Dynamic Energy Budget model is an energy balance that can be formulated for any living organism (Kooijman 2010). The model can be classified as a structured one because the biomass of modeled organisms can be divided into two main groups of compartments: structures, which require energy to maintain, and reserves which do not. According to the model, an organism obtains energy from the environment through its surface and stores it in reserves. Assimilated energy can then be used for different purposes. A fixed fraction κ of mobilized energy is used with priority for somatic maintenance and for the increase in the mass of structures that define growth in the DEB context. Maturation, maturity maintenance, and reproduction are powered by the remaining fraction of mobilized energy (1 − κ) (Kooijman 2010). The reserves play an important role in DEB models as they enable to include metabolic memory, to smooth out fluctuations in substrate availability, and capture changes in the chemical composition of organisms (Kooijman 2010). Moreover, it is possible to build different DEB models with multiple reserves and structures when it is needed. More details about the DEB model and theory can be found in Kooijman (2010).

In terms of DEB theory, the bacteria culture in a liquid medium can be modeled as one organism. It means that state variables like volume, mass of structures, and mass of reserves of single bacteria can be simply added to each other to obtain values for the whole population living in the bioreactor. Such model organism grows by cell divisions, increasing its volume and surface area proportionally to the number of cells. Therefore, it has to be considered a V1-morph, an organism which surface area is proportional to its volume during growth (Grossowicz et al. 2017; Kooijman 2010; Livanou et al. 2019; Lorena et al. 2010). Some physiological processes like nutrient uptake are proportional to the surface area of the organism, and others like maintenance costs are proportional to the volume of the organism (Kooijman 2010). The ratio between surface area and volume has a huge influence on the organism’s metabolism (Kearney 2021; Kooijman 2010).

In the present study, we conducted the bioprocess with a sole carbon source of glucose and a sole nitrogen source of NH4+. In that case, the carbon and nitrogen assimilation pathways in E. coli are basically independent (Kim and Gadd 2019; Reitzer 2003; Willey et al. 2020). The carbon is assimilated through glycolysis and the citric acid cycle and the nitrogen through the reductive amination pathway or glutamine synthetase-glutamate synthetase system (Kim and Gadd 2019; van Heeswijk et al. 2013; Willey et al. 2020). The strong homeostasis assumption in DEB theory implies that the chemical composition of the reserve or structure does not change in living organisms (Kooijman 2010). Therefore, to model two or more independent assimilation pathways, we need to consider the multi-reserve system, with distinct reserves for each pathway. We assume that the assimilation of substrates is conducted directly from the culture medium, which simplified the model by omitting the feeding rate (Grossowicz et al. 2017; Kooijman 2010; Livanou et al. 2019; Lorena et al. 2010). The scheme of the model is presented in Fig. 1, and the symbols of parameters with units are listed in Table 1; fluxes are given in C moles or moles of certain substances per hour.

Scheme of DEB model for microorganism population with two reserves and one structure. The dotted line represents the cell surface, and the arrows represent mass fluxes and are described by the appropriate symbols (only for substrate A and reserve A path, as the path for B can be described analogously). SU stands for the synthesizing unit

The specific assimilation flux \(__A}\) of substrates C and N are given by Monod-type equation:

$$__A}=__Am}\frac_}_+_}$$

(1)

where \(i\) is the index denotes substrate C or N; \(__Am}\) is the maximum specific assimilation flux, \(_\) is substrate concentration (C or N) [C-molSC L−1] or [molSN L−1]; \(_\) is the half saturation concentration for C or N. Assimilated substrates are stored in reserves. The reserves are mobilized, which is represented by the mobilization flux \(__C}\). Mobilization specific flux is proportional to the reserves density \(__}\) in [C-molEC molV−1] or [molEN molV−1]:

$$__C}=__}\left(}_-\dot\right)$$

(2)

where \(}_\) is the reserve turnover rate; \(\dot\) is the specific growth rate \(\frac_}\frac_}\) where \(_\) is structural mass in [C-molMV]. The first-order dynamics follow directly from the assumptions of DEB theory (Kooijman 2010). Somatic and maturity maintenance \(__M}\) are paid from mobilization flux \(__C}\). The remaining part of \(__C}\) goes to growth flux \(__G}\); therefore, it can be expressed as:

The maturation process is not modeled in the case of microorganisms (Grossowicz et al. 2017; Kooijman 2010; Livanou et al. 2019; Lorena et al. 2010). The synthesizing unit SU is responsible for integrating growth fluxes from different reserves, in this study from C reserves and N reserves.

The specific growth fluxes \(__G}\) are parallel and complementary and they interact with each other in SU to create specific growth flux \(_\) (Grossowicz et al. 2017; Kooijman 2010; Livanou et al. 2019; Lorena et al. 2010):

$$_=_}^+\sum___G}}__V}}\right)}^-\frac__G}}__V}}\right)}^\right]}^$$

(4)

For more details, please see Supporting Information (Fig. S9). Assuming that the specific rate of product formation \(_\) is much larger than the fluxes entering the SU, the equation can be simplified:

$$_=__G}}__V}}\right)}^-\frac__G}}__V}}\right)}^\right]}^$$

(5)

The \(__V}\) represents the yield coefficient or conversion factor used to convert reserves flux \(__G}\) into corresponding structures flux \(_\) and vice versa. The chemical composition of structures cannot be changed; therefore, an excess of \(__G}\) is rejected as \(__R}\) which can be partially returned into reserves \(__}j}__R}\), or excreted from the cells \(__}\right)j}__R}\). The rejected flux \(__R}\) can be thus expressed as:

The structures can be used to pay maintenance costs in case when \(__C}\) is insufficient. Each of the component fluxes can be expressed by the Switch Model (Grossowicz et al. 2017; Kooijman 2010; Livanou et al. 2019; Lorena et al. 2010):

$$__}=\left(__M}-min\left(__C},__M}\right)\right)__V}}^$$

(7)

and the general \(_\) flux as:

$$_=\sum\nolimits___}$$

(8)

The specific growth rate is simply the difference between \(_\) and \(_\):

However, in this study, we assume a negligible decrease in structure and therefore \(\dot=_\). The mass balance for substrates \(_\), reserves \(__},\) and structure \(_\) is given by the set of equation:

$$\frac_=\left[-__A}+__}\left(__}\right)j}__R}\right]__}\fracM}_+\frac}__}(only\;for\;_)$$

(10)

$$\frac__}=__A}-__C}+__}j}__R}-\dot__}$$

(11)

$$\frac\frac_=\dot\frac_$$

(12)

where \(V\) denotes the volume of growth medium in the bioreactor (here \(V\) = 1L). Moreover, ammonia can be produced as a metabolite (overheads of assimilation and growth, dissipation) and excreted from the cell increasing the total concentration of ammonia in the medium. It has to be included in mass balance and calculations by introducing \(}__}\) [mol h−1] which is the total flux of metabolite \(}_\) (see SI for more details about mass balance).

To eliminate the unphysical model solution, two constraints were defined. First, if \(__C}<__M}\), the maintenance is not paid and the organism dies. The death means that all metabolic fluxes are equal 0, the substrates are not being uptake anymore, minerals are not produced, the reserves are not mobilized, and the growth does not continue. We assume that up to a few hours after death the bacterial cells do not undergo destruction and can be detected by our analytical methods. Note that maintenance can also be filled from structures; however, in this study, we assumed that \(_\) = 0 mainly because we did not obtain sufficient experimental evidence of this process. Second, if the concentration of substrate \(i\) is equal to or lower than 0, the assimilation flux \(__A}\) is also equal to 0. This constraint prevents solving the model equations for unrealistic negative values of substrate concentrations, and is important in the case of cell transfer to limitation media (Chapter 3 in SI).

The derived DEB model was compared to often-used growth kinetics given by the set of three equations:

$$\frac_}=-\frac__}}\mu X$$

(14)

$$\frac_}=-\frac__}}\mu X$$

(15)

where \(X\) is biomass concentration [C-mol L−1], and \(_\) and \(_\) carbon and nitrogen substrate concentration in the medium [C-mol L−1] or [mol L−1]. Note that the yields of biomass production for C and N substrates \(__}\) and \(__}\) are assumed to be constant during the process. The specific growth rate \(\mu\) was given by the Monod equation (Monod 1949) with an extension to two limiting substrates (Zinn et al. 2004):

$$\mu = _\frac_}_+_}\frac_}_+_}$$

(16)

where \(_\) and \(_\) are half saturation constants for C and N substrates, respectively.

Escherichia coli strain (PCM 2057) was obtained from Ludwig Hirschfeld Institute of Immunology and Experimental Therapy-Polish Academy of Science. E. coli cultures were prepared in sterile Erlenmeyer flasks by transferring 1 ml of inoculum (see SI) to 100 ml of appropriately modified M9 liquid medium (Tab. S1 and S2).

In order to determine the growth kinetics in carbon-limiting conditions, three media were prepared (Tab. S2). The concentration of NH4Cl was 0.075 gL−1 (150% of NH4Cl concentration in the basic medium), while the concentration of glucose was reduced to 25, 50, and 75% of the basic concentration (1, 2, and 3 gL−1, respectively). Such conditions provide unlimited access to NH4Cl with the simultaneous depletion of glucose after some time of culturing.

In the case of experiments determining growth limitation by nitrogen, the basic concentration of glucose was increased by 50% to ensure its excess, while the concentration of NH4Cl was reduced to 0.075, 0.125, and 0.25 gL−1 (15, 25, and 50% of NH4Cl basic concentration) (Tab. S2). The concentration of NH4Cl was lowered more than the concentration of glucose because nitrogen limitation is less severe for the bacteria and growth stops much later.

The cultures were incubated at 37 °C on a rotary shaker (at 160 rpm, IKA KS 4000, Germany). The samples were collected in triplication, every hour while the culture was being grown. The concentrations of bacteria cells (optical density at 550 nm), glucose concentration (enzymatic test, Biomaxima, Tab. S3), and nitrogen concentration (colorimetric cuvette test, Spectroquant, Merck) were measured in each sample (please see SI for details, Fig. S1-S5).

The batch culture was chosen mainly because most industrial bioprocesses are conducted in fed-batch not in continuous reactors (Luo et al. 2021). Moreover, the control and optimization of batch processes using mechanistic models seems to be one of the challenges of modern industrial microbiology (Luo et al. 2021; Rathore et al. 2021).

In order to determine shock limitation, the basic M9 medium (4 gL−1 glucose and 0.05 gL−1 NH4Cl) was inoculated and cultured as described above (section “Growth kinetics”) for 4 h. The concentrations of bacteria cells, glucose, and nitrogen were controlled. The culturing was sopped in the mid-log growth phase, the bacterial suspensions were centrifuged (Hettich Universal 320R, 4 min, 4000 rpm), and the cell pellet was separated from the supernatant. The cells were transferred to two different fresh (sterile) mediums.

The first flask consisted of 100 mL of basic M9 medium deprived of glucose and the second one was without NH4Cl. The whole transfer procedure lasted app. 30 min. Then, the cultures were again incubated at 37 °C with shaking at 160 rpm on a rotary shaker. The samples were collected in triplication every 30 min. As previously, the concentrations of bacteria cells, glucose, and nitrogen were measured.

C, H, N, and S content was determined separately in cultures where growth was stopped due to lack of carbon or nitrogen source. The C-limited culture was conducted in an M9 medium with glucose and NH4Cl in initial concentrations of 1.0 g L−1 and 0.75 gL−1, respectively. The N-limited culture was prepared analogically with the initial concentration of glucose 6 gL−1 and NH4Cl 0.075 gL−1. The medium was inoculated and incubated for 8 h in previously described conditions (section “Growth kinetics”). After this time, when the logarithmic growth phase was finished and the stationary phase began, the culturing was stopped. The bacterial suspensions were centrifuged (Hettich Universal 320R, 20 min, 9000 rpm), and the pellet was separated from the supernatant and dried for 24 h (Memmert, 45 °C). The content of C, H, N, and S in dry bacterial pellets was determined by CHNS Elemental Analyzer Vario EL Cube, with acetanilide as a standard (in the external laboratory). The content of C, H, and N in biomass was used to determine the molecular weights and elemental composition of structure and biomass in different limitations conditions. The stoichiometry of C-limited and N-limited biomass was used to calculate the reserves densities in both cases. See Supporting Information for details.

The DEB model parameters (\(__Am}\), \(__M}\)) and Monod’s model parameters (\(_\), \(__}\), and \(__}\)) were fitted simultaneously to all datasets containing time-depending mean values of biomass, glucose, and ammonium concentrations obtained for different C and N limitation conditions (see more details in SI). The reserve turnover rate \(}_\) was estimated separately from C and N limitations data, and its mean value was used (see SI). The weighted nonlinear least-square method was used. The loss function, weighted sum of squared errors (WSSE), for three different variables biomass \(_\), C substrate concentration \(_\), and N substrate concentration \(_\) in six different limitation scenarios (\(j\) = 1,2,…,6) was given by the equation:

$$WSSE=\sum_^\left(_\sum_^_\left(_,\mathbf\right)-_\right]}^+__j}\sum_^_\left(_,\mathbf\right)-__j}\right]}^+__j}\sum_^_\left(_,\mathbf\right)-__j}\right]}^\right)$$

where \(n\) is the number of data points for each variable in each scenario (here unchanged between scenarios); \(_\), \(__j}\), and \(__j}\) represent data points for \(i\) th time and \(j\) th scenario; \(_\), \(__j}\), and \(__j}\) are the weight coefficients for each variable dataset in each limitation scenario. The weight coefficient was defined as a reciprocal of mean variance of measured values of one variable in a certain limitation scenario.

Note that the variance \(^}_\) is calculated for each variable at each time of incubation as they were measured in three repetitions. The loss function was minimized using the Nelder-Mead Simplex Method in Matlab (Lagarias et al. 1998). The set of DEB model and Monod’s model differential equations were solved using the Runge–Kutta method in each step of loss function minimization. The goodness of fit was described by the adjusted coefficient of determination (\(^}_\)), and root mean square error (RMSE). More details about the parameter values estimation procedure can be found in SI.