabolization. The absorption of extracellular carbon is very simple: it is directly metabolized into intracellular organic carbon. The two input limiting components are denoted and. Moreover, to describe the two absorption phenomena, the following variables are chosen: the extracellular nitrogen substrate, the intracellular inorganic nitrogen, the intracellular organic nitrogen, the extracellular carbon substrate, the intracellular organic carbon. We consider the following biochemical mechanisms. storage mortality growth and respiration Let us focus on the mechanism of growth and respiration. The nitrogen metabolization and the carbon metabolization are both represented. More precisely, the nitrogen metabolization is explained by one unit of intracellular inorganic nitrogen metabolized into one unit of intracellular organic nitrogen with the rate of reaction ; the carbon metabolization is characterized by more than one unit of extracellular inorganic carbon metabolized into one unit of intracellular organic carbon with the rate of reaction, the rest being rejected by the respiration with the same rate of reaction. The cell mortality describes the possible cell death before its division; indeed, the dilution rate cannot justify alone the total biomass disappearance. It is modeled by a constant rate ; let us remark that the dynamics of this dead cells (, 5

6 , ) is not defined in this model. In the literature, it can be chosen to add a delay phenomenon in the extracellular inorganic components for the recycling of the dead cells ; indeed, the conversion of organic components into inorganic component is not immediate [2, 3]. Moreover, we take into account another phenomenon, which is not described by the biochemical mechanisms: the conversion of atmospheric carbon into dissolved carbon (extracellular inorganic carbon) with a rate (. Let us remark that this rate can be negative. Adding the physical part due to the flow passing through the chemostat, this description leads to the following model, denoted (S): (4) (5) (6) (7) ( (8) Biological part Physical part The variable represents the available stored nitrogen per organic biomass unit, since, in biology, the usual measure unit of organic biomass is the organic carbon. We make the same hypotheses as for the Monod model (H2.1) on the qualitative behavior of the growth rates. Hypotheses H 3.1 ' and is a increasing bounded function; "! ' and is a increasing bounded function.. Let us remark that the function is not defined for. This is not a problem because this function describe the evolution of the intracellular inorganic components, then has not a biological sense for this. 3.2 Model reduction In this section, we explain why the equations (4), (5), (7) are enough to describe the dynamics of the model (S). First, consider the variable carbon unit and nitrogen unit., means that a cell is composed by one 6

7 The dynamical equation associated to is The variable and have the same asymptotic behavior. More precisely, the equilibrium (i.e. ), is globally asymptotically stable. We deduce that, asymptotically, the organic nitrogen and the organic carbon are linked by a fixed ratio. This fact is often observed at equilibrium on biological data. Thus, the dynamical equation (6) does not give more informations than the one (7). The dynamical equation (9) only depends on the dynamics of (7), (5) and itself; moreover, all the other equations do not rely on the carbon dynamics. Then, in the sequel, we consider the reduced system: (9) 3.3 Interpretation of the limiting nutrient quota The intracellular quota of limiting nutrient is proposed with respect to experimental observations in the Droop model. It represents the intracellular inorganic limiting nutrient quantity per unit of biomass. With the variables of our model, the quota is the total nitrogen quantity in the cell per biomass unit, meaning that is given by (. The dynamical equation of ( is then as follows: ( (10) Compare this equation with the Droop equation (3). The model (2) has been built under equilibria conditions and without mortality phenomenon (i.e. ). If we consider the model (S) under equilibria conditions and with, we obtain the following conditions: and Thus, the quota equation (10) becomes: ( ( ( Though we have built model (9) in a different manner Droop had, we obtain the same formulation for the intracellular quota. Moreover, we are able to explain the 7

8 clear biological meaning of the Droop minimum intracellular quota ( defined by the minimum stored nutrient of a cell to stay alive and estimated from data: it is the meaning of the constant given in the previous subsection. 4 Model analysis The asymptotic behavior of the two versions of the model (9) can be studied: without any mortality rate, or with a fixed positive mortality rate. The second one is very similar to the Droop model. 4.1 Existence of steady states and invariant set First, we give some general properties of the system (9) with. For this system, is not acceptable since the function is not well defined at this point. Then, to make a complete study of this system, we consider the change of variables: which defines a diffeomorphism in the set. We obtain the new system: (11) This new system is well defined in the set Note that it can be seen as an extended system of (9) since it is defined for. Thus, we only consider initial conditions for the state variables belonging to the set which only has a biological sense for (11). Moreover, let us remark that this new system is a loop system [11]: for To ensure the existence of the non trivial equilibrium, we make the following hypotheses. Hypotheses H 4.1 "!. "!. 8

9 Proposition 4.2 Under the hypotheses (H3.1) and (H4.1), there exists two admissible stationary points for the system (11): the washout point and the non trivial equilibrium. Proposition 4.3 Under the hypotheses (H3.1), is invariant for the system (11). 4.2 Local analysis For the local behavior of the system (11), we use the classical method of linearization. Thus, we compute the associated Jacobian of system (11) at a generic point : Let us remark that as the number of non positive off diagonal terms is odd, the system is similar to a competitive system taking [14]. Proposition 4.4 Under the hypotheses (H3.1) and (H4.1), the washout point is unstable. Proposition 4.5 Under the hypotheses (H3.1) and (H4.1), the non trivial equilibrium is locally asymptotically stable. 4.3 Global analysis without a mortality rate We consider the system (11) in the case. We obtain the new system: (12) Proposition 4.6 Under hypotheses (H3.1) and (H4.1), the mass principle is verified and the state variables are bounded. Proof : Consider the total mass in the chemostat. The dynamical equation of verifies equation: $# 9

10 Meaning that: "! #. Thus, the mass principle is verified and the variables are bounded. To prove the global stability of this equilibrium, we use the same techniques as Lange et al. [9, 13] for the study of the classical Droop model. Proposition 4.7 Under hypotheses (H3.1) and (H4.1), the non trivial equilibrium is globally asymptotically stable (wrt. the positive orthant). 5 Observers It is often the case that the quota (either for the Droop model, or for our model) is not measured or is difficult to measure. It can be useful to build observers, that will estimate this quota from the available measurements. The problem here is that some parts of the model can also be not so well known, for example the growth rate. One possibility in this case is to built a kind of observer (sometimes called asymptotic observer, see [1]) independent on this unknown part. Let us suppose that and are the measurements. Then,! verifies the equation and the simple estimator: estimated by. converges towards. The quota can be High gain observers can also be built, if the model is well known (cf. [4]). We will present here an intermediate case (subject of ongoing research) where the model is not well known, but bounds are available. The idea is to build a hybrid observer, similar to a high-gain observer when far of the solution, and switching to an asymptotic observer when closer to the solution (cf. [10]). Let us suppose that the substrate is the only measurement and that the growth rate is such that a bound is known: With the variable, the system is: and an hybrid observer will be: (13) 10

11 & & If, the observer is an asymptotic observer; for, the gains can be adjusted to obtain a high gain observer (for known ). The switch is done when the output error is smaller than. A convergence towards the solution faster than the asymptotic one can be attained. 6 Conclusion In this paper, a model similar to the Droop model has been considered. The new structured model is characterized by biochemical mechanisms which describe the cell storage and how the cell uses this storage to grow. Thus, we obtain a descriptive model using variables which have a more biological sense and which do not come from experimental data only. However, if the same experimental conditions as in the Droop model are considered (equilibria conditions, no mortality rate), then the same formulation for the intracellular quota is obtained. This structured model contains, therefore, more informations than the Droop one. The building of observers for such models is the subject of ongoing research. References [1] G. Bastin and D.Dochain. On-line Estimation and Adaptative Control of Bioreactors, volume 1. Elseiver, Amsterdam, [2] E. Beretta and Y. Takeuchi. Qualitative properties of chemostat equations with time delays: boundedness, local and global asymptotic stability. Differential Equations and Dynamical Systems, 2(1):19 40, [3] E. Beretta and Y. Takeuchi. Qualitative properties of chemostat equations with time delays ii. Differential Equations and Dynamical Systems, 2(4): , [4] O. Bernard, G. Sallet, and A. Sciandra. Nonlinear observers for a class of biological systems. Application to validation of a phytoplanktonic growth model. IEEE Trans. Aut. Control, 43(8): ,

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