CET vol 99


 
 
 
 
 
 
 
 
 
 
                                                                                                                                                                 DOI: 10.3303/CET2399099 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Paper Received: 21 December 2022; Revised: 29 March 2023; Accepted: 26 April 2023 
Please cite this article as: Kemesis B., Urniezius R., Vaitkus V., Survyla A., Simutis R., Babilius P., Zlatkus L., 2023, Modeling the Glucose 
Concentration for the Recombinant E.coli Bioprocesses, Chemical Engineering Transactions, 99, 589-594  DOI:10.3303/CET2399099 
  

 CHEMICAL ENGINEERING TRANSACTIONS  
 

VOL.  99, 2023 

A publication of 

 

The Italian Association 

of Chemical Engineering 

Online at www.cetjournal.it 

Guest Editors: Sauro Pierucci, Flavio Manenti 

Copyright © 2023, AIDIC Servizi S.r.l. 

ISBN 978-88-95608-98-3; ISSN 2283-9216 

Modeling the Glucose Concentration for the Recombinant 

E.coli Bioprocesses 

Benas Kemesis, Renaldas Urniezius*, Vygandas Vaitkus, Arnas Survyla, Rimvydas 

Simutis, Povilas Babilius, Lukas Zlatkus 

Kaunas University of Technology, Dept of Automation, Kaunas LT-51368, Lithuania 

renaldas.urniezius@ktu.lt 

This article presents the sophisticated-to-date carbon mass balance for fed-batch E.coli bioprocesses. The 

model originates from the distribution of carbon mass from glucose in biomass, off-gas, and hypothetical solutes. 

The suggested model complements Pirt's equation as a particular case scenario. The approach uses the linear 

relationship between biomass carbon content per carbon grams in glucose and average cell population age. 

The carbon balance brings two potential practical benefits. First, it has the potential to assess the type of cell 

metabolism pathway and to have a soft sensor for the concentration of dissolved products such as acetates. 

The measure of glucose concentration suggests another finding, assuring the reliance on off-gas information 

only. The paper introduces an average carbon content ratio in biomass and off-gas, with numerical values of 

0.5 in growth-limiting experiments and 0.27 in nonlimiting ones, which may serve as a decision-making criterion 

for metabolic pathway detection in the future. 

1. Introduction 

Modeling glucose concentration in nonstationary fed-batch processes remains desired in research and 

development institutions and industrial installations. Therefore, corresponding dedicated online control 

techniques are unavailable. Currently, the regulators are present for bioreactor-related fed-batch setpoints 

(Levisauskas et al., 2019) and growth-limiting substrate feed for fed-batch (Galvanauskas et al., 2021) 

cultivations. Up until now, the state variables’ estimation is accessible through off-gas-based analytics routines 

providing soft sensors for microbial biomass (Urniezius and Survyla, 2019), targeting non-soluble recombinant 

product (Urniezius et al., 2021), and the specific biomass growth rate (Survyla et al., 2021). All mentioned 

methods are noninvasive and rely on exhaust gas mixture information. 

Moreover, they share a common benefit: such approaches do not suffer issues when the nutrition medium is 

nonhomogeneous, making them acceptable for industrial scale-up efforts. The reason is that the exhaust gas 

presents cumulative knowledge with the trade-off of unavoidable estimation delay due to the bioreactor outlet 

channel length and broth‘s changing volume. Therefore, the analytical optimization and resolution of crucial 

state parameters in fed-batch recombinant bioprocesses are still challenging, involving multiple dilutions and 

time constants due to asynchronous observations from different external sensors. 

So far, the specific glucose consumption rate behavior was modeled offline in both growth-limiting and repeated 

batch microbial cultivations (Survyla et al., 2022). Glucose concentration modeling efforts returned accumulated 

errors due to minor deviations in the substrate concentration and flow rate (Schubert et al., 1997). Building the 

parametric relationship with physiological meaning is ongoing. Therefore, we share an approach for glucose 

concentration modeling in fed-batch aerobic bacterial cultivations. The aim is to resolve the challenges of 

synchronizing different state variables: glucose concentration, the carbon content in biomass, and bacteria 

population age linked to the off-gas biosynthesis information. 

589



2. Metabolic Pathways 

The oxidative metabolism path is the desired metabolic pathway in aerobic fed-batch cultivations of recombinant 

E. coli. The general equation for aerobic respiration (Patel et al., 2022) 

𝐶6𝐻12𝑂2 + 6𝑂2 → 6𝐶𝑂2 + 6𝐻20 + 𝐸𝑛𝑒𝑟𝑔𝑦. (1) 

From a fed-batch process perspective, the latter equation expands by introducing the additional chemical 

compounds involved: 

𝐶6𝐻12𝑂2 + 𝑘𝑂2
∙ 6𝑂2 + 𝑁𝑢𝑡𝑟𝑖𝑒𝑛𝑡 𝑚𝑒𝑑𝑖𝑢𝑚 → 𝑘𝐶𝑂2

∙ 6𝐶𝑂2 + 6𝐻20 + 𝐵𝑖𝑜𝑚𝑎𝑠𝑠 + 𝑆𝑜𝑙𝑢𝑡𝑒𝑠, (2) 

where oxygen and carbon dioxide coefficients 𝑘𝑂2
 and 𝑘𝐶𝑂2

 relate to the respiratory quotient RQ.  

Correspondingly, the equation for the overflow metabolism (Deng et al., 2021):  

𝐶6𝐻12𝑂2 + 𝑘𝑂2
∙ 2𝑂2 → 2𝐶2𝐻3𝑂𝑂𝐻 + 𝑘𝐶𝑂2

∙ 2𝐶𝑂2 + 2𝐻2𝑂 + 𝐸𝑛𝑒𝑟𝑔𝑦. (3) 

In most cases, the respiratory quotient helps to determine the current cell metabolic pathway (Heyman et al., 

2020). This paper proposes that the respiratory quotient numeric value is close to one for all metabolic pathways 

in E.coli fed-batch bioprocesses 

𝑅𝑄 =
6𝐶𝑂2

6𝑂2
=

𝑘𝑂2

𝑘𝐶𝑂2

≅ 1. (4) 

Experimental proof confirms this proposition when measuring the off-gas carbon production rate 𝐶𝑃𝑅 and 

oxygen uptake rate 𝑂𝑈𝑅 no matter the chosen overflow mode, i.e., glucose feed can be limiting or non-limiting 

the biomass growth. 

2.1 Carbon Mass Balance 

Looking at Eq(2) through the perspective of the carbon element, the total carbon balance emerges 

𝑇𝐶𝐶6𝐻12𝑂2 =  ∑ 𝑇𝐶𝑖

𝑚

𝑖=1

, (5) 

where total carbon 𝑇𝐶𝐶6𝐻12𝑂2 in fed glucose and nutrient medium is equal to the sum of total carbon 𝑇𝐶𝑖 in each 

of the products m, produced from the aerobic process 

∑ 𝑇𝐶𝑖

𝑚

𝑖=1

= 𝑇𝐶𝑋 + 𝑇𝐶𝑠𝑜𝑙 + 𝑇𝐶𝐶𝑃𝑅 . (6) 

The sum of total carbon 𝑇𝐶 involves the carbon content in biomass 𝑇𝐶𝑋, soluble byproducts 𝑇𝐶𝑠𝑜𝑙 (e.g., acetates) 

and produced off-gas 𝑇𝐶𝐶𝑃𝑅. Given this, Eq(2) turns into the mass balance equation for carbon 

𝑚𝐶6𝐻12𝑂2 ∙ 𝐶𝑠 = 𝐶𝑋 ∙ ∆𝑋𝑡𝑜𝑡𝑎𝑙 + 𝑇𝐶𝑠𝑜𝑙 + 𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅 ∙ 𝑑𝑡, (7) 

where the first term 𝑚𝐶6𝐻12𝑂2 represents the total mass of carbon in glucose, 𝐶𝑠 = 0.03 and 𝐶𝑋 = 0.04 

coefficients are the carbon content in moles per gram of glucose 𝑠 and total biomass 𝑋𝑡𝑜𝑡𝑎𝑙, respectively (Xu et 

al., 1999). The constant 𝐶𝐶𝑃𝑅 = 0.272 relates to the weight ratio of carbon and oxygen in carbon dioxide. 

Data analysis showed that the coefficient (𝐶𝑋) is not a process constant. It instead represents a linear 

relationship with the average age of the cell population 𝐴𝑔𝑒̅̅ ̅̅ ̅, which is a function of biomass concentration X 

(Urniezius et al., 2021) 

𝐴𝑔𝑒̅̅ ̅̅ ̅(t)  ≅  
∑ (𝑡𝑖 − 𝑡𝑗)∆𝑋(𝑡𝑗)𝑖

𝑗=0

𝑋𝑖
. 

 
(8) 

The variable 𝑚𝐶6𝐻12𝑂2 consists of carbon masses in feeding glucose 𝑚𝑓𝑒𝑑,𝐶6𝐻12𝑂2 , initial glucose 𝑚0 and 

glucose concentration 𝑚𝑠 as follows 

𝑚𝐶6𝐻12𝑂2 = 𝑚𝑓𝑒𝑑,𝐶6𝐻12𝑂2 + 𝑚0 − 𝑚𝑠. (9) 

It leads to an equivalent form of Eq(7), solved for boundary conditions 

590



(
(𝐹(𝑡) − 𝐹0) ∙ 𝑆𝑓

𝑆𝑑
+ (𝑠0 ∙ 𝑊0 − 𝑠(𝑡) ∙ 𝑊(𝑡))) ∙ 𝐶𝑠

= 𝐶𝑋(𝑡) ∙ (𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) − 𝑋0) + 𝑇𝐶𝑠𝑜𝑙,𝑡𝑜𝑡𝑎𝑙(𝑡) + 𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗
𝑡

𝑡0

, 

(10) 

In Eq(10), the variable 𝐹 is the total fed glucose, and 𝑊 stands for the total volume of the cultivation medium. 

The initial values of the state variables after inoculation into the bioreactor are as follows 

𝐹0 ≡ 𝐹(𝑡 = 0), 

𝑊0 ≡ 𝑊(𝑡 = 0), 

𝑋0 ≡ 𝑋𝑡𝑜𝑡𝑎𝑙(𝑡 = 0), 

𝑠0 ≡ 𝑠(𝑡 = 0). 

(11) 

The coefficient 𝑆𝑓 is the concentration of glucose in the feed, while the coefficient (𝑆𝑑) is the glucose density 

which ensures the balance is in grams. 

Expressing biomass growth ∆𝑋𝐺 = (𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) − 𝑋0) and dividing Eq(10) by total biomass 𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) and ∆𝑡 ≡ 𝑡 −

𝑡0 produces specific rate terms concerning total biomass and time 

(
(𝐹(𝑡) − 𝐹0) ∙ 𝑆𝑓

𝑆𝑑
+ (𝑠0 ∙ 𝑊0 − 𝑠(𝑡) ∙ 𝑊(𝑡)))

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) ∙ ∆𝑡
∙ 𝐶𝑠 = 𝐶𝑋(𝑡) ∙

∆𝑋𝐺 

∆𝑡
∙

1

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡)
+ 

 
𝑇𝐶𝑠𝑜𝑙,𝑡𝑜𝑡𝑎𝑙(𝑡)

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) ∙ ∆𝑡
+ 𝐶𝐶𝑃𝑅 ∙

∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) ∙ ∆𝑡
, 

(12) 

where the biomass growth term becomes the average specific growth rate �̅�(𝑡) 

�̅�(𝑡) =
∆𝑋𝐺 

∆𝑡
∙

1

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡)
. (13) 

Extending the idea of specific rates, Xu (1999) presented specific rates of glucose consumption for the oxidative 

pathway 𝑞𝑆𝑜𝑥,𝑎𝑛 and cell maintenance 𝑞𝑚. Taking this into account, Eq(12) takes a different form 

(𝑞𝑆𝑜𝑥,𝑎𝑛 + 𝑞𝑚) ∙ 𝐶𝑠 = 𝐶𝑋(𝑡) ∙ �̅�(𝑡) +  
𝑇𝐶𝑠𝑜𝑙,𝑡𝑜𝑡𝑎𝑙(𝑡)

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) ∙ ∆𝑡
+ 𝐶𝐶𝑃𝑅 ∙

∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

𝑋𝑡𝑜𝑡𝑎𝑙(𝑡) ∙ ∆𝑡

→ 𝐶𝑋(𝑡) ∙ �̅�(𝑡) +  𝑇𝐶𝑠𝑜𝑙𝑋
̅̅ ̅̅ ̅̅ ̅̅ (𝑡) + 𝐶𝐶𝑃𝑅 ∙ 𝐶𝑃𝑅𝑋

̅̅ ̅̅ ̅̅ ̅(𝑡). 

(14) 

Rearranging and dividing Eq(14) by the average specific growth rate �̅�(𝑡) and the coefficient 𝐶𝑠 yields 

𝐶𝑋(𝑡)

𝐶𝑆
=

(𝑞𝑆𝑜𝑥,𝑎𝑛 + 𝑞𝑚)

�̅�(𝑡)
+

𝑇𝐶𝑠𝑜𝑙𝑋
̅̅ ̅̅ ̅̅ ̅̅ (𝑡) + 𝐶𝐶𝑃𝑅 ∙ 𝐶𝑃𝑅𝑋

̅̅ ̅̅ ̅̅ ̅(𝑡)

𝐶𝑆 ∙ �̅�(𝑡)
, (15) 

where the summand relating to the total carbon content in off-gas and solutes turns into 

𝑇𝐶𝑠𝑜𝑙𝑋
̅̅ ̅̅ ̅̅ ̅̅ (𝑡) + 𝐶𝐶𝑃𝑅 ∙ 𝐶𝑃𝑅𝑋

̅̅ ̅̅ ̅̅ ̅(𝑡)

𝐶𝑆 ∙ �̅�(𝑡)
=  

𝑇𝐶𝑠𝑜𝑙(𝑡)

𝑋(𝑡) ∙ ∆𝑡 ∙  
∆𝑋𝐺 

∆𝑡
∙

1
𝑋(𝑡)

∙ 𝐶𝑆

+
𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

𝑋(𝑡) ∙ ∆𝑡 ∙
∆𝑋𝐺 

∆𝑡
∙

1
𝑋(𝑡)

∙ 𝐶𝑆

→
𝑇𝐶𝑠𝑜𝑙(𝑡)

∆𝑋𝐺  ∙ 𝐶𝑆
+

𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

∆𝑋𝐺 ∙ 𝐶𝑆
. 

(16) 

Combining Eq(15) and Eq(16) results in: 

𝐶𝑋(𝑡)

𝐶𝑆
=

(𝑞𝑆𝑜𝑥,𝑎𝑛 + 𝑞𝑚)

�̅�(𝑡)
+

𝑇𝐶𝑠𝑜𝑙(𝑡)

∆𝑋𝐺  ∙ 𝐶𝑆
+

𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

∆𝑋𝐺 ∙ 𝐶𝑆
. (17) 

In Eq(17), the carbon term for growth 𝐺 of biomass has a different expression when maintenance energy is zero 

(Pirt, 1965). It relates to the aerobic yield coefficient (𝑌𝑋

𝑆
,𝑜𝑥

) exclusive of maintenance, which is assumed 

constant throughout the process (Xu et al., 1999) 

591



𝑞𝑆𝑜𝑥,𝑎𝑛

�̅�(𝑡)
=

∆𝑇𝑆𝐺

∆𝑋𝐺 
=

1

𝑌𝑋
𝑆

,𝑜𝑥

, (18) 

where ∆𝑇𝑆 is the total fed glucose.  

Joining Eq(17) and Eq(18) yields 

𝐶𝑋(𝑡)

𝐶𝑆
=

𝑞𝑚

�̅�(𝑡)
+

1

𝑌𝑋
𝑆

,𝑜𝑥

+
𝑇𝐶𝑠𝑜𝑙(𝑡)

∆𝑋𝐺  ∙ 𝐶𝑆
+

𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

∆𝑋𝐺 ∙ 𝐶𝑆
. (19) 

Assuming that the observed growth yield of cells 𝑌(𝑡) is inversely proportional to carbon balance terms 

1

𝑌(𝑡)
=

∆𝑇𝑆

∆𝑋
≅

𝐶𝑋(𝑡)

𝐶𝑆
 (20) 

concludes that the Pirt equation is a particular case of Eq(20) that does not account for the carbon content in 

solubles and off-gas carbon dioxide (Pirt, 1965): 

1

𝑌(𝑡)
=

𝑞𝑚

�̅�(𝑡)
+

1

𝑌𝑋
𝑆

,𝑜𝑥

. (21) 

A simplified illustration of proposed model: 

 

Figure 1: Simplified proposition of the paper – Pirt equation is a special case of the full carbon mass balance  

2.2 Glucose Estimation 

The glucose concentration expression solved from Eq(10):  

𝑠(𝑡) =

(𝑠0 ∙ 𝑊0 +
(𝐹(𝑡) − 𝐹0) ∙ 𝑆𝑓

𝑆𝑑
− (

𝐶𝑋(𝑡) ∙ (𝑋(𝑡) − 𝑋0) + 𝐶𝐶𝑃𝑅 ∙ ∫ 𝐶𝑃𝑅(𝑡∗) ∙ 𝑑𝑡∗𝑡

𝑡0

𝐶𝑠
))

𝑊𝑖
. 

(22) 

Numerical integration of Eq(22) leads to intermediate values of glucose concentration �̂�𝑖
∗ : 

�̂�𝑖
∗ =

(𝑠0 ∙ 𝑊0 +
(𝐹𝑖 − 𝐹0) ∙ 𝑆𝑓

𝑆𝑑
− (

𝐶𝑋,𝑖 ∙ (𝑋𝑖 − 𝑋0) + 𝐶𝐶𝑃𝑅 ∙ ∑ 𝐶𝑃𝑅𝑖 ∙ ∆𝑡𝑗𝑖
𝑗=1

𝐶𝑠
))

𝑊𝑖
. 

(23) 

The final estimated form (�̂�𝑖) is then dependent on a fuzzy rule 

�̂�𝑖 = {
�̂�𝑖 = 0,             𝑖𝑓 �̂�𝑖

∗ ≤ 0  

�̂�𝑖 =  �̂�𝑖
∗, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒

 , (24) 

3. where the logical condition assures the validity of the boundary condition, i.e., the 
computed glucose concentration is always non-negative.Experimental Verification 

The mean absolute error (MAE) operated as evaluation criteria to compare glucose estimation results 

592



𝑀𝐴𝐸 =
∑ |𝑠�̂� − 𝑠𝑖

∗|𝑛
𝑖=1

𝑛
, (25) 

where the glucose concentration (𝑠𝑖
∗) had discrete values observed every 15-60 minutes during the cultivation 

process. The total number of samples 𝑛 = 171 covered 5 experiments, of which 3 had growth-limiting feeding. 

The experimental data came from one site with Escherichia coli BL21(DE3) pETM-11+EGFP strain. 

Figure 2 provides examples of linear regression models of the coefficient 𝐶𝑋 for each experiment.

 

Figure 2: Carbon content in grams per gram of biomass 𝐶𝑋 dependence on average cell population age 

presenting slope, offset and 𝑅2 numerical values.  

The final model of glucose concentration used a shared set of fitted gain and slope parameters 

𝐶𝑋(𝑡) = 0.0813 ∙ 𝐴𝑔𝑒̅̅ ̅̅ ̅(t) − 0.0803. (26) 

Experimental data had no information about the concentrations of secreted compounds in the culture medium. 

Therefore, an average ratio of carbon content cross-verified the hypothesis by using 

(𝑠𝑛
∗ − 𝑠�̂�) ∝ (

(𝐶𝑃𝑅 ∙ 𝐶𝐶𝑃𝑅) ∙ d𝑡

d(𝑋𝑡𝑜𝑡𝑎𝑙 ∙ 𝐶𝑋)

̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅
). (27) 

A decrease in the average ratio between the exhaust carbon and the carbon in the biomass itself implies that 

the remaining carbon accumulates in the growth medium. The terms in Eq(27) support this assumption by 

resulting in a correlation coefficient being 0.949. Hence, the estimation errors in Figure 3 represent accumulated 

byproducts with their total carbon content. 

Table 1: The results of model fitting on experimental data 

Experiment 

No.  

MAE, 

g/L 

n Growth Type 𝑅𝑎𝑡𝑖𝑜𝐶𝑃𝑅/𝑋
̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅ ̅̅        (𝑠𝑛

∗ − 𝑠�̂�), g/L  

1 0.621 25 

Limiting 

1.765 -0.622 

2 1.434 27 2.725 -0.06 

3 0.927 29 2.376 -0.927 

4 3.967 30 
Nonlimiting 

1.824 -3.967 

5 6.642 60 0.917 -6.642 

 

Figure 3: Estimated and measured glucose concentrations for all five experiments. 

593



4. Conclusions 

The article presents a development of the Pirt equation considering the dynamical carbon balance in a fed-batch 

bioprocess. The newly introduced terms of carbon contents in off-gas and byproducts allowed us to advance to 

a complete mathematical model that models the glucose concentration during the bioprocess. Experimental 

data of five fed-batch bioprocesses, three growth-limited, verified the model, with glucose concentration 

estimation errors ranging from 0.6 to 6.7 g/L. The article also presents the discovered linear relationship between 

carbon content in grams per gram of biomass and the average age of the cell population, which reduces the 

model parameter count. A presented model verification criterion can also indicate the cellular metabolic 

pathway, thus allowing bioprocess operators to make appropriate decisions. The glucose estimation errors 

indicate that the model's current state requires further development. The lack of produced solubles 

measurements also affects the model's prediction accuracy, especially in non-limiting growth experiments. 

Overall, the described carbon balance hypothesis further expands the potential of off-gas for non-invasive 

estimation of bioprocess state variables. It is especially relevant when monitoring glucose concentration, where 

an adequate model eliminates or reduces the need for excessive sampling. 

Acknowledgments 

This project received funding from the European Regional Development Fund (project no. 01.2.2-LMT-K-718- 

03-0039) under a grant agreement with the Research Council of Lithuania (LMTLT). 

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https://doi.org/10.1002/elsc.201900121
http://www.ncbi.nlm.nih.gov/books/NBK531494/
https://doi.org/10.1016/j.csbj.2021.10.015
https://doi.org/10.3390/e21121221
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	Modeling the Glucose Concentration for the Recombinant E.coli Bioprocesses