Methodology for Bioprocess Analysis: Mass Balances, Yields and Stoichiometries

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.


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 a n a l y s i 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 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.

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(Solomon et al., , 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, 1980Roels, , 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.

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.

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 de l e m e n t ES dt =− 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: 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 www.intechopen.com 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.

Mathematical description of the equation system
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 unknown stoichiometric coefficients α i : 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: 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: 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" thanks to a substitution method. 2. More information than d is available; the system must be solved with the use of a reconciliation method as well as the use of statistical results. 3. With not enough information, the system cannot be solved.

Mathematical method for data reconciliation
Let the resulting linear system of constraints be written as follows: 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 Ŷ exp (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, Y r (m) is the column vector of the yields values obtained after data reconciliation. The following matricial expression is written: 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: 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 Y r (m) and K (p). Similarly, Ĥ exp (m+p) is the column vector formed by concatenation of Ŷ exp (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: 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: i.e.
Φ is a dimensionless number if W is used.

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In this study the experimental values are known, and both the confidence interval (i y ) and the variance (σ y ) of the model are given: 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: 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: Calculation of components of vector α is therefore obtained by the classical method of Lagrange multipliers. The Lagrangian function to be minimized is: 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: is a matrix derivation which leads to the n following equations : This leads to: and: considering that: if we denote ψ = M t W M, α is given by: Summarizing and combining computed matrices, the calculation leads to: The solution is given by: The standard deviations of the calculated coefficients are estimated by diagonal elements of matrices of covariances: Covariance on estimates of the model is: 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.

First application: Stoichiometric analysis of Fibrobacter succinogenes growth 3.1 Introduction
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 theFibrobacter 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.

Culture conditions
The strain used was Fibrobacter succinogenes S85 (ATCC 19169), and was grown anaerobically under 100% CO 2 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 Na 2 CO 3 (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 CO 2 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 000g, 5 min), and used to perform the colorimetric assays of ammonium ions, proteins and total www.intechopen.com 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: cell dry weight = 0.482 (± 0.034) OD 600 nm (31) where the cell dry weight is expressed in g.L -1 (correlation coefficient 0.989).
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.

Experimental data: Biomass formulae and element recoveries
The average molar biomass formulae established after elemental analysis during steadystate are presented in Table 1. 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 www.intechopen.com 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 CO 2 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.

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: 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 (Ŷ exp ) 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 Ŷ exp (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.
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 (Y r ) 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). 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 Ŷ exp (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.

Discussion
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 Y r /Y exp 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 Y r and Y exp on ammonium sulfate was explained by N-balances that were not satisfactorily assessed at these dilution rates. Y r 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.

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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.

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: 1. analysis of a single batch experiment with the minimal experimental data required for the stoichiometry (i.e. simple mass balance approach) 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 the experimental results. 3. analysis of several batch experiments (repeatability of experiments) for establishing the stoichiometry, including the statistical analysis of the repetition.

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 CO 2 (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 www.intechopen.com 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 O 2 and CO 2 molar fractions ( 2 out O y 2 out CO y ) which are measured by an Oxymat 5E (Siemens) and an infrared CO 2 analyser (Schlumberger), respectively.

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: 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. CO 2 production and O 2 consumption have been computed from the integration of the instantaneous CO 2 and O 2 respiration rates (r CO2 , r O2 ) acquired from the online gas balance measurements. The respiratory quotient (RQ = r CO2 / r O2 ) is also computed and its value above 1 is an indicator of the growth fermentative metabolism.  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 (Y A/B ) is calculated by a linear regression (figure 2) using the concentrations obtained in batch: 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). 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 Y X/? 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.

Stoichiometry for a minimal set of experimental data (simple elemental balance approach)
If it is chosen to solve the system using the minimal set of 5 experimental data (d=5) and without using reconciliation methods, it is crucial to use strictly independent relations. In the case of the Exp. 1, for instance, it is impossible to select simultaneously the biomass coefficient to 1 and to use the yields Y x/N-NH3 and Y x/AA . The 3 information are linked through the N balance (biomass, Amino Acid and NH 3 are the 3 compounds involved in the N-balance) in such a way that only two independent information are really introduced.
The most simple way to solve the stoichiometric equation with the list of yields presented in table 6 is to choose fixing the biomass coefficient to 1, considering 4 independent results within the list of the yield Y x/? .
Taking for example Y x/Eth , we can write : Thus, the ethanol stoichiometric coefficient is experimentally fixed and can be used as a relation to set up one of the system degree of freedom..
In the same way, taking the 3 other yields Y x/AA , Y x/gly and Y x/O2 , the following stoichiometry can be established : . By taking other experimental data, a different stoichiometric equation is obtained. For example using the experimental yields Y x/CO2 instead of Y x/Eth (it can be noticed in table 6 that the two yields were calculated with quite the same accuracy), the stoichiometric equation becomes: .
In this last expression of the stoichiometry, the coefficient for ethanol is 20% higher and the one of acetoin is hundred times lower than in the previous one. This illustrates the influence of the experimental date choice used to establish the stoichiometric equation, as with the minimal set of information necessary to solve the system, all the weight of the constraint for the C H O N conservation remaining on the coefficients that are not given experimentally.

Stoichiometry using the full set of experimental data
It was previously outlined that even if using only the required experimental data to fulfil the degree of freedom is the easiest methodology, this approach is very sensitive to the experimental data choice and to their accuracy. A way to solve this problem is to use all the available experimental information, with an accuracy evaluation of these data. Such an approach requires the reconciliation method presented in 2.2.2.
With the 8 experimental yields Yx/? of table 6 (and their standard deviation), and by fixing the biomass coefficient to 1, a system of 13 equations is built for computing the 10 stoichiometric coefficients of the reaction.
If the variance of Y x/? is computed as: the variance of 1/ Y x/Eth can be estimated to be : Thus, assuming a variance of 0 for the fixed reference stoichiometric coefficient (the one of the biomass), the variance of each stoichiometric coefficient used as a new experimental relation (as detail previously for ) can be calculated as The system is computed using the Matlab® (or Octave®) script presented in figure 3.
The resulting stoichiometric equation coefficients are determined, and their confidence interval is calculated (table 7).
The results are consistent with those computed with the previous classical approach. It is noticed that the main uncertainties concern the compounds that have the highest experimental uncertainty (i.e. acetoin and amino acid). This is one of the reconciliation method objectives: to have a high weight on the data that have been obtained with a high accuracy. The standard deviation for the coefficients is lower than the experimental ones but remains in the same order of magnitude.

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: 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 obtained for each experiment. 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 coefficients. 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 standard deviation computed for the stoichiometric coefficients.

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. 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. r 2 is the correlation coefficient of the linear regression used to calculate the yield.

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 to 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. -5.7560 -0.04% Table 10. Theoretical yields computed with reconciliation with all the experimental data "Yx/?" obtained with the 3 repetitions of the same batch culture.
One of the main feature of these computation remains in the fact that the computed coefficients satisfy elements conservation; this provides the robustness of the stoichiometric model; on the other hand, it must also be kept in mind that the model, even robust, intrinsically assumes (i) that all substrates and products have really been identified and taken in the stoichiometry and (ii) that all yields remain constant during the observation time (culture duration) taken for estimation of experimental yields. This is certainly the main difficulty here, both for the estimation of experimental yields from linear regression and for imposing a single stoichiometry model that summarizes three different experiments obtained in batch cultures. Any experimental deviation from the assumption of constant yields creates a lack of adequacy of the model that results in an increase of the variance of the model.
Nevertheless, it can be concluded that the results obtained, within their confidence intervals (table 10), fairly represent the experimental data obtained from the 3 different experiments, knowing that it is not clear if a more complex model (with more coefficients to identify) would be justified by the available degree of experimental information. www.intechopen.com

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.  (1996). The effect of growth and starvation on the lysis of the ruminal cellulolytic bacterium Fibrobacter succinogenes. Applied andEnvironmental Microbiology, Vol. 62, No.4, pp.1342-1346. www.intechopen.com The aim of this book is to provide an overview of the importance of stoichiometry in the biomedical field. It proposes a collection of selected research articles and reviews which provide up-to-date information related to stoichiometry at various levels. The first section deals with host-guest chemistry, focusing on selected calixarenes, cyclodextrins and crown ethers derivatives. In the second and third sections the book presents some issues concerning stoichiometry of metal complexes and lipids and polymers architecture. The fourth section aims to clarify the role of stoichiometry in the determination of protein interactions, while in the fifth section some selected experimental techniques applied to specific systems are introduced. The last section of the book is an attempt at showing some interesting connections between biomedicine and the environment, introducing the concept of biological stoichiometry. On this basis, the present volume would definitely be an ideal source of scientific information to researchers and scientists involved in biomedicine, biochemistry and other areas involving stoichiometry evaluation.