Introduction

The number of industrial bioprocesses conducted by host microorganisms seems to be growing (Nielsen et al. 2022). Each of these processes, regardless if it is a continuous, semi-continuous, or batch process, needs to be designed, controlled, and optimized. The simulations and predictions obtained from mathematical models of these processes are not overvalued in these aspects (Luo et al. 2021; Rathore et al. 2021).

There are two main approaches to bioprocess modeling (Luo et al. 2021). First, it is quite popular nowadays based on data-driven models including artificial intelligence, and second based on mechanistic models. Data-driven models as the name suggests need a lot of data to be calibrated and used to simulate and optimize different processes (Luo et al. 2021). This is one important disadvantage of this approach as some of the process variables are difficult or even impossible to be measured. Furthermore, they do not explain the mechanism of the process, unlike mechanistic models. Mechanistic models are derived from known physical, chemical, and biological laws and/or phenomenological relationships and are bringing a better process understanding and the possibility of simulating unmeasurable variables (Luo et al. 2021). In this group of models, we can distinguish between flux-based and kinetic-based models.

Flux-based models can be used to describe the biochemical pathways in living organisms. A good example is a method called flux balance analysis (FBA) (Orth et al. 2010). It was shown that it can be used to modify and optimize culture medium composition or even bacterial strain phenotype by genetic modifications in order to obtain high bioprocess efficiency (Lee et al. 2006; Raman and Chandra 2009; Swayambhu et al. 2020). These complex models can provide a huge amount of information about microorganisms’ physiology; however, in most cases, they are based on steady-state assumption, and they cannot be used to model growth kinetics (Orth et al. 2010; Stryjewski et al. 2015). On the other hand, there are kinetic-based models which are often based on Monod’s equation with different modifications (Monod 1949). This type of modeling often found a lot of applications including organic substances, ammonia nitrogen, and phosphorous removal by activated sludge (Hauduc et al. 2013; Henze et al. 2000; Hu et al. 2007).

In these kinds of models, unlike the flux-based models, the biomass of bacteria is treated as a single component, and the whole complexity of living organisms is brought to one variable. This simplification is reasonable in many cases; however, in many others, more sophisticated models have to be applied to catch observed phenomena (Jager et al. 2006; Stryjewski et al. 2015). On the other hand, too complex modes can have a large number of variables and parameters that can be difficult to measure or estimate (Luo et al. 2021). Therefore, the model complexity has to be chosen according to specific needs arising from certain bioprocess features.

The basic idea behind increasing the complexity of the model is to divide microorganisms’ cells into compartments and describe them with different variables. These kinds of models are often called structured models (Morchain & Fonade, 2009; Stryjewski et al. 2015). One of the first two-compartment models used to explain such phenomena as changes in size and composition of cells at higher growth rates, and cell division after nutrients were removed, and many others, was proposed by Williams (Williams 1967).

It seems that the application of structured models in the case of processes with substrate limitations is a reasonable idea. Multiple substrate limitations can influence the kinetics and stoichiometry of bacteria growth and therefore it can have a large impact on the efficiency of bioprocess (Zinn et al. 2004). Depending on the limiting substrate concentrations and substitutability, different effects on the growth rate can be observed (Zinn et al. 2004). Moreover, the elemental composition of microorganisms, DNA, RNA, enzymes, and other intracellular components contents can be changed (Pramanik and Keasling 1997; Zinn et al. 2004).

Therefore, in this study, we evaluate the Dynamic Energy Budget (DEB) model used to describe E. coli growth in a carbon and nitrogen substrate limitation. DEB theory and model is a complex mass and energy balance prepared for living organisms (Kooijman 2010). It describes how an organism obtains mass and energy from the environment and uses it to maintain life, grow, and reproduce. It can be classified as a structured model because it assumes that organisms can be divided into structures that require energy to be maintained and reserves that do not. DEB model was successfully used to investigate the influence of food availability, climate changes, and toxic substances on different species, modeling population dynamics, or organizing principles for metabolism (Jager 2020; Jager et al. 2016; Kearney 2021; Marques et al. 2018).

Escherichia coli is a Gram-negative, rod-shaped bacteria that is capable of conducting aerobic respiration in the presence of oxygen and fermentation when oxygen is absent. We choose this species as its morphology, physiology, and genome are well-studied and it plays a huge role in biotechnology and industrial microbiology as a model and host organism (Baniasad and Amoozgar 2015; Blount 2015). It is used in the industry for the production of for instance insulin, erythropoietin, and other recombinant therapeutic proteins as well as biofuels and industrial chemicals like phenol or mannitol (Blount 2015).

Materials and methods

Model derivation

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.

Fig. 1
figure 1

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

Full size image
Table 1 Symbols of DEB model and Monod’s model parameters with their estimated or fixed values
Full size table

The specific assimilation flux \({j}_{{E}_{i}A}\) of substrates C and N are given by Monod-type equation:

$${j}_{{E}_{i}A}={j}_{{E}_{i}Am}\frac{{S}_{i}}{{K}_{i}+{S}_{i}}$$
(1)

where \(i\) is the index denotes substrate C or N; \({j}_{{E}_{i}Am}\) is the maximum specific assimilation flux, \({S}_{i}\) is substrate concentration (C or N) [C-molSC L−1] or [molSN L−1]; \({K}_{i}\) 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 \({j}_{{E}_{i}C}\). Mobilization specific flux is proportional to the reserves density \({m}_{{E}_{i}}\) in [C-molEC molV−1] or [molEN molV−1]:

$${j}_{{E}_{i}C}={m}_{{E}_{i}}\left({\dot{k}}_{E}-\dot{r}\right)$$
(2)

where \({\dot{k}}_{E}\) is the reserve turnover rate; \(\dot{r}\) is the specific growth rate \(\frac{1}{{M}_{V}}\frac{d{M}_{V}}{dt}\) where \({M}_{V}\) is structural mass in [C-molMV]. The first-order dynamics follow directly from the assumptions of DEB theory (Kooijman 2010). Somatic and maturity maintenance \({j}_{{E}_{i}M}\) are paid from mobilization flux \({j}_{{E}_{i}C}\). The remaining part of \({j}_{{E}_{i}C}\) goes to growth flux \({j}_{{E}_{i}G}\); therefore, it can be expressed as:

$${j}_{{E}_{i}G}= {j}_{{E}_{i}C}-{j}_{{E}_{i}M}$$
(3)

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

$${j}_{G}={\left[{{j}_{P}}^{-1}+\sum_{i\in (C,N)}{\left(\frac{{j}_{{E}_{i}G}}{{y}_{{E}_{i}V}}\right)}^{-1}-{\left(\sum\nolimits_{i\in (C,N)}\frac{{j}_{{E}_{i}G}}{{y}_{{E}_{i}V}}\right)}^{-1}\right]}^{-1}$$
(4)

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

$${j}_{G}={\left[\sum\nolimits_{i\in (C,N)}{\left(\frac{{j}_{{E}_{i}G}}{{y}_{{E}_{i}V}}\right)}^{-1}-{\left(\sum\nolimits_{i\in (C,N)}\frac{{j}_{{E}_{i}G}}{{y}_{{E}_{i}V}}\right)}^{-1}\right]}^{-1}$$
(5)

The \({y}_{{E}_{i}V}\) represents the yield coefficient or conversion factor used to convert reserves flux \({j}_{{E}_{i}G}\) into corresponding structures flux \({j}_{G}\) and vice versa. The chemical composition of structures cannot be changed; therefore, an excess of \({j}_{{E}_{i}G}\) is rejected as \({j}_{{E}_{i}R}\) which can be partially returned into reserves \({{\kappa }_{{E}_{i}}j}_{{E}_{i}R}\), or excreted from the cells \({\left({1-\kappa }_{{E}_{i}}\right)j}_{{E}_{i}R}\). The rejected flux \({j}_{{E}_{i}R}\) can be thus expressed as:

$${j}_{{E}_{i}R}={j}_{{E}_{i}G}-{y}_{{E}_{i}V}{j}_{G}$$
(6)

The structures can be used to pay maintenance costs in case when \({j}_{{E}_{i}C}\) 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):

$${j}_{{VM}_{i}}=\left({j}_{{E}_{i}M}-min\left({j}_{{E}_{i}C},{j}_{{E}_{i}M}\right)\right){{y}_{{E}_{i}V}}^{-1}$$
(7)

and the general \({j}_{VM}\) flux as:

$${j}_{VM}=\sum\nolimits_{i\in (C,N)}{j}_{{VM}_{i}}$$
(8)

The specific growth rate is simply the difference between \({j}_{G}\) and \({j}_{VM}\):

$$\dot{r}={j}_{G}-{j}_{VM}$$
(9)

However, in this study, we assume a negligible decrease in structure and therefore \(\dot{r}={j}_{G}\). The mass balance for substrates \({S}_{i}\), reserves \({m}_{{E}_{i}},\) and structure \({M}_{V}\) is given by the set of equation:

$$\frac{d}{dt}{S}_{i}=\left[-{j}_{{E}_{i}A}+{{\kappa }_{{S}_{i}}\left({1-\kappa }_{{E}_{i}}\right)j}_{{E}_{i}R}\right]{{y}_{{SE}_{i}}\frac{1}{V}M}_{V}+\frac{1}{V}{\dot{J}}_{{NH}_{3}}(only\;for\;{S}_{N})$$
(10)
$$\frac{d}{dt}{m}_{{E}_{i}}={j}_{{E}_{i}A}-{j}_{{E}_{i}C}+{{\kappa }_{{E}_{i}}j}_{{E}_{i}R}-\dot{r}{m}_{{E}_{i}}$$
(11)
$$\frac{1}{V}\frac{d}{dt}{M}_{V}=\dot{r}\frac{1}{V}{M}_{V}$$
(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 \({\dot{J}}_{{NH}_{3}}\) [mol h−1] which is the total flux of metabolite \({\text{NH}}_{3}\) (see SI for more details about mass balance).

To eliminate the unphysical model solution, two constraints were defined. First, if \({j}_{{E}_{i}C}<{j}_{{E}_{i}M}\), 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 \({j}_{VM}\) = 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 \({j}_{{E}_{i}A}\) 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).

Monod’s equation

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

$$\frac{dX}{dt}=\mu X$$
(13)
$$\frac{d{S}_{C}}{dt}=-\frac{1}{{Y}_{X{S}_{C}}}\mu X$$
(14)
$$\frac{d{S}_{N}}{dt}=-\frac{1}{{Y}_{X{S}_{N}}}\mu X$$
(15)

where \(X\) is biomass concentration [C-mol L−1], and \({S}_{C}\) and \({S}_{N}\) 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 \({Y}_{X{S}_{C}}\) and \({Y}_{X{S}_{N}}\) 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):

$$\mu = {\mu }_{max}\frac{{S}_{C}}{{K}_{S}+{S}_{C}}\frac{{S}_{N}}{{K}_{N}+{S}_{N}}$$
(16)

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

Growth kinetics

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).

Shock limitation

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.

Biomass composition

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.

Parameter estimation

The DEB model parameters (\({j}_{{E}_{i}Am}\), \({j}_{{E}_{i}M}\)) and Monod’s model parameters (\({\mu }_{max}\), \({Y}_{X{S}_{C}}\), and \({Y}_{X{S}_{N}}\)) 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 \({\dot{k}}_{E}\) 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 \({X}_{j}\), C substrate concentration \({S}_{Cj}\), and N substrate concentration \({S}_{Nj}\) in six different limitation scenarios (\(j\) = 1,2,…,6) was given by the equation:

$$WSSE=\sum_{j=1}^{m}\left({W}_{Xj}\sum_{i=1}^{n}{\left[{X}_{j}\left({t}_{i},\mathbf{p}\right)-{Y}_{** the system dynamic and thermodynamically constrained. These features of the presented model can be used for optimization, prediction, and control of different bioprocesses conducted by microorganisms.