Original article Biomath 4 (2014), 1407121, 1–10 B f Volume ░, Number ░, 20░░ BIOMATH ISSN 1314-684X Editor–in–Chief: Roumen Anguelov B f BIOMATH h t t p : / / w w w . b i o m a t h f o r u m . o r g / b i o m a t h / i n d e x . p h p / b i o m a t h / Biomath Forum On the Mathematical Modelling of EPS Production by a Thermophilic Bacterium Nadja Radchenkova1, Margarita Kambourova1, Spasen Vassilev1, Rene Alt2, Svetoslav Markov3 1 Institute of Microbiology, Bulgarian Academy of Sciences 2 Sorbonne Universities, LIP6, UPMC, CNRS UMR7606 3 Institute for Mathematics and Informatics, Bulgarian Academy of Sciences Received: 24 January 2014, accepted: 12 July 2014, published: 23 July 2014 Abstract—This paper presents experimental data coming from a batch fermentation process and theoretical models aiming to explain various aspects of these data. The studied process is the production of exopolysaccharides (EPS) by a thermophilic bac- terium, Aeribacillus pallidus 418, isolated from the Rupi basin in South-West Bulgaria. The modelling approach chosen here is: first, biochemical reaction schemes are formulated, comprising several reaction steps; then the reaction schemes are translated into systems of ordinary differential equations (ODE) using the mass action law; then the ODE systems are studied by means of numerical simulations. The lat- ter means that the ODE systems are parametrically identified in order to possibly fit the experimental data. A main peculiarity of the proposed reaction schemes, resp. models, is the assumption that the cell biomass consist of two dynamically interacting cell fractions (dividing and non-dividing cells). This as- sumption allows us to implement certain modelling ideas borrowed from enzyme kinetics. The proposed models are compared to a classical model used as reference. It is demonstrated that the introduction of the two cell fractions allows a much better fit of the experimental data. Moreover, our modelling approach allows to draw conclusions about the underlying biological mechanisms, formulating the latter in the form of simple biochemical reaction steps. Keywords-batch fermentation processes, ther- mophilic bacterium, reaction schemes, dynamic models, numerical simulations I. INTRODUCTION An increasing interest towards microbial ex- opolysaccharides (EPS) is determined by the wide variety of their properties as a result of diversity in their composition. The biodegradability of EPS has an impact on environmentally friendly pro- cesses. Thermophilic microorganisms offer short fermentation processes, better mass transfer, de- creased viscosity of synthesized polymer and of the corresponding culture liquid. EPS from thermophilic microorganisms are of special interest due to the advantages of the ther- mophilic processes and the non-toxic nature of the polymer allowing applications in food and phar- maceutical industries [4]. Low EPS production in thermophilic processes determines the importance of mass transfer optimization for further devel- opment of thermophilic processes in an industrial Citation: Nadja Radchenkova, Margarita Kambourova, Spasen Vassilev, Rene Alt, Svetoslav Markov, On the Mathematical Modelling of EPS Production by a Thermophilic Bacterium, Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 1 of 10 http://www.biomathforum.org/biomath/index.php/biomath http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... scale. The aim of the current work is to describe and study the dynamics of microbial growth and prod- uct synthesis by means of suitable mathematical models, which may give more understanding for the underlying biochemical mechanisms. Our main assumption from a biological perspective is the existence of two cell fractions: dividing and non- dividing cells, as proposed in [10]. Based on com- parisons with a classical model it is demonstrated that the introduction of the two cell fractions allows a much better fit of the experimental data. In addition, our modelling approach allows to formulate the underlying biological mechanisms in the form of simple biochemical reaction steps. To achieve our aim we propose several biochem- ical reaction schemes that conform with the basic assumption for the existence of two cell fractions. Applying the mass action law the reaction schemes lead to dynamical models that are parametrically identified in order to fit the observed data. We demonstrate that the proposed models are com- patible with classical Monod fermentation models which are based on different principles. In order to visualize the numerically computed theoretical results, the latter are graphically compared to experimental data. The graphics show that the proposed models reflect specific features of the mechanism of the fermentation process, which may suggest further experimental and theoretical work. We believe that using the proposed approach one can study the basic mechanisms underlying the dynamics of cell growth, substrate uptake and product synthesis. Finally, we hope that the present study will contribute to the optimization of mass transfer and an enhancement of EPS yield by Aeribacillus pallidus 418. The paper is structured as follows. The meth- ods for the experiments are described in Section II. Section III presents the experimental results and Section IV outlines the modelling approach. Section V is devoted to several theoretical mod- els whose solutions fit the experimental measure- ments. The underlying idea is that such a theo- retical approach can help biologists in choosing or rejecting a possible mechanism for the dynam- ics of microbial growth and EPS synthesis. The numerical solutions obtained with each proposed model are compared to the experimental data and commented. The conclusion explains the merits of the proposed approach and comments on the sug- gested biological assumptions. It is demonstrated that the proposed models based on the assumption of existence of two cell fractions (dividing and non-dividing cells) are compatible with classical Monod fermentation models. II. MATERIALS AND METHODS A. Strain, medium and cultivation Aeribacillus pallidus 418 was isolated from a hot spring at Rupi basin, South-West Bulgaria, and selected as an EPS producer among 38 ther- mophilic bacterial polymer producers [12]. B. Experimental set-up and operation mode A 1.5 L jacketed glass reactor (AK-02, Rus- sia) equipped with a hydrofoil impeller Narcissus (0.05 m diameter) was filled with l L of MSM and 35 mL of 18 h strain culture was added as an inoculum. Fermentation parameters such as temperature, pH, foaming and aeration were kept constant during the whole processes performed. Temperature was 550C; pH value was maintained at 7.0 due to buffer properties of the medium; air flow 0.8 vvm (volume per volume medium) was continuously supplied. C. Determination of growth and EPS production The influence of agitation on the growth and EPS production was followed at 100, 400, 500, 600 and 800 rpm. Growth was determined by mea- suring of turbidity at 660 nm. A correlation curve reflecting the proportionality between turbidity and dry weight was obtained for the chosen strain. One unit OD corresponds to 1.05 mg ml−1 dry cells of Aeribacillus pallidus 418. EPS was recovered from the culture supernatant samples as previously described [4]. Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 2 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... III. EXPERIMENTAL RESULTS FOR BATCH CULTIVATION A. Influence of agitation speed Comparison of growth curves at different agi- tation conditions show that a longest exponential phase (8 h) and lowest biomass yield reached at stationary phase is observed at a lowest speed of stirring (100 rpm) (Figure 1). Exponential phase duration of 4–5 h is observed for all other agitation speeds. An increase in biomass accumulation and polymer production is observed with increase of stirring speed being highest at 500 rpm for growth and 600 rpm for EPS, after that it decreased. The lowest speed of stirring (100rpm) unfavorable for growth and EPS production and the speed that provide best growth (500 rpm) and best product synthesis (600 rpm) have been chosen for further modelling. Investigations on the influence of agitation speed on EPS production (Figure 2) reveal that the observed polymer production is connected with the biomass accumulation. First quantities are reg- istered in the early exponential phase and highest levels in the stationary phase. The concentration of the measured polymer is highest at 600 rpm (124 µg mL−1), it is five fold higher than that at 100 rpm (24 µg mL−1). 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 B io m a ss m g /m l Time in hrs Fermentor 1litre; Experimental Biomass, various rpm, Air 0.8:1 100 rpm 400 rpm 500 rpm 600 rpm 800 rpm Fig. 1. Experimental biomass for different rpm 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0 5 10 15 20 E P S m g /m l Time in hrs Fermentor 1litre; Experimental EPS, various rpm, Air 0.8:1 100 rpm 400 rpm 500 rpm 600 rpm 800 rpm Fig. 2. Experimental EPS for different rpm The experimentally obtained average values are presented in Figure 1 for the biomass and in Figure 2 for EPS. The figures show clearly that the best production of EPS appears for a speed of 600 revolutions per minute. The experimental results corresponding to maximum EPS yield (at 600 rpm) are presented in Table I. The measurement error is estimated to be ≤ 15% for the biomass data (≤ 18% for the last three data) and ≤ 5% for the EPS data. Taking into account the measurement error, the experimental data of Table 1 are used in the modelling process as interval data [9]. For example, biomass data for hour seven is 1.0, hence the resp. interval value is 1.0±15% = [0.85,1.15]. The interval data are visualized as vertical seg- ments in the presented figures. Time (hours) 0 1 2 3 4 Biomass (mg/ml) 0 0 0 0 0 Time (hours) 5 6 7 8 12 Biomass (mg/ml) 0.16 0.45 1.0 1.3 1.2 EPS (mg/ml) 0.006 0.037 0.07 0.112 0.117 Time (hours) 14 16 18 21 24 Biomass (mg/ml) 1.2 1.2 1.1 1.1 1.1 EPS (mg/ml) 0.119 0.120 0.124 0.124 0.124 TABLE I EXPERIMENTAL VALUES OF THE BIOMASS AND EPS Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 3 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... IV. MODELLING APPROACH The experimental measurements for the cell growth and EPS production together with the esti- mated measurement errors have been used for test- ing and studying several dynamical models aiming to explain the nature of the involved metabolic reactions. To this end an original modelling ap- proach similar to the one used in (bio)chemistry has been used. Instead of starting the construction of the model with formulating systems of ordi- nary differential equations (ODE) and specifying reaction terms in the right-hand side of the sys- tem’s equations as usually done in classical cell growth models, we start with proposing suitable reaction schemes. The origin of such an approach is seen in the works [2], [6], where reaction steps involving variable reaction rate coefficients are admitted. Our reaction schemes differ from the ones used in these works in that all reac- tion rate coefficients are numerical constants—as (commonly) in biochemistry and more specifically in Henri-Michaelis-Menten enzyme kinetics [10], [14], [16]. The proposed reaction schemes are then translated into systems of ODE’s using the mass action law, and the ODE systems are studied by means of numerical simulations. Another peculiarity of our modelling approach is the partitioning of the cell biomass into two dynamically interacting fractions of dividing and non-dividing cells. This allows to implement ideas borrowed from enzyme kinetics—there enzymes are subdivided into two fractions: free and bound. Bio-reactor models using fractions (compartments) for the biomass are known in the literature [6], [7], [17]. The proponents of the modelling approach using fractions of the cell biomass note that such structured models allow a better fit of the cell growth dynamics in the lag phase in comparison to the one obtained with classical Monod type models [17]. In our proposed models the cell population is conditionally partitioned into two fractions: di- viding and non-dividing cells. This allows to relax the rather restrictive assumption for synchronized states of individual cells that follows from (or is incorporated in) classical models using only one variable for the cell biomass. Thus structured models allow us to assume that cells are not all simultaneously in a specific state—something that is observed in reality. The terms dividing and non-dividing states refer here to individual cells as opposite to the terms lag, log and stationary phases that refer to the cell community. Classical Monod type models consider a population of a microbial species as being in one of the mentioned phases at any given moment. We consider the dividing cell property as a property of the individual cell which may be inherited from the mother cell [5], [13]. This property may change in time from dividing to non-dividing or vice versa depending on the environmental conditions but also on individual cell properties. Individual cell states are close to the biomass phases but do not coincide. If a cell population has been in a lag or stationary phase for a long time, then most (or all) of the individual cells will be in non-dividing state, conversely, if the cell community is in an established log phase, then most (or all) of the individual cells are in a dividing state. However, at the time when the environmental conditions change (from favorable to unfavorable or vice versa), then individual cells do not change their state simultaneously and there is a time interval when cells of both states coexist. Thus whenever the cell community passes from a lag or stationary phase to a log phase (or conversely) there are cells in dividing state, and others in non-dividing state. As will be demonstrated in this work struc- tured models based on the dividing property effec- tively contribute to overcoming the deficiency of classical methods to adequately describe the bio- processes during the intermediate time intervals when biomass phases change; we further refer to this problem as “intermediate-phase-deficiency” problem. One more peculiarity of our modelling approach is that we fit the theoretical solutions of the proposed models into the interval experimental data (data ± upper bounds for the estimated mea- surement errors), which contributes to the model verification process [9]. In other words, our aim Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 4 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... in the parameter identification process is to make the theoretical solutions pass through the interval experimental data. Appropriate parameters for the various models have been obtained using an op- timization method for minimizing the sum of the squared differences between the observed experi- mental data and the computed theoretical solution. But one cannot assert that these parameters are optimal as there are several local minima and dif- ferent sets of parameters may lead to almost iden- tical solutions. That is why we chose“manually” appropriate sets of parameters with the additional requirement to make the theoretical solutions pass within as many as possible interval experimental data (measurements plus/minus estimated upper bound for the measurement errors). V. MATHEMATICAL MODELLING Several mathematical models are proposed in the sequel and their parameters are numerically identified in order to fit with the observed experi- mental results as given in Table I. The first model is a classical Monod type model, whereas the other models conform with the idea of a structured bac- terial biomass and the reaction-scheme approach as proposed in [10]. Classical Model 1 is used as a reference in order to be compared to the remaining structured models following the reaction-scheme approach. A. Model 1 Model 1 is a classical Monod type cell growth model [11]. The model is a modification of a model for CGTase production by bacteria of the species B. circulans ATCC21783 [3]. The model is mathematically represented by the following differential system: dx/dt = (µ−γ) x ds/dt = −(µ/δ) x dp/dt = (α µ + β) x, (1) wherein: µ(s) = (µmax s/(Ks + s + Kis 2), with initial conditions s(0) = s0, x(0) = x0, p(0) = p0. Typically model (1) makes use of a growth rate parameter µ which is not a constant but a function; in this case µ is a function on the substrate con- centration s, namely the familiar Andrews/Haldane specific growth rate function µ(s). The meaning of the numeric parameters in model (1) is given in [3]. Computer simulations provide the following set of parameters: µmax = 5, α = 93, β = 0, γ = 0.005, Ks = 0.25, Ki = 30, δ = 13 with initialization: x0 = 0.05, s0 = 0.1, p0 = 0. The experimental data and the computed the- oretical solutions of model (1) are visualized in Figure 3. The vertical intervals correspond to the experimental values as given in Table 1 together with the measurement errors in the data (the same pattern is followed in the remaining figures). The aim has been to identify the parameters of system (1) in such a way that the computed solutions fit the experimental data in the sense that they possibly pass through the experimental interval data. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 25 30 B io m a ss a n d E P S Time in hrs Aeribacillus pallidus 418, Fermentor1; 1 l; 600 rpm,Air 0.8:1 Theor. biomass Theor. product EPS . Fig. 3. Solution of model (1), 600 rpm Monod type models describe quite well bio- processes under favorable conditions and hence allow a good fit of the log (exponential) phase. However, such models do not generally allow good fit for the intermediate phases, e. g. between the Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 5 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... lag and log phases, as is observed on Fig 3. Note that all experimental values for the biomass in the first four hours are zero whereas the computed theoretical solution for the biomass increases quite rapidly and cannot be fitted well for any values of the model parameters to the experimental interval data for t ≤ 5. Thus model (1) demonstrates the inherent “intermediate-phase-deficiency” of classi- cal Monod type models. A characteristic feature of Monod models is the use of the specific growth rate function µ = µ(s). Some authors formulate reaction schemes that make use of such a reaction rate function (instead of coefficient!) [2], [6]. It has been demonstrated that Monod type models cannot be induced from reaction schemes with constant reaction rate coef- ficients by means of the mass action law [10]. It has been also discussed in [10] that Monod type models are special cases of appropriate structured models under specific restrictions (providing ex- ponential growth). In order to overcome the “intermediate-phase- deficiency” problem and to obtain a better fit for the experimental interval data two original models are proposed and numerically tested using the same experimental data. These models follow the modelling approach of building biochemical reaction schemes with constant reaction rate coef- ficients [10]. The proposed models are expected to throw some light on the related cell growth mechanisms; both proposed models are based on the idea of structured biomass and the use of an enzyme-kinetics-like reaction scheme as a starting point. B. Model 2 Here the total biomass has been split into two fractions of non-dividing X-cells and dividing Y - cells. According to the reaction-scheme modelling methodology we first formulate suitable reac- tion steps underlying the biological (biochemical) mechanism behind the cell growth and cell produc- tion processes. The corresponding reaction scheme involves six reaction steps: a double growth step (RSG), two reproduction steps (RSR), a producing step (RSP) and a decay step (RSD). As shown in [10] the growth step (RSG) reflects the transition of X-cells into Y -cells and substitutes the use of the “specific growth rate function” µ(s) proposed by Monod [11]. This step also reflects the conver- sion of nutrient substrate S into metabolic products P used for the growth of the cells and for their preparation to pass into dividing Y -state. Note that P represents the total amount of metabolites obtained as result of the transformation of the nutrient substrate S. The two reproduction steps (RSR) describe the transfer of nutrient metabolite P partially into X- and Y -cells. The production step (RSP) describes the transition of the interme- diate metabolites P into (useful) product P1 (EPS). The decay step (RSD) reflects the decay of X-cells assuming that the disintegrated cells transform into a waste product Q. S + X k1−→ Y k2−→ P + X, (RSG) P + Y β −→ 2Y, (RSR) P + Y α−→ X + Y, (RSR) P + Y γ −→ Y + P1, (RSP) X kd−→ Q, (RSD) Applying the mass action law to the above reaction steps as in chemical and enzyme kinetics we obtain the following system of ODE’s where p1 represents the product EPS: ds/dt = −k1xs dx/dt = −k1xs + k2y + αpy −kdx dy/dt = k1xs−k2y + βpy dp/dt = k2y − (α + β + γ)py dp1/dt = γpy. (2) Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 6 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... 0 0.5 1 1.5 2 0 5 10 15 20 B io m a ss a n d E P S , m g /m l Time in hrs Fermentor 1litre, 600 rpm, Air 0.8:1 interval experim.EPS X cells Y cells Product EPS Total biomass Fig. 4. Solution of model (2), 600 rpm The values of the parameters numerically iden- tified in model (2) are as follows: k1 = 5.8,k2 = 2.8,kd = 0.005,α = 9,β = 9, γ = 1.7,s(0) = 0.9,x(0) = 0.011,y(0) = p(0) = p1(0) = 0. The graphs of the solutions fitting the experimental data are visualized in Fig. 4. A very good fit of the biomass, as well as of the product EPS, can be observed. Model (2) has been fitted for the measurement data obtained with other agitation rates. The re- sults are always good within the accuracy of the experimental measurements. Figures 5 and 6 are examples of the results obtained for 500 rpm and 100 rpm. 0 0.5 1 1.5 2 0 5 10 15 20 B io m a ss a n d E P S m g /m l Time in hrs Fermentor 1litre, 500 rpm, Air 0.8:1 Interval experim. Biomass interval experim.EPS X cells Y cells Product EPS Total biomass Fig. 5. Solution of model (2), 500 rpm 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 5 10 15 20 B io m a ss a n d E P S m g /m l Time in hrs Fermentor, 1litre; 100 rpm, Air 0.8:1 Interval experim. Biomass interval experim.EPS X cells Y cells Product EPS Total biomass Fig. 6. Solution of model (2), 100 rpm The coefficients obtained for the best fit at various agitation rates reported in Figure 1 are given in Table II. rpm k1 k2 α β γ s0 x0 100 83 5.5 17 17 2.7 0.32 0.002 400 13 2.8 9 9 1.7 0.6 0.01 500 5.5 2.7 9 9 1.5 0.9 0.01 600 5.8 2.8 9 9 1.7 0.9 0.01 800 9 2.8 9 9 2.5 0.6 0.01 TABLE II COEFFICIENTS OF MODEL (2) FOR VARIOUS AGITATION RATES It can be seen from Table II that the two coeffi- cients α and β are always equal. An interpretation could be that newborn cells belong wth equal probability to one of the two fractions (of dividing or non-dividing cells). Concerning the other coeffi- cients k1,k2,γ, they have minimal values for 500– 600 rpm. Note that these are the agitation rates corresponding to a maximum product synthesis. For model (2) another “good” set of parameters is found to be the following k1 = 3,k2 = 1.4,α = 6.5,β = 6.5,γ = 1.2,kd = 0.005,s0 = 1.65,x0 = 0.011,y0 = p0 = 0. The explanation of this phenomenon is the fact that the fitting of the mathematical solutions to the experimental data has been done with an optimization method Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 7 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... which minimizes the sum of the squared differ- ences between the experimental values and the mathematical values provided by the model (least square approximation) and the objective function has several local minima leading to several sets of parameters. To conclude, Model (2) provides a very good representation of the experimental data. The model is a modification of an ODE model proposed in [1]. C. Model 3 Model 3 is a particular case of model (2). Here we assume that the rate constant α = 0, that is the first reproduction reaction step P +Y α−→ X+Y is omitted. Consider then the following reaction scheme: S + X k1−→ Y k2−→ P + X, (RSG) P + Y β −→ 2Y, (RSR) P + Y γ −→ Y + P1, (RSP) X kd−→ Q, (RSD) It implies the following system of ODE’s: ds/dt = −k1xs dx/dt = −k1xs + k2y −kdx dy/dt = k1xs−k2y + βpy dp/dt = k2y − (β + γ)py dp1/dt = γpy (3) The values of the parameters in model (3) are identified as follows: k1 = 20,k2 = 5,β = 1.5,γ = 0.15,kd = 0.005;s0 = 2.2,x0 = 0.02,y0 = 0,p0 = p1(0) = 0. The graphs of the solutions fitting real experimental data are visualized in Figure 7. The fit of the biomass and EPS to the experimental data is almost as good as in model (2). This shows that the reaction step P+Y α−→ X+Y does not influence much the final solutions and can be omitted. The omission of this step can be interpreted as postulating that newborn cells are always in dividing state, which is biolog- ically relevant [5]. Of course the division process can be inhibited in case that meanwhile environ- mental conditions become unfavorable. Model (3) is a modification of a model proposed in [10]. 0 0.5 1 1.5 2 0 5 10 15 20 B io m a ss a n d p ro d u ct m g /m l Time in hrs Aeribacillus pallidus 418, Fermentor 1 litre X -cells Y - cells Product EPS Total biomass x+y Fig. 7. Solution of model (3), 600 rpm VI. CONCLUSION We described the dynamics of microbial growth and EPS synthesis using several mathematical models. The first model is a classical one, while models 2 and 3 are original structured models formulated in terms of reaction schemes, thereby the reaction growth step is borrowed from Henri- Michaelis-Menten enzyme kinetics. As shown in [10] a reaction scheme using such a reaction step generalizes classical models like Model 1. The suggested models differ in the choice of the reaction steps. Compared to classical Model 1 the proposed models produce a better fit for the cell biomass experimental data. To explain this, note that classical Monod type models make use of the so-called “specific growth rate function” µ(s). Recall that this function co- incides (up to a multiplier) with the nutrient substrate uptake rate µ(s) in Michaelis-Menten enzyme kinetics. However, Michaelis-Menten sub- strate uptake function µ(s) is an approximation of the substrate rate function induced by the Henri- Michaelis-Menten mechanism under the assump- tion that the enzyme/substrate ratio is small. The latter may not be true in vivo, as this ratio is not small in the cell cytoplasm [14], [15]. Note that our proposed models keep close to Henri-Michaelis- Menten mechanism, which is independent on the Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 8 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... value of the enzyme/substrate ratio. Bacterial cells are metabolically active not only in the dividing state but in the non-dividing state as well [13]. Hence, when formulating reaction steps corresponding to these two states one should take into account such metabolic activities. Our dynamical models based on reaction schemes are first steps in this direction. We hope that the proposed models throw some light on the related cell growth mechanisms. Indeed, the reaction steps used in our models presume metabolic activity of the cell in its non-dividing state. Our proposed models conform with the hypoth- esis from [10] saying that the reaction terms in cell growth dynamical systems can be deduced from appropriate biochemical reaction schemes with constant reaction rates via mass action law. Such an approach contributes to the interpretation of the underlying biological mechanism of the cell growth phenomena. The biological experiments have been per- formed with highest possible precision in order to serve further for accurate numerical simulations. The overall goal is to verify as much as possible the reaction scheme approach for cell growth and product synthesis modelling and possibly deter- mine a set of adequate reaction steps. A most useful feature of the proposed reaction scheme approach is its methodological value. This ap- proach allows the biologists to focus on the model formulation in terms of reaction schemes as done in biochemistry; thus biologists may not need to formulate their models directly in terms of differ- ential equations. Once the model is formulated in terms of reaction schemes, then the remaining part of the modelling process can be rather automated. Of course a theoretical (mathematical) study of the model, e.g. with respect to stability, may also be of scientific interest. A most important conclusion (in our opinion) is that the modelling approach based on the subdivi- sion of the cell population into two fractions (of dividing and non-dividing cells) proves as practi- cally useful. This biological paradigm is different from the classical one based on the biomass phases (lag, log, etc) which implies simultaneous transi- tions of all cells from one biomass phase to the other. The classical approach is oriented towards properties of the whole bio-population, whereas our paradigm is oriented towards the individual cell cycles. Individual cell cycles are considered in [6], [7] under the hypothesis that a certain part of the cell lifetime is devoted to maturing and another part to division. This interpretation is in accord with recent findings of cell biology [5] and is confirmed by the proposed models. We hope that our models contribute to a clar- ification of the biological mechanisms for the transition of an individual cell from a non-dividing to a dividing state (or conversely). Assume that the biomass has been inhibited or stressed for a sufficiently long period and (almost) all cells are in non-dividing state. When the environmen- tal conditions become favorable, the cells start to divide, however this process does not happen simultaneously in time for all cells. The precise mechanism for this transition is not known; it is suggested that cell signalling (quorum sensing) plays an important role in this process. Some cells are able to quickly respond to environmental changes and then transmit signals to other cells. In certain time periods there exist significant numbers of both cell populations and this biologically re- alistic assumption is confirmed by the proposed models involving two cell fractions. Future work. Future work is foreseen in the fol- lowing directions: A. To study the biological rel- evance of the proposed reaction steps and the so- lutions of the corresponding dynamic systems; B. To consider continuous fermentation experimental data and appropriate modelling within the outlined methodology; C. To provide suitable fermentation experimental data for studying mathematically in- hibition phenomena; D. To study mathematically the proposed dynamic systems with respect to stability. Biomath 4 (2014), 1407121, http://dx.doi.org/10.11145/j.biomath.2014.07.121 Page 9 of 10 http://dx.doi.org/10.11145/j.biomath.2014.07.121 N. Radchenkova et al., On the Mathematical Modelling of EPS Production ... ACKNOWLEDGMENT The authors are grateful to the National Fund for Scientific Research, Bulgaria, for financial support of this work (Contract DTK 02/46). They are also extremely grateful to both anonymous reviewers for their deep, detailed, very competent, construc- tive and beneficial comments and critical remarks. REFERENCES [1] Alt, R., S. Markov, Theoretical and computational stud- ies of some bioreactor models, Computers and Mathe- matics with Applications 64 (2012), 350–360. http://dx.doi.org/10.1016/J.Camwa.2012.02.046 [2] Bastin G., D. Dochain, On-line Estimation and Adap- tive Control of Bioreactors. Process measurement and control, Elsevier, 1990. http://dx.doi.org/10.1002/cite.330630220 [3] Burhan, N., Ts Sapundzhiev, V. 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