**Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries**

Farges Bérangère, Poughon Laurent, Pons Agnès and Dussap Claude-Gilles *Clermont Université, Université Blaise Pascal, Laboratoire de Génie Chimique et Biochimique, Clermont-Ferrand France* 

### **1. Introduction**

The stoichiometry of a chemical reaction provides basic information about the nature and the quantities of chemical species consumed and produced. It also intrinsically contains all the information on transformation yields. Such information is useful and necessary for the design of any biotechnological process. In the case of microbiological reactions that support microbial growth, this information deals with the carbon and energy sources consumed, the terminal electron acceptor utilized, other metabolic products formed, as well as the quantity of the biomass produced.

In the first part, general mass balances principles/methodology for the stoichiometric analysis of a bioprocess will be presented. These methods will lead to stoichiometric coefficients estimation including statistical analysis and data reconciliation in case of redundant information. Redundant information means more information than the minimum required to calculate all the conversion yields with a suitable approach of mass balances; this concept of minimum information required is presented (degree of freedom or number of unknowns of the system); conversely over-determined systems, in the case of experimental data in excess, are associated to a statistical treatment.

In the second part, the previous stoichiometry principles will be applied to two practical examples. The first examined case is the continuous anaerobic cultures of *Fibrobacter succinogenes*. This strictly anaerobic bacterium was grown in continuous culture in a bioreactor at different dilution rates (0.02 to 0.092 h-1) on a fully synthetic culture medium with glucose as carbon source. Robustness of the experimental information is checked by C and N balances estimations. A detailed overall stoichiometry analysis of the process for each dilution rate examined, including all substrates and products of the culture, is proposed. The mass balances involved in stoichiometric equations were solved using data reconciliation and linear algebra methods in order to take into account errors measurements.

In the last part, a second practical case,i.e. batch aerobic cultures of *Saccharomyces cerevisiae*, is presented. In this example, the bioprocess is analysed using different methodologies: (i) a

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 263

Typically, chemical reactions are written as a single or a system of stoichiometric equations into which elements should be conserved. For a given element, such as C, it can shift from one compound to another due to chemical reaction, but the total amount is conserved. For

**rate of accumulation of elements = rate of input of element - rate of output of element** 

*d element E S*

As a consequence, at steady state, the net rate of accumulation being zero, the net production/consumption of all elements is zero in the system. Moreover, for each reaction, the net balance for all elements must also be equal to 0. This can be written for the different elements as a linear system of equations, a special set of equation being representative of a

, [ ]

, [ ]

*k i i*

where αk,i is the relational coefficient of compound *i* in the reaction k and [X]i the

Usually one reaction is represented by stoichiometric reaction. With respect of the elemental

1 2 [ ][ ] [ ] 12 5 <sup>5</sup> .... .......

In the previous equation, the stoichiometric coefficients (α) have opposite sign for reactants and products. By convention the products are given positive stoichiometric coefficients,

When multiple reactions occur in a system, an overall stoichiometric equation can be written as a linear combination of the stoichiometric equations. This is possible if, and only if, the ratios between the rates of all reactions remain at constant values whatever the external conditions. This means that if all the rates remain proportional, the total set of reactions can be replaced by one single stoichiometric equation. This resulting stoichiometric equation is established by summation of the different stoichiometric coefficients of substrates and products with suitable proportionality constants calculated from the ratios between the rates. When dealing with biochemical systems stoichiometric equations are used for describing reactions in biochemical pathways, as well as for depicting complex systems such as the conversion of nutrients into cells or organisms. Considering that biochemical

*k i i*

,1

,1

*i* α*H* =

*n*

*i* α

*n*

( )

( )( )

0

0

 α*CHON CHON a bc d a bc d* + + ⎯⎯→ + *CHON a bc d* (4)

*element element*

*dt* = − (2)

*C* = (3)

**Elemental Mass balance** 

the system, it is written:

particular stoichiometric equation:

balance the general expression is:

**Multiple stoichiometries** 

composition in element [X] in the compound *i*.

αα

while substrates stoichiometric coefficients are negative.

or

simple mass balance with a minimal set of experimental data, (ii) a mass balance with a large set of experimental data including error measurement and (iii) a mass balance with experimental data obtained with 3 repetitions of the batch culture.

### **2. Main principles and methods of biochemistry stoichiometry**

The quantitative description of a microbial growth and product formation has maturated considerably during the past few decades due on one part to major improvements in the controlled technical equipments and on the other part to important theoretical progress thanks to the application of elemental mass balancing methods.

The mass balance of a fermentation has become more and more recognized as a valuable tool for analytical data validation, for detection of measurement errors and/or unnoticed products (Wang and Stephanopoulos, 1983) for estimation of variables for which no direct analytical methods are available (Humphrey, 1974), and for improving the accuracy and reliability of fermentation parameter estimation (Solomon *et al*., 1982, 1984). The mass balancing theory principles were developed in the late seventies by Minkevich, Erickson, Roels and others (Minkevich and Eroshin, 1973; Erickson *et al*., 1978; Minkevich, 1983; Roels, 1980, 1983). The theory uses the formalism of linear algebra to express the relationships between measurable (macroscopic) flows. Practical applications are mostly in the field of numerical procedures to solve a system for unknown flows or to calculate maximum likelihood estimators in case of over-determined systems. Thus, it is now recognized that biological reactions stoichiometry is mandatory for analysing biological processes.

### **2.1 Mass-balance equations**

The universal principle (Lavoisier principle) that matter cannot be created or destroyed (unless there is a nuclear reaction) holds in biochemical systems. In any closed system the total mass of every element, C, N, O, H, P, etc, is constant over time.

### **Mass balance of a compound**

As a first step in analyzing a biochemical reaction system, the system, its boundary and the surroundings must be defined. A physical entity, such as a cell or a bioreactor, is often defined as a system, and material and energy balance is performed on it. In other cases the balance is done on a set of reactions that is a representative subset of the reaction network in the cell. Material balance can be performed on either a compound (a chemical specie) or an element (Nielsen *et al*, 2003). For a chemical species *ί*, the material balance in a system can be written as:

### **Accumulation of** *i* **= rate of inputs of** *i* **– rate of outputs of** *i* **+ net rate of production/consumption of** *i* **from all reaction.**

or

$$\frac{d\langle i \rangle}{dt} = E\_{\langle i \rangle} - S\_{\langle i \rangle} + \sum\_{k=1}^{n} R\_{k\langle i \rangle} \tag{1}$$

with *Rk i*( ) being the rate of consumption/production of *i* in the kth reaction.

### **Elemental Mass balance**

Typically, chemical reactions are written as a single or a system of stoichiometric equations into which elements should be conserved. For a given element, such as C, it can shift from one compound to another due to chemical reaction, but the total amount is conserved. For the system, it is written:

### **rate of accumulation of elements = rate of input of element - rate of output of element**

or

262 Stoichiometry and Research – The Importance of Quantity in Biomedicine

simple mass balance with a minimal set of experimental data, (ii) a mass balance with a large set of experimental data including error measurement and (iii) a mass balance with

The quantitative description of a microbial growth and product formation has maturated considerably during the past few decades due on one part to major improvements in the controlled technical equipments and on the other part to important theoretical progress

The mass balance of a fermentation has become more and more recognized as a valuable tool for analytical data validation, for detection of measurement errors and/or unnoticed products (Wang and Stephanopoulos, 1983) for estimation of variables for which no direct analytical methods are available (Humphrey, 1974), and for improving the accuracy and reliability of fermentation parameter estimation (Solomon *et al*., 1982, 1984). The mass balancing theory principles were developed in the late seventies by Minkevich, Erickson, Roels and others (Minkevich and Eroshin, 1973; Erickson *et al*., 1978; Minkevich, 1983; Roels, 1980, 1983). The theory uses the formalism of linear algebra to express the relationships between measurable (macroscopic) flows. Practical applications are mostly in the field of numerical procedures to solve a system for unknown flows or to calculate maximum likelihood estimators in case of over-determined systems. Thus, it is now recognized that

biological reactions stoichiometry is mandatory for analysing biological processes.

total mass of every element, C, N, O, H, P, etc, is constant over time.

The universal principle (Lavoisier principle) that matter cannot be created or destroyed (unless there is a nuclear reaction) holds in biochemical systems. In any closed system the

As a first step in analyzing a biochemical reaction system, the system, its boundary and the surroundings must be defined. A physical entity, such as a cell or a bioreactor, is often defined as a system, and material and energy balance is performed on it. In other cases the balance is done on a set of reactions that is a representative subset of the reaction network in the cell. Material balance can be performed on either a compound (a chemical specie) or an element (Nielsen *et al*, 2003). For a chemical species *ί*, the material balance in a system can be

**Accumulation of** *i* **= rate of inputs of** *i* **– rate of outputs of** *i* **+ net rate of** 

( ) *<sup>n</sup>*

with *Rk i*( ) being the rate of consumption/production of *i* in the kth reaction.

*d i ES R dt* <sup>=</sup>

() () () 1

=−+ (1)

*i i ki k*

experimental data obtained with 3 repetitions of the batch culture.

**2. Main principles and methods of biochemistry stoichiometry** 

thanks to the application of elemental mass balancing methods.

**2.1 Mass-balance equations** 

**Mass balance of a compound** 

**production/consumption of** *i* **from all reaction.** 

written as:

or

$$\frac{d(element)}{dt} = E\_{\text{(element)}} - S\_{\text{(element)}} \tag{2}$$

As a consequence, at steady state, the net rate of accumulation being zero, the net production/consumption of all elements is zero in the system. Moreover, for each reaction, the net balance for all elements must also be equal to 0. This can be written for the different elements as a linear system of equations, a special set of equation being representative of a particular stoichiometric equation:

$$
\sum\_{i,1}^{n} \alpha\_{k,i} \left[ \mathbf{C} \right]\_i = \mathbf{0} \tag{3}
$$

$$
\sum\_{i,1}^{n} \alpha\_{k,i} \left[ H \right]\_i = \mathbf{0}
$$

where αk,i is the relational coefficient of compound *i* in the reaction k and [X]i the composition in element [X] in the compound *i*.

Usually one reaction is represented by stoichiometric reaction. With respect of the elemental balance the general expression is:

$$a\_1 \left[ \mathbb{C}\_a H\_b O\_c N\_d \right]\_1 + a\_2 \left[ \mathbb{C}\_a H\_b O\_c N\_d \right]\_2 + \dots \longrightarrow a\_5 \left[ \mathbb{C}\_a H\_b O\_c N\_d \right]\_5 + \dots \tag{4}$$

In the previous equation, the stoichiometric coefficients (α) have opposite sign for reactants and products. By convention the products are given positive stoichiometric coefficients, while substrates stoichiometric coefficients are negative.

### **Multiple stoichiometries**

When multiple reactions occur in a system, an overall stoichiometric equation can be written as a linear combination of the stoichiometric equations. This is possible if, and only if, the ratios between the rates of all reactions remain at constant values whatever the external conditions. This means that if all the rates remain proportional, the total set of reactions can be replaced by one single stoichiometric equation. This resulting stoichiometric equation is established by summation of the different stoichiometric coefficients of substrates and products with suitable proportionality constants calculated from the ratios between the rates. When dealing with biochemical systems stoichiometric equations are used for describing reactions in biochemical pathways, as well as for depicting complex systems such as the conversion of nutrients into cells or organisms. Considering that biochemical

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 265

α*<sup>i</sup>*[ ] *O <sup>i</sup>*

α*<sup>i</sup>*[ ] *N <sup>i</sup>*

<sup>=</sup> <sup>0</sup>

<sup>=</sup> <sup>0</sup>

*i*=1

*n*

*i*=1

*n*

In order to know if the system can be solved by linear algebra, it is necessary to analyse its **degree of freedom d** at first. Normally the previous equations are linearly independent. The

In order to solve the system, it is necessary to provide d more independent information resulting from experimental yields determination and/or theoretical assumptions. Three

1. The minimum of experiments or theoretical independent information for solving the system is available; that is to say that d experiments or theoretical independent information are available; thus, the resolution of the system can be realised "by hand"

2. More information than d is available; the system must be solved with the use of a

S K α

**α** (n) is the aforementioned column vector of stoichiometric coefficients (n rows, negative for the substrates, positive for the products). **S** (p, n) contains the elemental formula of all compounds (4 rows for 4 elements) and additional constraints (additional rows) such as the stoichiometric coefficient choice, that is fixed to 1, and eventually other constraints equations between the coefficients. Therefore, p is the total number of equations. **K** (p) is the known column vector of resulting constraints, for example 0 for corresponding conservation equations, 1 for the fixed stoichiometric coefficient and 0 for the linear constraints equation

If a detailed analysis of a process leads to consider a set of n compounds, an overall stoichiometric equation with (n-1) unknown coefficients is established, knowing that one coefficient is arbitrarily fixed to a value of 1. To determine these (n-1) stoichiometric coefficients, experimental data are needed, and it is necessary to keep in mind that calculations must be performed by meeting the constraint of Lavoisier principle of elements

d = n – c – 1 (7)

= (8)

O

N

α

In all cases, one unknown coefficient ( ) is fixed. This leads to:

reconciliation method as well as the use of statistical results. 3. With not enough information, the system cannot be solved.

Let the resulting linear system of constraints be written as follows:

kernel matrix dimension is n – c.

thanks to a substitution method.

**2.2.2 Mathematical method for data reconciliation** 

between the coefficients (Urrieta-Saltijeral *et al*, 2001).

cases are thus possible:

conservation.

pathways are proceeding at constant ratios between the biochemical rates (due to metabolic regulation and enzymatic control), the formulation of traditional chemical reactions and biochemical reactions are virtually identical.

Conversely, when within a set of reactions the ratio between the reactions rates do not remain constant, the resulting stoichiometry may vary with the external conditions, *i.e.* the substrates/products concentrations, time, development phases...etc.. This leads to a variable stoichiometry, *i.e.* variable conversion yields. An instantaneous stoichiometry may be calculated as previously as a rates-weighted sum of the different stoichiometric equations leading to account for instantaneous conversion yields. But, in any case, the balance equations for the compounds must be written by using the reaction rates of all equations separately. This is easily understandable by considering the previous Equation 1, noticing that the instantaneous stoichiometry and the instantaneous overall reaction rate that are established from the set of elementary equations cannot be used directly without considering a second-order term for accounting of yields variations.

### **Pseudo- stoichiometry**

When using stoichiometric equations for non-chemically defined compounds, the elemental composition of which being non-completely defined, such as macro-compounds (proteins, biomass), transaction will only consider C, H, O, N, and P in most cases. The other elements participate only in a small fraction of all biochemical reactions and only slightly contributeto the biomass. Therefore, stoichiometric equations can be written for reactions that occur inside a biological system, such as a cell whether they are enzyme, catalysed or not; stoichiometric equation are also used for characterising cell growth and compounds production inside a reactor. Whatever the case, the material balance involves input and output flows and the reaction rates in the system. In the case of a pseudo-stoichiometry it must be kept in mind that the elemental composition accuracy of macro-compounds is limited by experimental measurements errors (generally not exceeding 10-3 relative error), generating a systematic inaccuracy in balance equations.

### **2.2 Method for stoichiometric coefficients estimation and statistical analysis**

### **2.2.1 Mathematical description of the equation system**

Considering the following stoichiometric equation, involving n compounds and c elements:

$$\alpha\_1 \text{[CHON]}\_1 + \alpha\_2 \text{[CHON]}\_2 + \dots \longrightarrow \alpha\_5 \text{[CHON]}\_3 + \dots \tag{5}$$

The elemental balances imply a system of c equations (here c = 4: C, H, O and N) and n unknown stoichiometric coefficients αi:

$$\begin{aligned} \mathbb{C} \quad & \sum\_{i=1}^{n} \alpha\_i [C]\_i = 0 \\\\ \mathbb{H} \quad & \sum\_{i=1}^{n} \alpha\_i [H]\_i = 0 \end{aligned} \tag{6}$$

pathways are proceeding at constant ratios between the biochemical rates (due to metabolic regulation and enzymatic control), the formulation of traditional chemical reactions and

Conversely, when within a set of reactions the ratio between the reactions rates do not remain constant, the resulting stoichiometry may vary with the external conditions, *i.e.* the substrates/products concentrations, time, development phases...etc.. This leads to a variable stoichiometry, *i.e.* variable conversion yields. An instantaneous stoichiometry may be calculated as previously as a rates-weighted sum of the different stoichiometric equations leading to account for instantaneous conversion yields. But, in any case, the balance equations for the compounds must be written by using the reaction rates of all equations separately. This is easily understandable by considering the previous Equation 1, noticing that the instantaneous stoichiometry and the instantaneous overall reaction rate that are established from the set of elementary equations cannot be used directly without

When using stoichiometric equations for non-chemically defined compounds, the elemental composition of which being non-completely defined, such as macro-compounds (proteins, biomass), transaction will only consider C, H, O, N, and P in most cases. The other elements participate only in a small fraction of all biochemical reactions and only slightly contributeto the biomass. Therefore, stoichiometric equations can be written for reactions that occur inside a biological system, such as a cell whether they are enzyme, catalysed or not; stoichiometric equation are also used for characterising cell growth and compounds production inside a reactor. Whatever the case, the material balance involves input and output flows and the reaction rates in the system. In the case of a pseudo-stoichiometry it must be kept in mind that the elemental composition accuracy of macro-compounds is limited by experimental measurements errors (generally not exceeding 10-3 relative error),

**2.2 Method for stoichiometric coefficients estimation and statistical analysis** 

Considering the following stoichiometric equation, involving n compounds and c elements:

The elemental balances imply a system of c equations (here c = 4: C, H, O and N) and n

α*<sup>i</sup>*[ ] *C*

α*<sup>i</sup>*[ ] *H <sup>i</sup>*

*<sup>i</sup>*<sup>=</sup> <sup>0</sup>

<sup>=</sup> <sup>0</sup>

*i*=1

*n*

*i*=1

*n*

 α*CHON CHON* + +⎯.... ⎯→ + *CHON* ....... (5)

H (6)

1 2 [ ][ ] [ ] 12 3 <sup>5</sup>

C

considering a second-order term for accounting of yields variations.

generating a systematic inaccuracy in balance equations.

**2.2.1 Mathematical description of the equation system** 

αα

unknown stoichiometric coefficients αi:

biochemical reactions are virtually identical.

**Pseudo- stoichiometry** 

$$\begin{aligned} \text{O} \quad & \sum\_{i=1}^{n} \alpha\_i [O]\_i = 0 \\\\ \text{N} \quad & \sum\_{i=1}^{n} \alpha\_i [N]\_i = 0 \end{aligned}$$

In order to know if the system can be solved by linear algebra, it is necessary to analyse its **degree of freedom d** at first. Normally the previous equations are linearly independent. The kernel matrix dimension is n – c.

In all cases, one unknown coefficient ( ) is fixed. This leads to: α

$$\mathbf{d} = \mathbf{n} - \mathbf{c} - \mathbf{1} \tag{7}$$

In order to solve the system, it is necessary to provide d more independent information resulting from experimental yields determination and/or theoretical assumptions. Three cases are thus possible:


### **2.2.2 Mathematical method for data reconciliation**

Let the resulting linear system of constraints be written as follows:

$$\mathbf{S} \,\alpha = \mathbf{K} \,\tag{8}$$

**α** (n) is the aforementioned column vector of stoichiometric coefficients (n rows, negative for the substrates, positive for the products). **S** (p, n) contains the elemental formula of all compounds (4 rows for 4 elements) and additional constraints (additional rows) such as the stoichiometric coefficient choice, that is fixed to 1, and eventually other constraints equations between the coefficients. Therefore, p is the total number of equations. **K** (p) is the known column vector of resulting constraints, for example 0 for corresponding conservation equations, 1 for the fixed stoichiometric coefficient and 0 for the linear constraints equation between the coefficients (Urrieta-Saltijeral *et al*, 2001).

If a detailed analysis of a process leads to consider a set of n compounds, an overall stoichiometric equation with (n-1) unknown coefficients is established, knowing that one coefficient is arbitrarily fixed to a value of 1. To determine these (n-1) stoichiometric coefficients, experimental data are needed, and it is necessary to keep in mind that calculations must be performed by meeting the constraint of Lavoisier principle of elements conservation.

Let us introduce experimental data in a column vector **Ŷexp** (m) of m mass yields, all calculated with the same compound as reference, glucose for example. Let us consider **α** (n) the column vector of the stoichiometric coefficients, with the relevant value for the reference compound, again glucose for example, being set to 1. Finally, **Yr** (m) is the column vector of the yields values obtained after data reconciliation. The following matricial expression is written:

$$\mathbf{A} \ \alpha = \mathbf{Y}\_{\mathbf{r}} \tag{9}$$

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 267

In this study the experimental values are known, and both the confidence interval (iy) and

y 1- /2 y y y 1- /2

If the number of points for the estimation is sufficiently large, t 0,975 ≈ 2 (α = 0.05), and the interval is a confidence interval of 95%. The weights for all measured yields are given by:

> y j 2 2 1 4 <sup>W</sup> σ

Estimation of the vector α components can be directly obtained through matrix calculus from the previous equation. However, a direct inversion does not guarantee that the obtained set of stoichiometric coefficients (vector α) will obey the Lavoisier principle (elements conservation). The originality of the proposed method is to add as constraints the first element conservation equations. The constraints to fulfil are given as previously by:

> S K α

Calculation of components of vector α is therefore obtained by the classical method of

L =− − Φ Λ (S K) α

The p Lagrange multipliers and the n stoichiometric coefficients are obtained by solving the

<sup>L</sup> 0 ( equations)

α

S K (p equations)

<sup>L</sup> <sup>0</sup> α

<sup>∂</sup> <sup>=</sup> ∂

<sup>∧</sup> <sup>∂</sup> <sup>=</sup> Λ =

t tt <sup>t</sup> 2 (M W M) 2 M W H S

exp

α

t t <sup>t</sup> exp L - 2 M W (H - M ) - S 0

α

∧

∂ (19)

= + Λ (20)

Lagrange multipliers. The Lagrangian function to be minimized is:

where **Λ** is the unknown row vector (p) of the Lagrange multipliers.

α

∂ = ∂

α

is a matrix derivation which leads to the n following equations :

α

=

(p + n) system of equations formed by:

This leads to:

y y

Δ = Δ = =

σ

y y i 2 i 2 t i / t

∧ = ±

y

α

α

σ

(14)

*<sup>j</sup> <sup>y</sup> <sup>j</sup> yi* = = (15)

= (16)

(17)

(18)

the variance (σy) of the model are given:

**A** (m, n) is the matrix enabling to build the mass yields values knowing the stoichiometric coefficients.

The element conservation balances are written once the elemental formula of all compounds are known. This is a constraint for the identification, as well as the need that one stoichiometric coefficient is fixed to 1.

The complete linear system for the stoichiometric coefficients calculation is obtained by concatenation of the previous expressions, leading to:

$$\mathbf{M}\,\alpha = \mathbf{H} \tag{10}$$

where **M** is a (m + p, n) matrix composed of matrix **A** and **S**, and **H** is a (m + p) column vector formed by concatenation of **Yr** (m) and **K** (p). Similarly, **Ĥexp** (m+p) is the column vector formed by concatenation of **Ŷexp** (m,) and **K** (p). A weighted diagonal matrix **W** (m + p, m +p) is also be filled with the inverse of experimental variances for the n experimental data and with 1 for the constraint relationships. **W** enables to account for the difference in experimental errors of the measurement yields. The system **M α** = **H** can be solved by direct inversion of the matrix **M**, if matrix rank and (m + p) are equal to α (determined system). In such case:

$$
\alpha = \mathbf{M}^{\cdot 1} \text{ H} \tag{11}
$$

In the case of redundant information (more independent rows than columns, *i.e.* (m + p) > α), a data reconciliation method must be used to solve the constrained and over-determined system. We propose to use a Lagrange method for solving this problem of optimisation, with the assumption that errors measurements (*i.e* variances) follow a Gaussian law (Himmelblau, 1970). The method is developed as follows.

The variable to be minimised is the least square estimate (Φ) given by:

$$\boldsymbol{\Phi} = \begin{pmatrix} \stackrel{\wedge}{\mathbf{H}}\_{\exp} & \neg \mathbf{H} \end{pmatrix}^{t} \begin{pmatrix} \mathcal{W} \end{pmatrix} \begin{pmatrix} \stackrel{\wedge}{\mathbf{H}} - \mathcal{H} \end{pmatrix} \tag{12}$$

i.e.

$$\boldsymbol{\Phi} = (\stackrel{\wedge}{\mathcal{H}}\_{\exp} \text{ - } \mathcal{M} \; \boldsymbol{\alpha})^{\mathsf{t}} \text{ (W) } (\stackrel{\wedge}{\mathcal{H}}\_{\exp} - \mathcal{M} \; \boldsymbol{\alpha}) \tag{13}$$

Φ is a dimensionless number if **W** is used.

In this study the experimental values are known, and both the confidence interval (iy) and the variance (σy) of the model are given:

$$\begin{aligned} \mathbf{y} &= \mathbf{\hat{y}} \pm \mathbf{i\_y} \\ \mathbf{\hat{A}\_y} &= \mathbf{2} \, \mathbf{i\_y} \\ \mathbf{\hat{A}\_y} &= \mathbf{2} \, \mathbf{t\_{1-\alpha/2}} \sigma\_\mathbf{y} \\ \sigma\_\mathbf{y} &= \mathbf{i\_y} / \, \mathbf{t\_{1-\alpha/2}} \end{aligned} \tag{14}$$

If the number of points for the estimation is sufficiently large, t 0,975 ≈ 2 (α = 0.05), and the interval is a confidence interval of 95%. The weights for all measured yields are given by:

$$\mathbf{W}\_{\mathbf{y}\mathbf{j}} = \frac{1}{\sigma\_{\mathbf{y}\mathbf{j}}^2} = \frac{4}{\mathbf{i}\_{\mathbf{y}\mathbf{j}}^2} \tag{15}$$

Estimation of the vector α components can be directly obtained through matrix calculus from the previous equation. However, a direct inversion does not guarantee that the obtained set of stoichiometric coefficients (vector α) will obey the Lavoisier principle (elements conservation). The originality of the proposed method is to add as constraints the first element conservation equations. The constraints to fulfil are given as previously by:

$$\mathbf{S} \,\alpha = \mathbf{K} \tag{16}$$

Calculation of components of vector α is therefore obtained by the classical method of Lagrange multipliers. The Lagrangian function to be minimized is:

$$\mathbf{L} = \Phi - \Lambda \text{ (S } \alpha - \mathbf{K} \text{)}\tag{17}$$

where **Λ** is the unknown row vector (p) of the Lagrange multipliers.

The p Lagrange multipliers and the n stoichiometric coefficients are obtained by solving the (p + n) system of equations formed by:

$$\begin{aligned} \frac{\partial}{\partial \alpha} &= 0 \qquad \text{( $\alpha$  equations)}\\ \text{S } \alpha &= \text{K} \qquad \text{( $\text{p equations}$ )} \end{aligned} \tag{18}$$

$$\frac{\partial \mathcal{L}}{\partial \alpha} = 0$$

is a matrix derivation which leads to the n following equations :

$$\frac{\partial}{\partial \alpha} \frac{\mathbf{L}}{\alpha} = -2 \, \mathbf{M}^{\dagger} \, \mathbf{W} \left( \hat{\mathbf{H}}\_{\text{exp}} \mathbf{-} \mathbf{M} \, \alpha \right) \cdot \mathbf{S}^{\dagger} \, \mathbf{A}^{\dagger} = 0 \tag{19}$$

This leads to:

266 Stoichiometry and Research – The Importance of Quantity in Biomedicine

Let us introduce experimental data in a column vector **Ŷexp** (m) of m mass yields, all calculated with the same compound as reference, glucose for example. Let us consider **α** (n) the column vector of the stoichiometric coefficients, with the relevant value for the reference compound, again glucose for example, being set to 1. Finally, **Yr** (m) is the column vector of the yields values obtained after data reconciliation. The following matricial expression is

> A Y α

**A** (m, n) is the matrix enabling to build the mass yields values knowing the stoichiometric

The element conservation balances are written once the elemental formula of all compounds are known. This is a constraint for the identification, as well as the need that one

The complete linear system for the stoichiometric coefficients calculation is obtained by

M H α

where **M** is a (m + p, n) matrix composed of matrix **A** and **S**, and **H** is a (m + p) column vector formed by concatenation of **Yr** (m) and **K** (p). Similarly, **Ĥexp** (m+p) is the column vector formed by concatenation of **Ŷexp** (m,) and **K** (p). A weighted diagonal matrix **W** (m + p, m +p) is also be filled with the inverse of experimental variances for the n experimental data and with 1 for the constraint relationships. **W** enables to account for the difference in experimental errors of the measurement yields. The system **M α** = **H** can be solved by direct inversion of the matrix **M**, if matrix rank and (m + p) are equal to α (determined system). In

M H -1

In the case of redundant information (more independent rows than columns, *i.e.* (m + p) > α), a data reconciliation method must be used to solve the constrained and over-determined system. We propose to use a Lagrange method for solving this problem of optimisation, with the assumption that errors measurements (*i.e* variances) follow a Gaussian law

> <sup>t</sup> Φ ( H -H) (W) ( H H) exp ∧ ∧

<sup>t</sup> Φ ( H -M ) (W) ( H M ) exp α

∧ ∧

α

= r (9)

= (10)

= (11)

= − (12)

 αexp

= − (13)

written:

coefficients.

such case:

i.e.

stoichiometric coefficient is fixed to 1.

concatenation of the previous expressions, leading to:

(Himmelblau, 1970). The method is developed as follows.

Φ is a dimensionless number if **W** is used.

The variable to be minimised is the least square estimate (Φ) given by:

$$2\left(\mathbf{M}^{\mathrm{t}}\,\mathrm{W}\,\mathrm{M}\right)\,\mathrm{at}\,\,=\,\,\,\,\mathrm{2}\,\mathrm{M}^{\mathrm{t}}\,\,\mathrm{W}\,\,\stackrel{\wedge}{\mathrm{H}\_{\mathrm{exp}}}\,\,+\,\,\mathrm{S}^{\mathrm{t}}\,\,\mathrm{A}^{\mathrm{t}}\tag{20}$$

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 269

*Fibrobacter succinogenes* is one of the main fibrolytic bacteria in the bovine rumen (Hungate, 1950). It's a strictly anaerobic bacterium with enzymatic equipment well adapted to the degradation of vegetable fibers and plants, especially when these are highly branches and lignified. The degradation steps lead to the production of cellobiose and glucose that are further metabolized by the bacterium. The fermentative metabolism of this bacterium has been studied and leads to the production of succinate, acetate and formate. *Fibrobacter succinogenes* is also able to store intracellular glycogen, even in cells of young cultures (Gaudet *et al*, 1992) and it can produce and release oligosaccharides (Nouaille *et al*, 2005).

However, there is little information on the global stoichiometric description of this metabolism. Such quantitative information is necessary for further understanding the growth of *Fibrobacter succinogenes,* for example through a global stoichiometric approach

The aim of this work was to establish the overall stoichiometry of the*Fibrobacter succinogenes* S85 growth, cultivated in a standardized continuous anaerobic culture process on a fully synthetic culture medium with glucose as carbon source for different dilution rates (Guiavarch *et al*, 2010). Linear algebra and data reconciliation methods previously developed were applied to solve the overdetermined system obtained from the large

The strain used was *Fibrobacter succinogenes* S85 (ATCC 19169), and was grown anaerobically under 100% CO2 in a synthetic medium with glucose as carbon source. The reactor was a B.BRAUN culture unit (Biostat ED, B.BRAUN Germany). The working volume was 5 L and the stirring speed 100 rpm. Temperature was controlled at 39°C and the pH was maintained at 6.3 by automatic addition of Na2CO3 (70 g.L-1). The culture vessel was fed with fresh medium completed with various glucose concentrations (from 8.2 to 19.1 g.L-1) at three volumetric flow rates (99, 255 and 464 mL.h-1) corresponding to three dilution rates (D = 0.02, 0.051 and 0.092 h-1). Culture vessel and all tanks were interconnected by a gas system and the pressure was maintained at 0.2 bars above atmospheric pressure. The whole gas system was continuously flushed with 5 sccm of sterile oxygen-free CO2 during the culture to preserve anaerobic conditions. This flow rate was controlled using a mass flow controller (0-5 sccm, Tylan), while the gas flow at the exit of the reactor was measured with a mass

Samples were taken at regular time intervals during the experiment. Microscopic observations showed that the culture was always axenic. For each sample, the absorbance was measured at 600 nm. HPLC apparatus was used to determine glucose and organic acids concentrations (Agilent 1100 series fitted with two Phenomenex Rezex ROA columns, 7.8

Culture supernatants were obtained after centrifugation of an aliquot (10 000*g*, 5 min), and used to perform the colorimetric assays of ammonium ions, proteins and total

**3. First application: Stoichiometric analysis of** *Fibrobacter succinogenes*

**growth** 

**3.1 Introduction** 

prior to metabolic flux modelling.

number of collected experimental data.

**3.2 Culture conditions** 

flow meter (0-20 sccm, Brooks).

mm diameter and 300 mm length).

then:

$$\boldsymbol{\alpha} = \left(\boldsymbol{\mathsf{M}}^{\mathrm{t}} \,\, \mathbf{W} \,\, \mathbf{M}\right)^{\mathrm{-1}} \,\, \mathbf{M}^{\mathrm{t}} \,\, \mathbf{W} \,\, \hat{\boldsymbol{\Pi}}\_{\mathrm{exp}} \,\, \left(\boldsymbol{\mathsf{M}}^{\mathrm{t}} \,\, \mathbf{W} \,\, \mathbf{M}\right)^{\mathrm{-1}} \,\, \mathbf{S}^{\mathrm{t}} \,\, \boldsymbol{\Lambda}^{\mathrm{t}} \tag{21}$$

and:

$$\text{S (M}^{\text{t}}\text{ W M)}^{\text{-1}}\text{ M }^{\text{t}}\text{W }\hat{\text{H}}\_{\text{exp}}^{\text{-}} + \text{ 1/2 S (M}^{\text{t}}\text{W M)}^{\text{-1}}\text{ S }^{\text{t}}\text{ A}^{\text{t}} = \text{K} \tag{22}$$

considering that:

$$\Lambda^{\dagger} = 2 \left[ \mathbf{S} \left( \mathbf{M}^{\dagger} \mathbf{W} \, \mathbf{M} \right)^{\perp} \mathbf{S}^{\dagger} \right]^{1} \left[ \mathbf{K} - \mathbf{S} \left( \mathbf{M}^{\dagger} \mathbf{W} \, \mathbf{M} \right)^{\perp} \mathbf{M}^{\dagger} \, \mathbf{W} \, \hat{\mathbf{H}}\_{\exp} \, \mathbf{J} \right] \tag{23}$$

if we denote ψ = Mt W M, **α** is given by:

$$\boldsymbol{\alpha} = \boldsymbol{\upmu}^{\cdot 1} \, \mathbf{M}^{\mathrm{t}} \, \mathbf{W} \, \overset{\wedge}{\mathbf{H}\_{\mathrm{exp}}} \, \mathbf{+} \, \boldsymbol{\upmu}^{\cdot 1} \, \mathbf{S}^{\mathrm{t}} \, [\mathbf{S} \, \boldsymbol{\upmu}^{\cdot 1} \, \mathbf{S}^{\mathrm{t}}]^{\mathrm{T}} \, [\mathbf{K} \, \mathbf{-} \, \mathbf{S} \, \boldsymbol{\upmu}^{\cdot 1} \, \mathbf{M}^{\mathrm{t}} \, \mathbf{W} \, \overset{\wedge}{\mathbf{H}\_{\mathrm{exp}}} \, \mathbf{J}] \tag{24}$$

Summarizing and combining computed matrices, the calculation leads to:

$$
\mu\_- = \mathbf{M}^t \,\mathbf{W} \,\mathbf{M} \tag{25}
$$

$$\mathbf{M}^{\text{t}} = \mathbf{M}^{\text{t}} \mathbf{W}^{\text{h}} \stackrel{\wedge}{\mathbf{H}\_{\text{exp}}} \tag{26}$$

The solution is given by:

$$\alpha = \left\| \boldsymbol{\upmu}^{\cdot 1} \left[ \boldsymbol{\upOmega} + \boldsymbol{\up S}^{t} \left( \boldsymbol{\upSigma} \boldsymbol{\upmu}^{\cdot 1} \boldsymbol{\upTheta} \right)^{\cdot 1} \left( \boldsymbol{\upK} \cdot \boldsymbol{\upSigma} \boldsymbol{\upmu}^{\cdot 1} \boldsymbol{\upOmega} \right) \right] \tag{27}$$

The standard deviations of the calculated coefficients are estimated by diagonal elements of matrices of covariances:

$$\text{Covar} \left( \stackrel{\wedge}{\alpha} \right) = \left( \text{M}^{\text{t}} \text{ W} \text{ M} \right)^{\text{-1}} = \text{\textasciicangled} \tag{28}$$

(if W is known from variances)

$$\text{Covar}\left(\stackrel{\wedge}{\alpha}\right) = \text{(M}^{\text{t}}\text{M)}\,\frac{\Phi}{\text{n}+\text{p}-\alpha}\tag{29}$$

(if W is not known)

Covariance on estimates of the model is:

$$\text{Covar(\stackrel{\wedge}{H}\_{\text{exp}})} = \text{M Covar}\left(\stackrel{\wedge}{a}\right)\text{M}^{\text{t}}\tag{30}$$

This set matrix equations determines the stoichiometric coefficients (vector α), the covariance vector of the estimated coefficients and the covariance of the model prediction. The constraint of elements conservation is intrinsically fulfilled, which is a prerequisite of any robust modeling including mass balance.

### **3. First application: Stoichiometric analysis of** *Fibrobacter succinogenes* **growth**

### **3.1 Introduction**

268 Stoichiometry and Research – The Importance of Quantity in Biomedicine

 (M W M) M W H 1/2 (M W M) S exp ∧

t -1 t t -1 t t S (M W M) M W H 1 2 S (M W M) S K exp

<sup>t</sup> t -1 t -1 t -1 t 2 [ S (M W M) S ] [K - S (M W M) M W H ] exp


ψ M W H exp ψ S [S ψ S ] [K - S ψ M W H ] exp

t M W Hexp ∧


The standard deviations of the calculated coefficients are estimated by diagonal elements of

<sup>∧</sup> <sup>Φ</sup> <sup>=</sup> <sup>+</sup>

<sup>t</sup> Covar(H ) M Covar ( ) M exp

This set matrix equations determines the stoichiometric coefficients (vector α), the covariance vector of the estimated coefficients and the covariance of the model prediction. The constraint of elements conservation is intrinsically fulfilled, which is a prerequisite of

∧ ∧

t -1 -1 Covar ( ) (M W M)

α

<sup>t</sup> Covar ( ) (M M)

α

∧

∧

Summarizing and combining computed matrices, the calculation leads to:

W M, **α** is given by:

α

t -1 t t -1 t t

= +Λ (21)

Λ = (23)

∧ ∧ = + (24)

+ Λ = (22)

<sup>t</sup> ψ M W M = (25)

Ω = (26)

= = Ψ (28)

= (30)

(29)

= Ω+ *S* Ω (27)

n p -

α

α

∧

then:

and:

considering that:

if we denote ψ = Mt

α

The solution is given by:

matrices of covariances:

(if W is not known)

(if W is known from variances)

Covariance on estimates of the model is:

any robust modeling including mass balance.

α

*Fibrobacter succinogenes* is one of the main fibrolytic bacteria in the bovine rumen (Hungate, 1950). It's a strictly anaerobic bacterium with enzymatic equipment well adapted to the degradation of vegetable fibers and plants, especially when these are highly branches and lignified. The degradation steps lead to the production of cellobiose and glucose that are further metabolized by the bacterium. The fermentative metabolism of this bacterium has been studied and leads to the production of succinate, acetate and formate. *Fibrobacter succinogenes* is also able to store intracellular glycogen, even in cells of young cultures (Gaudet *et al*, 1992) and it can produce and release oligosaccharides (Nouaille *et al*, 2005).

However, there is little information on the global stoichiometric description of this metabolism. Such quantitative information is necessary for further understanding the growth of *Fibrobacter succinogenes,* for example through a global stoichiometric approach prior to metabolic flux modelling.

The aim of this work was to establish the overall stoichiometry of the*Fibrobacter succinogenes* S85 growth, cultivated in a standardized continuous anaerobic culture process on a fully synthetic culture medium with glucose as carbon source for different dilution rates (Guiavarch *et al*, 2010). Linear algebra and data reconciliation methods previously developed were applied to solve the overdetermined system obtained from the large number of collected experimental data.

### **3.2 Culture conditions**

The strain used was *Fibrobacter succinogenes* S85 (ATCC 19169), and was grown anaerobically under 100% CO2 in a synthetic medium with glucose as carbon source. The reactor was a B.BRAUN culture unit (Biostat ED, B.BRAUN Germany). The working volume was 5 L and the stirring speed 100 rpm. Temperature was controlled at 39°C and the pH was maintained at 6.3 by automatic addition of Na2CO3 (70 g.L-1). The culture vessel was fed with fresh medium completed with various glucose concentrations (from 8.2 to 19.1 g.L-1) at three volumetric flow rates (99, 255 and 464 mL.h-1) corresponding to three dilution rates (D = 0.02, 0.051 and 0.092 h-1). Culture vessel and all tanks were interconnected by a gas system and the pressure was maintained at 0.2 bars above atmospheric pressure. The whole gas system was continuously flushed with 5 sccm of sterile oxygen-free CO2 during the culture to preserve anaerobic conditions. This flow rate was controlled using a mass flow controller (0-5 sccm, Tylan), while the gas flow at the exit of the reactor was measured with a mass flow meter (0-20 sccm, Brooks).

Samples were taken at regular time intervals during the experiment. Microscopic observations showed that the culture was always axenic. For each sample, the absorbance was measured at 600 nm. HPLC apparatus was used to determine glucose and organic acids concentrations (Agilent 1100 series fitted with two Phenomenex Rezex ROA columns, 7.8 mm diameter and 300 mm length).

Culture supernatants were obtained after centrifugation of an aliquot (10 000*g*, 5 min), and used to perform the colorimetric assays of ammonium ions, proteins and total

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 271

accounted for cellular protein, RNA, or DNA, and released large amounts of non ammonia

Redox potential (mV) Dissolved CO2 partial pressure (mbar)

(32)

N recovery (%)

0.02 101 98 -358 1167 0.051 98 62 -347 1191 0.092 97 82 -345 1193 Table 2. C and N element recoveries, redox potential and dissolved CO2 partial pressure for

All these results showed a rather good consistency of the experimental results obtained during the course of the culture, indicating that the data reconciliation technique can be

The following detailed stoichiometric equation was proposed to describe the culture (Guiavarch *et al*, 2008). The chemical dissociation imposed by the pH value (6.3) was taken into account in the elementary formulae of the organic acids. Nevertheless, it could be assumed that, in the range of pH supported by this bacterium (from 6.0 to 7.0), the relevant formulae remained rather identical. The stoichiometric equation included 13 compounds and therefore 12 stoichiometric coefficients had to be determined, the coefficient for glucose

6 12 6 1 4 2 4 2 2 3 3 3.69 6.76 2.66 0.25 0.010 (biomass, see table 1)

+ + +

+

At least 12 theoretical and/or experimental data were necessary. On-line and off-line parameters measured during the culture provided nine experimental mass yields (Ŷexp) related to ammonium sulfate, sodium carbonate, biomass, succinate, acetate, formate, carbohydrates, carbon dioxide and protein, each value being weighted by a standard deviation associated to the measured value (Table 3). For these data, a A (9, 13) matrix and Ŷexp (9) column vector could be built with molar mass of compounds and nine experimental mass yields. Elemental balances on C, H, O, N, S and Na provided 6 linear equations of constraints. Coefficient of glucose was here again fixed to 1, allowing to build a S (7, 13) matrix filled with chemical formulae of compounds and coefficient of glucose, and a K (7) column vector filled with 0 for conservation equations and 1 for fixed stoichiometric

+ + + +

4 4 4.1864 4 1.8136 (sodium succinate)

8 4.43 7.09 1.29 1.27 0.042 (proteins)

5 2 3 2 (sodium acetate) 6 2 (sodium formate)

7 6 10 5 (carbohy

+ drates)

α CHO NS α HO α NaHCO α CO α Na SO

9 2 10 3 11 2 12 2 4

α C H O Na α C H O Na α CHO Na α CH O

nitrogen that were not identified and quantified.

C recovery (%)

**3.4 Stoichiometric equation analysis and data reconciliation** 

CH O α (NH ) SO α Na CO α CHONS

+ +→

Dilution rate D ( h-1)

different dilution rates.

being, as already pointed out, set to 1:

valuably applied.

coefficient.

carbohydrates. Soluble carbohydrates production was calculated by difference between total carbohydrates and glucose concentrations.

The pellets resulting from centrifugation step were dried in an oven at 100°C for 24 h to obtain biomass dry weight. A typical correlation OD-cell dry weight was established from data collected during the exponential growth phase of a batch experiment carried out in the same culture conditions as:

$$\text{cell dry weight} = 0.482 \text{ (}\pm 0.034\text{) OD}\_{60\text{ nm}} \tag{31}$$

where the cell dry weight is expressed in g.L-1 (correlation coefficient 0.989).

At steady-state, one sample was centrifuged (10 000*g*, 15 min, 5°C), washed with 0.9% NaCl and dried under vacuum at 65°C (48 h) to determine an average biomass formula (CHONSP) by elemental analysis.

Gas at the exit of the bubble column was analyzed by gas chromatography (Hewlett Packard 5890 series II, fitted with a Thermal Conductivity Detector). Two 1.5 m length, 1/8" diameter stainless steel columns (Porapak Q and 5 Å molecular sieves) connected with a 6-port commutation valve were used.

### **3.3 Experimental data: Biomass formulae and element recoveries**

The average molar biomass formulae established after elemental analysis during steadystate are presented in Table 1.


Table 1. Average biomass formulae for different dilution rates.

The biomass formula changed weakly with the dilution rate, the most important modification concerning the nitrogen mass fraction that increases with the dilution rate. This modification could be explained by the variation in glycogen to protein ratio that has been evidenced when the growth rate is increased.

To calculate C-recovery, consumptions of sodium carbonate and glucose were both taken into account as carbon sources as well as production of biomass, soluble proteins, succinate, acetate, formate, and soluble carbohydrates. As reported in Table 2, C-balance was between 97 and 101%. Data considered in N-recovery were the nitrogen source consumption (ion ammonium), the nitrogen content in cell dry weight and the soluble proteins measured in the culture supernatant. At low dilution rate (0.02 h-1), N-balance was satisfactory with a value of 98% but at high dilution rates (0.051 and 0.092h-1), N-balances were low with respectively 62 and 82%.

N-recovery was dilution rate dependent and thus growth rate dependent. This dependence has already been shown in *Fibrobacter succinogenes* by Wells and Russell (1996). Growing cultures of *Fibrobacter succinogenes* were reported to assimilate more ammonia than could be

carbohydrates. Soluble carbohydrates production was calculated by difference between total

The pellets resulting from centrifugation step were dried in an oven at 100°C for 24 h to obtain biomass dry weight. A typical correlation OD-cell dry weight was established from data collected during the exponential growth phase of a batch experiment carried out in the

At steady-state, one sample was centrifuged (10 000*g*, 15 min, 5°C), washed with 0.9% NaCl and dried under vacuum at 65°C (48 h) to determine an average biomass formula

Gas at the exit of the bubble column was analyzed by gas chromatography (Hewlett Packard 5890 series II, fitted with a Thermal Conductivity Detector). Two 1.5 m length, 1/8" diameter stainless steel columns (Porapak Q and 5 Å molecular sieves) connected with a 6-port

The average molar biomass formulae established after elemental analysis during steady-

0.020 3.688 6.760 2.665 0.247 0.010 0.051 3.656 6.821 2.791 0.255 0.009 0.092 3.799 6.981 2.545 0.380 0.008

The biomass formula changed weakly with the dilution rate, the most important modification concerning the nitrogen mass fraction that increases with the dilution rate. This modification could be explained by the variation in glycogen to protein ratio that has been

To calculate C-recovery, consumptions of sodium carbonate and glucose were both taken into account as carbon sources as well as production of biomass, soluble proteins, succinate, acetate, formate, and soluble carbohydrates. As reported in Table 2, C-balance was between 97 and 101%. Data considered in N-recovery were the nitrogen source consumption (ion ammonium), the nitrogen content in cell dry weight and the soluble proteins measured in the culture supernatant. At low dilution rate (0.02 h-1), N-balance was satisfactory with a value of 98% but at high dilution rates (0.051 and 0.092h-1), N-balances were low with

N-recovery was dilution rate dependent and thus growth rate dependent. This dependence has already been shown in *Fibrobacter succinogenes* by Wells and Russell (1996). Growing cultures of *Fibrobacter succinogenes* were reported to assimilate more ammonia than could be

Average biomass formula C H O N S

where the cell dry weight is expressed in g.L-1 (correlation coefficient 0.989).

**3.3 Experimental data: Biomass formulae and element recoveries** 

Table 1. Average biomass formulae for different dilution rates.

evidenced when the growth rate is increased.

cell dry weight = 0.482 (± 0.034) OD 600 nm (31)

carbohydrates and glucose concentrations.

same culture conditions as:

(CHONSP) by elemental analysis.

commutation valve were used.

state are presented in Table 1.

Dilution rate D (h-1)

respectively 62 and 82%.


accounted for cellular protein, RNA, or DNA, and released large amounts of non ammonia nitrogen that were not identified and quantified.

Table 2. C and N element recoveries, redox potential and dissolved CO2 partial pressure for different dilution rates.

All these results showed a rather good consistency of the experimental results obtained during the course of the culture, indicating that the data reconciliation technique can be valuably applied.

### **3.4 Stoichiometric equation analysis and data reconciliation**

The following detailed stoichiometric equation was proposed to describe the culture (Guiavarch *et al*, 2008). The chemical dissociation imposed by the pH value (6.3) was taken into account in the elementary formulae of the organic acids. Nevertheless, it could be assumed that, in the range of pH supported by this bacterium (from 6.0 to 7.0), the relevant formulae remained rather identical. The stoichiometric equation included 13 compounds and therefore 12 stoichiometric coefficients had to be determined, the coefficient for glucose being, as already pointed out, set to 1:

$$\begin{aligned} \text{C}\_6\text{H}\_{12}\text{O}\_6 + \text{a}\_1\text{ (NH}\_4\text{)}\_2\text{SO}\_4 + \text{a}\_2\text{ Na}\_2\text{CO}\_3 &\rightarrow \text{a}\_3\text{ C}\_{1,66}\text{H}\_{6,76}\text{O}\_{2,66}\text{N}\_{0.25}\text{S}\_{0.010 (bonus, see table 1)}\\ &+ \text{a}\_4\text{ C}\_4\text{H}\_{4,1864}\text{O}\_4\text{Na}\_{18136 (sodium succinate)}\\ &+ \text{a}\_5\text{ C}\_2\text{H}\_3\text{O}\_2\text{Na}\_{(sodium acetate)}\\ &+ \text{a}\_6\text{ C}\text{HO}\_2\text{Na}\_{(sodium format)}\\ &+ \text{a}\_7\text{ C}\_6\text{H}\_{10}\text{O}\_5 \quad (\text{calcdydrates})\\ &+ \text{a}\_8\text{ C}\_{4,45}\text{H}\_{7,09}\text{O}\_{1.29}\text{N}\_{1.27}\text{S}\_{0.022}\text{ (protein)}\\ &+ \text{a}\_9\text{ H}\_2\text{O} + \text{a}\_{10}\text{ NaHCO}\_3\\ &+ \text{a}\_{11}\text{CO}\_2 + \text{a}\_{12}\text{ Na}\_2\text{SO}\_4 \end{aligned} \tag{??eins}$$

At least 12 theoretical and/or experimental data were necessary. On-line and off-line parameters measured during the culture provided nine experimental mass yields (Ŷexp) related to ammonium sulfate, sodium carbonate, biomass, succinate, acetate, formate, carbohydrates, carbon dioxide and protein, each value being weighted by a standard deviation associated to the measured value (Table 3). For these data, a A (9, 13) matrix and Ŷexp (9) column vector could be built with molar mass of compounds and nine experimental mass yields. Elemental balances on C, H, O, N, S and Na provided 6 linear equations of constraints. Coefficient of glucose was here again fixed to 1, allowing to build a S (7, 13) matrix filled with chemical formulae of compounds and coefficient of glucose, and a K (7) column vector filled with 0 for conservation equations and 1 for fixed stoichiometric coefficient.

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 273

At a dilution rate of 0.02h-1, reconciled molar yields of acid production and sodium carbonate consumption were equal to experimental yields (Table 3). Biomass and protein yields were slightly decreased which led to an increase of ammonium sulfate yield from - 0.060 (σ = 0.008) to -0.071 (σ = 0.004). These variations were of the same order of magnitude than the standard deviation. There was also a slight increase in soluble carbohydrates yield from 0.234 (σ = 0.139) to 0.260 (σ = 0.067) at this dilution rate (0.02 h-1) although the standard deviation associated to this value was high. So, this linear system was sufficient to obtain

At the other dilution rates of 0.051 and 0.092 h-1, this linear system did not give satisfactory results for the soluble carbohydrates yield since a negative value was calculated. This result would lead to consider that, under these conditions, soluble carbohydrates were a substrate. This idea could not be considered as realistic since glucose was the sole carbon source in the fresh medium. It was thus necessary to modify the linear system by adding a new constraint on soluble carbohydrates for dilution rates of 0.051 and 0.092 h-1. This supplementary

The new system also resulted in a new over-determined linear system of equations, then made of 8 relationships from experimental measurements, 6 constrained equations from elemental balances (C, H, O, N, S, Na) and 2 reference coefficients (glucose set to 1 and carbohydrate set to 0). **A**(8, 13) was the matrix of known constant coefficients and **Ŷexp**(8) was the column vector of experimental yields. These data allowed to build a matrix **S**(8, 13)

The average reconciled yields and the relevant variances were calculated from this second linear system using data reconciliation. Experimental and reconciled values were compared

This modified linear system gave rather satisfactory results for dilution rates of 0.051 and 0.092 h-1 with very close experimental and identified yield values. Particularly Yr/Yexp ratios were often close to 1 except for carbohydrates that had been set to zero by the supplementary constraint. The same major reconciliations were observed on carbon dioxide and ammonium sulfate yields. The discrepancy between Yr and Yexp on ammonium sulfate was explained by N-balances that were not satisfactorily assessed at these dilution rates. Yr took into account only ammonium sulfate used for cellular growth and protein production. During continuous culture, no significant carbon dioxide gas production or consumption had been measured and there was an important standard deviation on these measurements. Carbon dioxide gas production or consumption was calculated by difference between inlet and outlet gas. The whole gas system was flushed with a regular gas flow of 5 sccm of carbon dioxide minimum to preserve correct anaerobic conditions during the continuous culture. However, effluent and medium tank volumes were about 10 times higher than the reactor volume and were not regulated in temperature. Therefore, carbon dioxide solubility was permanently modified by ambient temperature variations that consequently led to unreliable carbon dioxide flow rate at the exit of the culture vessel. As carbon dioxide yield, carbohydrates yield was obtained indirectly by difference between total carbohydrates and

satisfactory results at a dilution rate of 0.02 h-1.

**3.5 Discussion** 

glucose concentrations.

in Table 3.

constraint was to fix the coefficient of soluble carbohydrates to zero.

filled with chemical formulae of compounds, and a **K**(8) column vector.

The resulting system of 16 linear equations was made of 9 relationships obtained from experimental measurements and 7 constraints relationships resulting from elemental balances (C, H, O, N, S, Na) and glucose coefficient fixed to 1. It was over-determined since there were only 12 unknown coefficients to calculate. The advantage of data reconciliation was to allow the use of all available information to reduce inaccurate data due to experimental errors. Reconciled molar yields (Yr) were thus estimated from the calculated stoichiometric coefficients.

At first , this linear system composed of 9 weighted relations from experimental measurements and 7 constraints from elemental balances was used to reconcile molar yields obtained at three different dilution rates (Table 3).


Table 3. Experimental mass yields values (Ŷexp mass) (g substrate or product. (g glucose)-1) and comparative values of experimental (Yexp molar) and reconciled (Yr) average molar yields (mol substrate or product. (mol glucose)-1) with the associated variances for different dilution rates.

At a dilution rate of 0.02h-1, reconciled molar yields of acid production and sodium carbonate consumption were equal to experimental yields (Table 3). Biomass and protein yields were slightly decreased which led to an increase of ammonium sulfate yield from - 0.060 (σ = 0.008) to -0.071 (σ = 0.004). These variations were of the same order of magnitude than the standard deviation. There was also a slight increase in soluble carbohydrates yield from 0.234 (σ = 0.139) to 0.260 (σ = 0.067) at this dilution rate (0.02 h-1) although the standard deviation associated to this value was high. So, this linear system was sufficient to obtain satisfactory results at a dilution rate of 0.02 h-1.

At the other dilution rates of 0.051 and 0.092 h-1, this linear system did not give satisfactory results for the soluble carbohydrates yield since a negative value was calculated. This result would lead to consider that, under these conditions, soluble carbohydrates were a substrate. This idea could not be considered as realistic since glucose was the sole carbon source in the fresh medium. It was thus necessary to modify the linear system by adding a new constraint on soluble carbohydrates for dilution rates of 0.051 and 0.092 h-1. This supplementary constraint was to fix the coefficient of soluble carbohydrates to zero.

The new system also resulted in a new over-determined linear system of equations, then made of 8 relationships from experimental measurements, 6 constrained equations from elemental balances (C, H, O, N, S, Na) and 2 reference coefficients (glucose set to 1 and carbohydrate set to 0). **A**(8, 13) was the matrix of known constant coefficients and **Ŷexp**(8) was the column vector of experimental yields. These data allowed to build a matrix **S**(8, 13) filled with chemical formulae of compounds, and a **K**(8) column vector.

### **3.5 Discussion**

272 Stoichiometry and Research – The Importance of Quantity in Biomedicine

The resulting system of 16 linear equations was made of 9 relationships obtained from experimental measurements and 7 constraints relationships resulting from elemental balances (C, H, O, N, S, Na) and glucose coefficient fixed to 1. It was over-determined since there were only 12 unknown coefficients to calculate. The advantage of data reconciliation was to allow the use of all available information to reduce inaccurate data due to experimental errors. Reconciled molar yields (Yr) were thus estimated from the calculated

At first , this linear system composed of 9 weighted relations from experimental measurements and 7 constraints from elemental balances was used to reconcile molar yields

> Standard deviation of Yexp molar

0.02 (NH4)2SO4 -0.044 -0.060 0.008 -0.071 0.004 0.008 1.19

0.051 (NH4)2SO4 -0.140 -0.191 0.017 -0.124 0.009 0.018 0.65

0.092 (NH4)2SO4 -0.165 -0.225 0.018 -0.159 0.009 0.018 0.71

Table 3. Experimental mass yields values (Ŷexp mass) (g substrate or product. (g glucose)-1) and comparative values of experimental (Yexp molar) and reconciled (Yr) average molar yields (mol substrate or product. (mol glucose)-1) with the associated variances for different

Na2CO3 -0.603 -1.024 0.038 -1.025 0.019 0.038 1.00 Biomass 0.207 0.387 0.047 0.343 0.024 0.048 0.89 Succinate 0.553 0.630 0.030 0.630 0.015 0.030 1.00 Acetate 0.147 0.322 0.028 0.322 0.014 0.028 1.00 Formate 0.004 0.108 0.012 0.108 0.006 0.012 1.00 Carbohydrates 0.237 0.235 0.139 0.260 0.067 0.134 1.11 CO2 0.000 0.000 0.939 0.389 0.423 0.846 - Proteins 0.026 0.048 0.004 0.047 0.002 0.004 0.97

Na2CO3 -0.553 -0.939 0.048 -0.940 0.024 0.048 1.00 Biomass 0.448 0.814 0.041 0.844 0.020 0.040 1.04 Succinate 0.502 0.572 0.017 0.569 0.009 0.018 0.99 Acetate 0.141 0.310 0.021 0.308 0.010 0.020 0.99 Formate 0.055 0.147 0.014 0.146 0.007 0.014 1.00 Carbohydrates 0.157 0.173 0.131 0.000 0.063 0.126 - CO2 0.007 0.027 1.297 0.537 0.541 1.082 20.0 Proteins 0.014 0.025 0.003 0.026 0.001 0.002 1.03

Na2CO3 -0.589 -0.999 0.027 -1.000 0.024 0.048 1.00 Biomass 0.496 0.906 0.038 0.751 0.020 0.040 0.83 Succinate 0.562 0.641 0.018 0.606 0.009 0.018 0.95 Acetate 0.168 0.369 0.036 0.340 0.010 0.020 0.92 Formate 0.058 0.152 0.005 0.149 0.007 0.014 0.98 Carbohydrates 0.038 0.042 0.347 0.000 0.063 0.126 - CO2 0.055 0.227 1.144 0.670 0.494 0.988 2.95 Proteins 0.014 0.026 0.005 0.025 0.003 0.006 0.98

Yr Standard deviation of Yr

Confidence interval after data reconciliation Yr/Yexp

stoichiometric coefficients.

Substrat and Product

Dilution rate D (h-1)

dilution rates.

obtained at three different dilution rates (Table 3).

Ŷexp mass

Yexp molar

> The average reconciled yields and the relevant variances were calculated from this second linear system using data reconciliation. Experimental and reconciled values were compared in Table 3.

> This modified linear system gave rather satisfactory results for dilution rates of 0.051 and 0.092 h-1 with very close experimental and identified yield values. Particularly Yr/Yexp ratios were often close to 1 except for carbohydrates that had been set to zero by the supplementary constraint. The same major reconciliations were observed on carbon dioxide and ammonium sulfate yields. The discrepancy between Yr and Yexp on ammonium sulfate was explained by N-balances that were not satisfactorily assessed at these dilution rates. Yr took into account only ammonium sulfate used for cellular growth and protein production. During continuous culture, no significant carbon dioxide gas production or consumption had been measured and there was an important standard deviation on these measurements. Carbon dioxide gas production or consumption was calculated by difference between inlet and outlet gas. The whole gas system was flushed with a regular gas flow of 5 sccm of carbon dioxide minimum to preserve correct anaerobic conditions during the continuous culture. However, effluent and medium tank volumes were about 10 times higher than the reactor volume and were not regulated in temperature. Therefore, carbon dioxide solubility was permanently modified by ambient temperature variations that consequently led to unreliable carbon dioxide flow rate at the exit of the culture vessel. As carbon dioxide yield, carbohydrates yield was obtained indirectly by difference between total carbohydrates and glucose concentrations.

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 275

oxygen is monitored (Ingold probe 34-100-3003), and temperature and pH are controlled (respectively at 30°C and pH 5). Bioreactor data (pH, PO2, temperature, stirring rate) are acquired by the digital control unit of the reactor (Micro DCU 300 – B.BRAUN, Germany) as

are measured by an Oxymat 5E (Siemens) and an infrared CO2 analyser (Schlumberger),

The growth medium is taken from Kristiansen (1994) (glucose 50 g.L-1, yeast extract 6 g.L-1, KH2PO4 5 g.L-1, (NH4)2SO4 2 g.L-1). Organic acids and ethanol products are measured by HPLC (ionic exclusion column Shodex SH1011, 300 x 7.8 mm). Glucose is measured by the 3,5 dinitrosalicylic acid (DNS) method of Summer and Howell (1935). Amino Acids concentrations (yeast extract content) are measured as a leucine equivalent by a colorimetric method after reaction with ninhydrin (Ruhemann's purple read at 570 nm). Ammonium ions (NH4+) are measured by the colorimetric method of Patton and Crouch (1977). The biomass growth is followed by optical density at 550 nm and dry mass

The main substrates and products identified and expected during the growth of *Sc. cerevisiae* are reported in table 4. It can be noticed that the previously detailed analyses allows quantifying all these compounds during the batch culture. The culture is performed during 8 hours with an initial glucose concentration of about 50 g.L-1, which leads to a growth with a glucose saturated metabolism of *Sc. cerevisiae* (Crabtree effect), *i.e.* a mixed oxidative/fermentative metabolism with ethanol production. It is also considered that the produced metabolites are primary metabolites, which in turns means that all metabolic rates, including biomass synthesis and metabolites production, remain proportional. This

Theoretically, the mass balance on the culture can be expressed with the following

ܿ݅݀ܣ Ǥ݉݅݊ܣସߙ ଷܪܰଷߙ ଶܱଶߙ ݁ݏܿݑ݈݃ଵߙ

 (33) ܱଶܪଵߙ ݅݊ݐܽܿ݁ଽߙ ݈ݎܿ݁ݕ଼݈݃ߙ ݈݄ܽ݊ݐ݁ߙ ଶܱܥߙ ݏݏܾ݉ܽ݅ହߙ՜ Considering the 4 elements (C H O N) balance and that one of the stoichiometric coefficients is arbitrary fixed to 1, the degree of freedom d = n - c - 1 = 10 – 4 - 1 is equal to 5. This means that a minimal set of 5 independent additional information are required to calculate the 10 stoichiometric coefficients. We will obtain these data from the experimental yields

All compounds, except water, can be experimentally measured during the culture. An example of the results obtained for the experiment 1 (Exp 1) is reported in table 5 and figure 1. CO2 production and O2 consumption have been computed from the integration of the instantaneous CO2 and O2 respiration rates (rCO2, rO2) acquired from the online gas balance measurements. The respiratory quotient (RQ = rCO2 / rO2) is also computed and its value

*out yO* <sup>2</sup>

*out yCO* ) which

well as the online analysis of the gas output O2 and CO2 molar fractions ( <sup>2</sup>

**4.1.2 Analysis of a batch culture: Results and theoretical stoichiometry** 

justifies that a single stoichiometry approach is applied.

calculated from the various experimental data acquired.

above 1 is an indicator of the growth fermentative metabolism.

respectively.

measurements.

stoichiometric equation:

These results showed that the stoichiometric equation was dilution rate dependent. When the dilution rate increased from 0.020 to 0.092h-1, biomass and ammonium sulfate yields significantly increased as well. The biomass yield was improved from 0.343 (σ = 0.024) to 0.751 (σ = 0.020) mol biomass (mol glucose)-1 (from 0.183 to 0.412 g biomass (g glucose)-1). An important result was to notice that no significant variations were observed on succinate, acetate, formate and sodium carbonate yields. In the range of the studied dilution rates, the rates of succinate, acetate, and formate production were proportional to the rate of glucose consumed into the system. This tended to demonstrate that the metabolism of *Fibrobacter succinogenes* was not limited by the production of these acids which were directly linked to energy metabolism.

These results showed that assays used to track products of fermentation, consumption of nitrogen and carbon source were efficient as well as analysis of the biomass. This information was reliable to establish a stoichiometric equation for each dilution rate. It should also be pointed out that soluble carbohydrates production should be measured using a more specific technique. This information must be accurately available to go further in the analysis of *Fibrobacter succinogenes* growth by the use of a metabolic flux model.

### **4. Application 2: Stoichiometry for an aerated batch culture of**  *Saccharomyces cerevisiae*

In this second example of application, stoichiometric equations were established using experimental data obtained for the growth of a strain of *Saccharomyces cerevisiae* in an aerated and controlled bioreactor. The method of data reconciliation to establish the stoichiometry is applied here considering 3 approaches for exploiting the experimental results:


### **4.1 Batch culture of** *Saccharomyces cerevisiae*

### **4.1.1 Culture conditions, monitoring and analysis**

A strain of *Saccharomyces cerevisiae* ATCC 7754 is grown in a controlled 6 liters bioreactor (Biostat A. B-Braun, Germany). *Sc. cerevisiae* is a facultative anaerobic microorganism and can metabolise glucose into ethanol (fermentative metabolism) or/and into CO2 (oxidative metabolism). It is also a glucose sensitive microorganism, as for glucose concentration above 0.1 g.L-1, the "Crabtree effect" can be observed. The Crabtree effect reflects the respiratory chain saturation, which is the main path for regenerating reduced co-factors, and thus even in an aerated system the alternate route to ethanol is used to regenerating the co-factors.

The bioreactor is operated during 8 hours in liquid batch conditions (4.4 L liquid volume), air flow rate (1.5 NL.min-1) and perfectly mixed (6 blades stirrer at 500rpm). The dissolved

These results showed that the stoichiometric equation was dilution rate dependent. When the dilution rate increased from 0.020 to 0.092h-1, biomass and ammonium sulfate yields significantly increased as well. The biomass yield was improved from 0.343 (σ = 0.024) to 0.751 (σ = 0.020) mol biomass (mol glucose)-1 (from 0.183 to 0.412 g biomass (g glucose)-1). An important result was to notice that no significant variations were observed on succinate, acetate, formate and sodium carbonate yields. In the range of the studied dilution rates, the rates of succinate, acetate, and formate production were proportional to the rate of glucose consumed into the system. This tended to demonstrate that the metabolism of *Fibrobacter succinogenes* was not limited by the production of these acids which were directly linked to

These results showed that assays used to track products of fermentation, consumption of nitrogen and carbon source were efficient as well as analysis of the biomass. This information was reliable to establish a stoichiometric equation for each dilution rate. It should also be pointed out that soluble carbohydrates production should be measured using a more specific technique. This information must be accurately available to go further in the

In this second example of application, stoichiometric equations were established using experimental data obtained for the growth of a strain of *Saccharomyces cerevisiae* in an aerated and controlled bioreactor. The method of data reconciliation to establish the stoichiometry is applied here considering 3 approaches for exploiting the experimental

1. analysis of a single batch experiment with the minimal experimental data required for

2. analysis of a single batch experiment and use of all experimental data for establishing the stoichiometry, including the experimental uncertainty (i.e. variance) estimation of

3. analysis of several batch experiments (repeatability of experiments) for establishing the

A strain of *Saccharomyces cerevisiae* ATCC 7754 is grown in a controlled 6 liters bioreactor (Biostat A. B-Braun, Germany). *Sc. cerevisiae* is a facultative anaerobic microorganism and can metabolise glucose into ethanol (fermentative metabolism) or/and into CO2 (oxidative metabolism). It is also a glucose sensitive microorganism, as for glucose concentration above 0.1 g.L-1, the "Crabtree effect" can be observed. The Crabtree effect reflects the respiratory chain saturation, which is the main path for regenerating reduced co-factors, and thus even in an aerated system the alternate route to ethanol is used to regenerating the co-factors.

The bioreactor is operated during 8 hours in liquid batch conditions (4.4 L liquid volume), air flow rate (1.5 NL.min-1) and perfectly mixed (6 blades stirrer at 500rpm). The dissolved

analysis of *Fibrobacter succinogenes* growth by the use of a metabolic flux model.

**4. Application 2: Stoichiometry for an aerated batch culture of** 

the stoichiometry (i.e. simple mass balance approach)

**4.1 Batch culture of** *Saccharomyces cerevisiae* **4.1.1 Culture conditions, monitoring and analysis** 

stoichiometry, including the statistical analysis of the repetition.

energy metabolism.

results:

*Saccharomyces cerevisiae*

the experimental results.

oxygen is monitored (Ingold probe 34-100-3003), and temperature and pH are controlled (respectively at 30°C and pH 5). Bioreactor data (pH, PO2, temperature, stirring rate) are acquired by the digital control unit of the reactor (Micro DCU 300 – B.BRAUN, Germany) as well as the online analysis of the gas output O2 and CO2 molar fractions ( <sup>2</sup> *out yO* <sup>2</sup> *out yCO* ) which are measured by an Oxymat 5E (Siemens) and an infrared CO2 analyser (Schlumberger), respectively.

The growth medium is taken from Kristiansen (1994) (glucose 50 g.L-1, yeast extract 6 g.L-1, KH2PO4 5 g.L-1, (NH4)2SO4 2 g.L-1). Organic acids and ethanol products are measured by HPLC (ionic exclusion column Shodex SH1011, 300 x 7.8 mm). Glucose is measured by the 3,5 dinitrosalicylic acid (DNS) method of Summer and Howell (1935). Amino Acids concentrations (yeast extract content) are measured as a leucine equivalent by a colorimetric method after reaction with ninhydrin (Ruhemann's purple read at 570 nm). Ammonium ions (NH4+) are measured by the colorimetric method of Patton and Crouch (1977). The biomass growth is followed by optical density at 550 nm and dry mass measurements.

### **4.1.2 Analysis of a batch culture: Results and theoretical stoichiometry**

The main substrates and products identified and expected during the growth of *Sc. cerevisiae* are reported in table 4. It can be noticed that the previously detailed analyses allows quantifying all these compounds during the batch culture. The culture is performed during 8 hours with an initial glucose concentration of about 50 g.L-1, which leads to a growth with a glucose saturated metabolism of *Sc. cerevisiae* (Crabtree effect), *i.e.* a mixed oxidative/fermentative metabolism with ethanol production. It is also considered that the produced metabolites are primary metabolites, which in turns means that all metabolic rates, including biomass synthesis and metabolites production, remain proportional. This justifies that a single stoichiometry approach is applied.

Theoretically, the mass balance on the culture can be expressed with the following stoichiometric equation:

$$a\_1 
glucose + a\_2 
 O\_2 + a\_3 
 NH\_3 + a\_4 
 Amino.Acid$$

$$\rightarrow a\_5 \text{ biomass} + a\_6 \text{CO}\_2 + a\_7 \text{ethanol} + a\_8 \text{glycine} + a\_9 \text{acetate} + a\_{10} \text{H}\_2\text{O} \tag{33}$$

Considering the 4 elements (C H O N) balance and that one of the stoichiometric coefficients is arbitrary fixed to 1, the degree of freedom d = n - c - 1 = 10 – 4 - 1 is equal to 5. This means that a minimal set of 5 independent additional information are required to calculate the 10 stoichiometric coefficients. We will obtain these data from the experimental yields calculated from the various experimental data acquired.

All compounds, except water, can be experimentally measured during the culture. An example of the results obtained for the experiment 1 (Exp 1) is reported in table 5 and figure 1. CO2 production and O2 consumption have been computed from the integration of the instantaneous CO2 and O2 respiration rates (rCO2, rO2) acquired from the online gas balance measurements. The respiratory quotient (RQ = rCO2 / rO2) is also computed and its value above 1 is an indicator of the growth fermentative metabolism.

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 277

Fig. 1. Glucose consumption, biomass and ethanol production measured for the growth of

02468

**Time (h)**

Some yields calculated for the experiment 1 are reported in table 6 with their standard deviation and correlation coefficient. It is possible to take the minimum of 5 relations required to solve the stoichiometric equation from these results. Besides, the C and N balance can be evaluated, being 0.91 and 0.78 (using YX/? yields), respectively. These experimental balances may be an indicator of the success for solving the stoichiometric

deviation

correlation coefficient

0

2

4

6

8

**Biomass ; Ethanol (g/L)**

10

12

14

*Sc. cerevisiae -* Experiment 1.

**Glucose (g/L)**

used.

equation, as the theory implies that balances are equal to 1.

Glucose (g/L)

Biomass (g/L)

Ethanol (g/L)

Mass yield (g.g-1) Standard

YAA/glu 0.0374 +/- 5.8% 0.965 \* YN-NH3/glu 0.0096 +/- 2.5% 0.993 \*

Yx/Eth 0.5498 +/- 4.2% 0.981 \* Yx/gly 13.6788 +/- 5.1% 0.973 \* Yx/acet 9.2216 +/- 11.8% 0.866 \* Yx/CO2 0.4193 +/- 4.8% 0.977 \* Yx/O2 6.0254 +/- 2.5% 0.994 \* Table 6. Yields calculated for Exp. 1. \* means that a linear regression without intercept was

Yx/glu -0.1806 +/- 3.4% 0.988

Yeth/glu -0.3798 +/- 5.6% 0.970 Ygly/glu -0.0156 +/- 5.7% 0.969 Yacet/glu -0.0076 +/- 34.2% 0.461 YCO2/glu -0.5245 +/- 3.4% 0.990 YO2/glu -0.0332 +/- 4.2% 0.984 Yx/AA -3.0418 +/- 8.1% 0.939 Yx/N-NH3 -14.9807 +/- 4.8% 0.977


Table 4. Compounds involved in the batch culture of *Sc. cerevisiae* and their characteristics. The biomass composition was obtained from elemental analysis of dry sample. The amino acids composition was obtained from average content of amino acids in yeast extract.


Table 5. Results obtained for the growth of *Sc. cerevisiae -* Experiment 1.

A first treatment of the results obtained for the batch culture consists in the calculation of the experimental yields. Considering all products as primary metabolites, the ratios between all production or consumption rates are constant (*i.e* constant yields). Consequently the yield between a compound A and a compound B (YA/B) is calculated by a linear regression (figure 2) using the concentrations obtained in batch:

$$\mathbf{A(t) = Y\_{A/B}B(t) + cst},$$

$$\mathbf{A(t) = Y\_{A/B}B(t)} \text{ (when initial values are 0: A(0) = B(0) = 0).}\tag{34}$$

The regression line slope is the yield, and calculation of the estimation variance gives the standard deviation. A complete statistical analysis leads to examine the reliability of the linearity assumption between the concentrations of A and B (constant yield and stoichiometry).

Table 4. Compounds involved in the batch culture of *Sc. cerevisiae* and their characteristics. The biomass composition was obtained from elemental analysis of dry sample. The amino acids composition was obtained from average content of amino acids in yeast extract.

0.1 0.82 42.88 1.808 0.4226 0.854 0 0.16 0.294 0.049 - 1.1 0.847 42.59 1.936 0.426 0.869 0.045 0.167 0.908 0.118 5.4 2.02 1.033 43.90 1.816 0.4111 1.352 0.06 0.224 2.011 0.212 7.1 3.01 1.486 41.71 1.599 0.394 1.886 0.088 0.242 2.855 0.267 10.2 3.5 1.925 37.65 1.513 0.3931 2.37 0.097 0.269 3.930 0.330 11.7 4 2.235 37.21 1.332 0.3652 3.498 0.115 0.367 5.207 0.404 12.7 4.5 2.61 34.12 1.264 0.3462 2.92 0.11 0.303 6.748 0.498 12.4 5 3.13 31.91 1.095 0.3102 5.015 0.204 0.41 8.588 0.617 11.5 5.51 3.875 27.78 0.544 0.2571 7.46 0.288 0.467 10.766 0.758 11.1 6 4.425 25.93 0.508 0.2129 7.814 0.305 0.426 13.361 0.916 11.4 6.5 5.515 18.90 0.541 0.1363 10.91 0.41 0.491 16.395 1.091 12.2 7.01 6.205 12.85 0.3131 0.0607 11.79 0.51 0.3 0.294 0.049 12.8

A first treatment of the results obtained for the batch culture consists in the calculation of the experimental yields. Considering all products as primary metabolites, the ratios between all production or consumption rates are constant (*i.e* constant yields). Consequently the yield between a compound A and a compound B (YA/B) is calculated by a linear regression

A(t)= ܻA/B B(t) + cst ,

 A(t)= ܻA/B B(t), (when initial values are 0: A(0)=B(0)=0). (34) The regression line slope is the yield, and calculation of the estimation variance gives the standard deviation. A complete statistical analysis leads to examine the reliability of the linearity assumption between the concentrations of A and B (constant yield and

Ethanol (g.L-1)

Glycerol (g.L-1)

Acetoin (g.L-1)

CO2 (g.L-1) O2 (g.L-1) RQ

N-NH3 (g.L-1)

Table 5. Results obtained for the growth of *Sc. cerevisiae -* Experiment 1.

(figure 2) using the concentrations obtained in batch:

Biomasse C6H1.62O0.52N0.15P0.01 24.35 49.28 8.6 Glucose C6H12O6 180 40 0 AA CH2.24O0.48N0.24 25.28 47.47 13.3 N-NH3 N 14 0 100 Ethanol C2H6O 46 52.17 0 Glycerol C3H8O3 92 39.13 0 Acetoine C4H8O2 88 54.55 0 CO2 CO2 44 27.27 0 O2 O2 32 0 0

**(g.mol-1)** 

**% Carbon content** 

**% N content** 

**Compound Formula (CHON) Molar mass** 

Time (h)

Biomass (g.L-1)

stoichiometry).

Glucose (g.L-1)

Amino. Ac.(g.L-1)

Fig. 1. Glucose consumption, biomass and ethanol production measured for the growth of *Sc. cerevisiae -* Experiment 1.

Some yields calculated for the experiment 1 are reported in table 6 with their standard deviation and correlation coefficient. It is possible to take the minimum of 5 relations required to solve the stoichiometric equation from these results. Besides, the C and N balance can be evaluated, being 0.91 and 0.78 (using YX/? yields), respectively. These experimental balances may be an indicator of the success for solving the stoichiometric equation, as the theory implies that balances are equal to 1.


Table 6. Yields calculated for Exp. 1. \* means that a linear regression without intercept was used.

$$Y\_{X/Eth} = \frac{\text{Mass Bonass}}{\text{Mass Ethanol}} = \frac{a\_5}{a\_7} \cdot \frac{\text{Molar mass blomassa}}{\text{Molar mass Ethanol}}$$

$$\alpha\_7 = \frac{a\_5}{Y\_{X/Eth}} \cdot \frac{\text{Molar mass blomassa}}{\text{Molar mass Ethanol}} = \frac{1}{Y\_{X/Eth}} \frac{24.35}{46} = 0.9628 \tag{35}$$

$$0.7492\text{ glucose} + 0.1263\text{ O}\_2 + 0.074\text{ NH}\_3 + 0.3167\text{ Amino}.\text{Acid} \rightarrow 1\text{ biomass} + 1.1305\text{ CO}\_2 + 0.9228\text{ ethanol} + 0.0193\text{ glycerol} + 0.1745\text{ aceetoin} + 0.7491\text{ H}\_2\text{O} \tag{36}$$

$$0.7492\text{ glucose} + 0.1263\text{ O}\_2 + 0.074\text{ N}\_3 + 0.3167\text{ Amino}.\text{Acid} \rightarrow 1\text{ biomass} + 1.1305\text{ CO}\_2 + 1.1928\text{ ethanol} + 0.0193\text{ glycerol} + 0.0121\text{ aceetoin} + 0.4651\text{ H}\_2\text{O} \tag{37}$$

$$
\sigma\_Y = \text{(std deviation } \% \,\, \* \, Y\_{X/2} \text{)}^2 \tag{38}
$$

$$
\sigma\_{1/Y} = \frac{1}{\left(\mathbb{Y}\_{\mathbf{X}/\mathbb{Y}}\right)^{\bullet}}.\tag{39}
$$

$$
\sigma\_{\alpha ?} = \frac{\text{Molar mass blomas}}{\text{Molar mass} \, ?} \, . \sigma\_{1/\text{y}} \tag{40}
$$

Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries 281

yields. It can also be observed that within the 3 repetitions there is about 20% of variation in

*C balance = 0.75 N balance = 0.71* 

Yx/glu **-0.1806** +/- 3.4% 0.988 **-0.1775** +/-10.8% 0.887 **-0.1495** +/-3.9% 0.983 YAA/glu **0.0374\*** +/- 5.8% 0.965 **0.0532 \*** +/-3.8% 0.983 **0.0266 \*** +/-6.1% 0.957 YN-NH3/glu **0.0096\*** +/- 2.5% 0.993 **0.0104 \*** +/-3.3% 0.987 **0.0079 \*** +/-3.3% 0.987 Yeth/glu **-0.3798** +/- 5.6% 0.970 **-0.3039** +/-11.2% 0.878 **-0.3047** +/-4.5% 0.978 Ygly/glu **-0.0156** +/- 5.7% 0.969 **-0.0222** +/-11.3% 0.876 **-** - - Yacet/glu **-0.0076** +/- 34.2% 0.461 **-0.0130** +/-14.7% 0.807 **-** - - YCO2/glu **-0.5245** +/- 3.4% 0.990 **-0.4288** +/-12.1% 0.873 **-0.4065** +/-2.8% 0.992 YO2/glu **-0.0332** +/- 4.2% 0.984 **-0.0410** +/-11.3% 0.888 **-0.0388** +/-4.0% 0.984 Yx/AA **-3.0418** +/- 8.1% 0.939 **-2.4486** +/-7.5% 0.942 **-4.0251** +/-6.4% 0.957 Yx/N-NH3 **-14.9807\*** +/- 4.8% 0.977 **-15.0498 \*** +/-7.1% 0.948 **-15.6974\*** +/-9.2% 0.916 Yx/Eth **0.5498\*** +/- 4.2% 0.981 **0.6869 \*** +/-3.8% 0.983 **0.5732 \*** +/-4.2% 0.979 Yx/gly **13.6788\*** +/- 5.1% 0.973 **10.4657 \*** +/-7.8% 0.932 **-** - - Yx/acet **9.2216\*** +/- 11.8% 0.866 **10.6540 \*** +/-6.0% 0.959 **-** - - Yx/CO2 **0.4193\*** +/- 4.8% 0.977 **0.5549 \*** +/-7.1% 0.947 **0.4441** +/-4.6% 0.978 Yx/O2 **6.0254\*** +/- 2.5% 0.994 **5.4944 \*** +/-5.3% 0.970 **4.7570** +/-5.7% 0.965

Table 8. Yields calculated for 3 repetitions of the batch growth of *Sc. cerevisiae*. . \* means that a linear regression without intercept was used. \*\* balance is obtained without glycerol and acetoin. r2 is the correlation coefficient of the linear regression used to

With the 22 experimental yields Yx/? of table 8 (and their standard deviation), and by fixing the biomass coefficient ߙହto 1, a system of 29 equations is built for computing the 10 stoichiometric coefficients of the reaction. The system is highly over determined and information given is redundant. The solution is computed using the same script (figure 3), by changing the file experimental matrix and imposing the element conservation

The stoichiometric equation computed is reported in table 9. The result obtained remains closed to the one obtained in table 7. It can also be observed that the standard deviations of all coefficients are significantly lower than the results presented in the table 8. In fact, repetitions of the experiment coupled with the over-determined system have increased the computation robustness (on a statistical point of view). Obviously, the stoichiometry remains theoretical and is an average of 3 experiments, but it is representative of the batch growth of *Sc. cerevisiae* and it can be also considered as reproducible for any growth of the

**4.4.2 Stoichiometric equation and comparison with experiments** 

r2 Yield (g.g-1)

*Experiment 2* 

Standard deviation

*Experiment 3* 

Standard deviation r2

*C balance = 0.72 \*\* N balance = 0.89* 

r2 Yield (g.g-1)

the mass yields calculated.

 Yield (g.g-1)

calculate the yield.

equations.

strain in the same condition.

*Experiment 1* 

Standard deviation

*C balance = 0.92 N balance = 0.78* 



Table 7. Stoichiometric equation with reconciliation with all the experimental data "Yx/?" obtained during a batch culture.

### **4.4 Stoichiometry using several repetition of the batch growth of** *Sc. cerevisiae* **in bioreactor**

It is generally recommended that an experiment is repeated several times in order to check its reproducibility and thus to confirm the results obtained. There are several ways for using the data obtained from the replication of the experiment:


### **4.4.1 Experimental yields of 3 repetition of the batch growth of** *Sc. cerevisiae*

The results obtained for 3 repetitions of the batch growth of *Sc. cerevisiae* are reported in table 8. The experimental carbon and nitrogen conservation balances reported in the table are calculated using the Yx/? yields (the balance can be different if calculated using the other yields). In experiment 2, the redundant information gives bad results in terms of conservation of elements. This is due to the large standard deviation calculated for the

Compound Stoichiometric coefficient Standard deviation Biomass 1.0000 0.0000 Glucose -0.7128 +/- 0.0145 (2.03%) AA -0.2166 +/- 0.0172 (7.94%) N-NH3 -0.0980 +/- 0.0041 (4.18%) Ethanol 1.0557 +/- 0.0287 (2.72%) Glycerol 0.0193 +/- 0.0010 (5.18%) Acetoin 0.0306 +/- 0.0036 (11.76%) CO2 1.2019 +/- 0.0286 (2.38%) O2 -0.1267 +/- 0.0032 (2.53%) H2O 0.5358 +/- 0.0096 (1.79%)

�������������� � ������ �\_� � ������ ��\_� � ������������ ���� � � ������� � ����������\_� � ������ ������� � ������ �������� � ������ ������� � ��������\_���

Table 7. Stoichiometric equation with reconciliation with all the experimental data "Yx/?"

**4.4 Stoichiometry using several repetition of the batch growth of** *Sc. cerevisiae* **in** 

It is generally recommended that an experiment is repeated several times in order to check its reproducibility and thus to confirm the results obtained. There are several ways for using

1. By analyzing the results of each experiment independently and comparing the final results. For this purpose, it is necessary to express the results of each experiment (as presented in table 5) into yields (as presented in table 6) and to compute the resulting stoichiometric equations by either simple mass balance or data reconciliation method. In such an approach the reproducibility is checked by comparing the equations

2. By analyzing the results of each experiment independently, but instead of computing one stoichiometric equation for each experiment, the yields calculated are used as a set of information to compute a single average stoichiometry for the repeated experiments. The reproducibility is tested either by comparing the results with the experimental yields or by analyzing the standard deviation computed for the stoichiometric

3. By compiling all the results of the experiments as a single set of data and calculating a single average yield from all experiments. The stoichiometric equation should then be computed using a reconciliation data method and the reproducibility is tested with the

The results obtained for 3 repetitions of the batch growth of *Sc. cerevisiae* are reported in table 8. The experimental carbon and nitrogen conservation balances reported in the table are calculated using the Yx/? yields (the balance can be different if calculated using the other yields). In experiment 2, the redundant information gives bad results in terms of conservation of elements. This is due to the large standard deviation calculated for the

standard deviation computed for the stoichiometric coefficients.

**4.4.1 Experimental yields of 3 repetition of the batch growth of** *Sc. cerevisiae*

obtained during a batch culture.

obtained for each experiment.

coefficients.

the data obtained from the replication of the experiment:

**bioreactor** 


yields. It can also be observed that within the 3 repetitions there is about 20% of variation in the mass yields calculated.

Table 8. Yields calculated for 3 repetitions of the batch growth of *Sc. cerevisiae*. . \* means that a linear regression without intercept was used. \*\* balance is obtained without glycerol and acetoin. r2 is the correlation coefficient of the linear regression used to calculate the yield.

### **4.4.2 Stoichiometric equation and comparison with experiments**

With the 22 experimental yields Yx/? of table 8 (and their standard deviation), and by fixing the biomass coefficient ߙହto 1, a system of 29 equations is built for computing the 10 stoichiometric coefficients of the reaction. The system is highly over determined and information given is redundant. The solution is computed using the same script (figure 3), by changing the file experimental matrix and imposing the element conservation equations.

The stoichiometric equation computed is reported in table 9. The result obtained remains closed to the one obtained in table 7. It can also be observed that the standard deviations of all coefficients are significantly lower than the results presented in the table 8. In fact, repetitions of the experiment coupled with the over-determined system have increased the computation robustness (on a statistical point of view). Obviously, the stoichiometry remains theoretical and is an average of 3 experiments, but it is representative of the batch growth of *Sc. cerevisiae* and it can be also considered as reproducible for any growth of the strain in the same condition.



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## **5. Conclusion**

Elements conservation laws are a prerequisite to evaluate a solid mass balance model in fermentation technology. As in Chemistry, the perfect vehicle for accounting for elements conservation is the stoichiometric equation. Application of this representation to biochemical systems (and particularly to microbial growth processes) presents some difficulties. The first one is linked to the fact that macromolecules and biomass constituents have seldom a well-determined elemental composition. This is a source of inaccuracy and variability. However general observation of biomass composition clearly shows that it is seldom highly variable (except in the case a product is accumulated into the cells in high quantities). The second obstacle is related to the fact that a single stoichiometry representation intrinsically assumes that the yields are constant. This is certainly a good assumption for continuous processes where biomass metabolism is confined in a constant environment. This is more inaccurate in the case of batch cultures.

Provided a complete statistical analysis is performed, including the calculation of both the coefficients and the model variances, the single stoichiometry approach can be applied for characterizing bioreactions.

We have presented here a data reconciliation technique coupled with the constraint of elements conservation. The main interest of this approach is that the coefficients are obtained with a flexible method applicable with linear algebra techniques, the result being "stoichiometrically" valid.

This technique has been applied to two cases. In the first case, results on the continuous culture of the rumen anaerobic bacteria *Fibrobacter succinogenes* lead to characterize the culture by a stoichiometric equation, slightly depending on the dilution rate. In the second case, batch experiments for the culture of *Saccharomyces cerevisiae* clearly indicate that a single stoichiometry approach is less accurate than for continuous cultures. Nevertheless, a stoichiometric equation has been obtained and realistic mean square deviations have been calculated for the coefficients. The technique has been applied to lump the experimental information from three independent experiments. This shows that this first-order stoichiometric model, including elements balance conservation, is certainly a valuable characterization of the biomass growth and of primary metabolites production. It must also be kept in mind that more complex models would involve more coefficients finally resulting in inaccurate predictions without creating more robustness.

### **6. References**


Elements conservation laws are a prerequisite to evaluate a solid mass balance model in fermentation technology. As in Chemistry, the perfect vehicle for accounting for elements conservation is the stoichiometric equation. Application of this representation to biochemical systems (and particularly to microbial growth processes) presents some difficulties. The first one is linked to the fact that macromolecules and biomass constituents have seldom a well-determined elemental composition. This is a source of inaccuracy and variability. However general observation of biomass composition clearly shows that it is seldom highly variable (except in the case a product is accumulated into the cells in high quantities). The second obstacle is related to the fact that a single stoichiometry representation intrinsically assumes that the yields are constant. This is certainly a good assumption for continuous processes where biomass metabolism is confined in a constant environment. This is more inaccurate in the case of

Provided a complete statistical analysis is performed, including the calculation of both the coefficients and the model variances, the single stoichiometry approach can be applied for

We have presented here a data reconciliation technique coupled with the constraint of elements conservation. The main interest of this approach is that the coefficients are obtained with a flexible method applicable with linear algebra techniques, the result being

This technique has been applied to two cases. In the first case, results on the continuous culture of the rumen anaerobic bacteria *Fibrobacter succinogenes* lead to characterize the culture by a stoichiometric equation, slightly depending on the dilution rate. In the second case, batch experiments for the culture of *Saccharomyces cerevisiae* clearly indicate that a single stoichiometry approach is less accurate than for continuous cultures. Nevertheless, a stoichiometric equation has been obtained and realistic mean square deviations have been calculated for the coefficients. The technique has been applied to lump the experimental information from three independent experiments. This shows that this first-order stoichiometric model, including elements balance conservation, is certainly a valuable characterization of the biomass growth and of primary metabolites production. It must also be kept in mind that more complex models would involve more coefficients finally resulting

Erickson, L.E., Minkevich, I.G. & Eroshin, V.K. (1978) Application of mass and energy

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balance regularities in fermentation. *Biotechnology and Bioengineering*, Vol.20, No.10,

*Fibrobacter succinogenes* as shown by in situ 1H NMR and 13C NMR investigations.

in inaccurate predictions without creating more robustness.

*European Journal of Biochemistry*, Vol.107, pp. 155-162.

**5. Conclusion** 

batch cultures.

characterizing bioreactions.

"stoichiometrically" valid.

**6. References** 

pp. 1595-1621.


**13** 

*Mexico* 

**Distribution Diagrams** 

*1Depto. de Química, Área de Química Analítica, Universidad Autónoma Metropolitana-Iztapalapa,* 

*Universidad Nacional Autónoma de México,* 

**and Graphical Methods to Determine** 

Alberto Rojas-Hernández1, Norma Rodríguez-Laguna1,

*2Facultad de Estudios Superiores-Cuautitlán, Lab. 10, UIM,* 

María Teresa Ramírez-Silva1 and Rosario Moya-Hernández2

**or to Use the Stoichiometric Coefficients** 

**of Acid-Base and Complexation Reactions** 

Graphical methods to study the behaviour of systems showing different chemical equilibrium are known and very used in several fields of Chemistry (particularly in Bioinorganic, Medicinal and Pharmaceutical Chemistry) in order to establish the chemical species that are related with drugs behaviour in different systems. Among these methods, the most commonly used are the distribution diagrams (Högfeldt, 1979), the titration curves (Asuero & Michałowski, 2011) and the molar ratio and continuous variations methods

In the present work we have selected two molecules used extensively like drugs. In order to exemplify some novelties related with distribution diagrams and titration curves for acidbase systems we have selected the case of oxine (HOX, 8-hydroxiquinoline) that has been used as antiseptic and disinfectant. On the other hand, we have selected the complexation interaction between Fe(III) and tenoxicam (Tenox) to show other novelties related with more complicated distribution diagrams and molar ratio and continuous variations methods, because tenoxicam has been extensively used as non-steroidal anti-inflammatory drug that may be complexed with several metal ions. The chemical developed formulae of these

Graphic representations of chemical systems have found wide application because a simple look at them allows for solve specific problems and have a panorama, qualitative and quantitative, for different problems and phenomena. Moreover, some of these representations also permit to graphically solve the stated problem with a predetermined error (Vicente-Pérez, 1985). Distribution diagrams of species are some of the most used

**2. Distribution diagrams for acid-base and complexation systems** 

**1. Introduction** 

(Hartley et al., 1980).

compounds are presented in Scheme 1.

Wells, J.E. & Russel, J.B. (1996). The effect of growth and starvation on the lysis of the ruminal cellulolytic bacterium *Fibrobacter succinogenes*. *Applied and Environmental Microbiology*, Vol. 62, No.4, pp.1342-1346.
