L ilse

where 6h(x) denotes the threshold value of x. Soft thresholding shrinks the wavelet coefficients which are greater than the threshold value towards zero by subtracting the threshold value from the wavelet coefficients as well:

Different methods for selecting the threshold value have been suggested in the literature by Donoho and co-workers [132]. These methods are grouped

Figure 3.13. Wavelet decomposition of a process signal (CO2 evolution rate).
wavelet denoised signal
Figure 3.14. Wavelet denoising of a process signal.

into two categories as global thresholding and level-dependent thresholding. A single threshold value is applied for all scales in global thresholding whereas for level-dependent thresholding, a different threshold value is selected for each scale. Level-dependent thresholding is suitable especially for data with non-stationary noise. Figure 3.14 illustrates the wavelet de-noising of a process variable using level-dependent hard thresholding. Haar wavelet was used for de-noising the data in three scales.

3.6 Theoretical Confirmation/Stoichiometry and Energetics of Growth

Data collected from a process should also be checked for consistency by using fundamental process knowledge such as stoichiometry and the energetics of growth. Cell growth involves consumption of nutrients for the synthesis of additional biomass. The nutrients that supply energy and raw materials for the biosynthesis should be compatible with the enzymatic machinery of the cell. Knowledge of the reaction stoichiometry provides a convenient way of obtaining various yield coefficients and consequently provides information for formulating a growth medium that will supply all the required nutrients in balanced amounts. This will in turn be very useful for (i) determining the other quantities by expressing one in terms of the others, (ii) monitoring the bioprocess, (iii) eliminating the measurements of compounds that are difficult to measure while keeping track of the easy to measure ones.

For many microorganisms, the energy and carbon requirements for growth and product formation can be met by the same organic compound. This considerably simplifies the analysis of cellular kinetics.

3.6.1 Stoichiometric Balances

To examine cell growth, it is important to know what the cells are made of, that is their chemical composition. Although there are many different biological species, it turns out that a very large fraction of their mass is made of a few elements, namely carbon (C), oxygen (O), nitrogen (N) and hydrogen (H). Minor elements in the cell include phosphorus, sulfur, calcium, potassium and sodium. Typically, 70% of cell mass is water and the remainder is dry matter. Therefore it is conventional to express cell composition on a dry basis. Nearly half of the dry matter in cells is carbon and the elements carbon, oxygen, nitrogen and hydrogen make up about 92% of the total dry mass. In different microbes, the carbon content varies from 46 to 50%, hydrogen from 6 to 7%, nitrogen from 8 to 14% and oxygen from 29 to 35%. These are small variations and they appear to depend on substrate and growth conditions. For many engineering calculations, it is reasonable to consider the cell as a chemical species having the formula of CH1.gO0.5N0.2-This engineering approximation is a good starting point for many quantitative analyses while a more carefully formulated empirical formula based on gravimetric techniques may be necessary for complete material flow analysis. The cell "molecular weight" for the generic molecular formula stated above is then 12+1.8 + 0.5(16) +0.2 (14) = 24.6. More generally, the elemental composition of the cell can be represented as CHaO{,Nc. Elemental composition of selected microorganisms can be found in [26].

When the cells grow in a medium (a source of all elements needed by the cells) in the presence of oxygen, they oxidize or respire some of the carbon to produce energy for biosynthesis and maintenance of cellular metabolic machinery. Furthermore, cells may produce extracellular products that accumulate in the broth. The overall growth process may therefore be represented simplistically as:

More Cells + Extracellular Products + C02 + H20 (3.36)

Carbon dioxide and water on the product side of the reaction (overall growth process) result from oxidation of carbon source (such as glucose) in the medium. Assuming that glucose and ammonia are the sole C and N sources, and the cell composition is represented as CH1.sO0.5N0.2, the overall cell growth may be described by

Here, CHxOyNz is the elemental composition of extracellular product and a, b, x, y, z ,a, /3, 7 and 5 are the parameters to be determined. In order to calculate these parameters, some additional information, such as yield coefficient (Yx/S), respiratory quotient (RQ) and degree of reductance (70), is needed.

Elemental balances, when applied to Eq. 3.37, lead to the following algebraic relations

Eqs. 3.38-3.41 reduce the degrees of freedom (parameters to be determined, such as a, b, x, y, z, a, /?, 7, and 5 by four. If the elemental composition of the extracellular product is available a priori, then additional information on this variables, such as cell mass yield {Yx/S and respiratory quotient (RQ), is needed. If the elemental composition of the extracellular product is not available, then information on five variables must be available from experiments for complete specification of Eq. 3.37.

Cell Mass Yield can be defined as the amount of cell mass produced per unit amount of substrate consumed,

C6H1206 + aNH:i + b02 -> aCH1AOÜANa.2 + ßCHxOyNz + jC02 + 6H20 . (3.37)

6 = a + ß + j 12 + 3a = 1.8a + xß + 25 6 + 2 b = 0.5a + yß + 2-y + 6 a = 0.2a 4- zß

amount of cell mass produced AX

amount of substrate consumed A S

The subscript x/s denotes that, cell yield (X) is based on substrate (S). This notation is especially important when there is more than one substrate which significantly influences cell mass yield. This definition of yield can be extended to non-biomass products (P) with the basis being substrate consumed or biomass produced:

Y amount of product produced AP ^ ^ amount of substrate consumed AS

Y _ amount of product produced _ AP _ v' x amount of cell mass produced AX

The cell mass yield based on oxygen (Yx/0) and yield of ATP (Adenosine triphosphate, YATFix) can be obtained in analogous manner.

Respiratory Quotient, RQ, is defined as the rate of carbon dioxide formation divided by the rate of oxygen consumption in aerobic growth.

rate of 02 consumption

This ratio can be calculated from on-line measurements of feed and exit CO2 and O2 using CO2 and O2 analyzers. If the nature of the major extracellular product(s) is known (i.e., x, y, z of CHxO^Nz), then it is possible to calculate the parameters a, P, 7 and <5 in Eq. 3.37 from experimental measurement of RQ and one other measurement. If no significant amount of extracellular product is formed, as in some cell growth processes, then it is evident from Eqs. 3.38-3.41 (P = 0) only one measurement such as RQ is needed to calculate stoichiometric coefficients.

Degree of Reductance of an organic compound is defined as the number of electrons available for transfer to oxygen upon combustion of the compound to CO2, N2 and H2O. It is also defined as the number of equivalents of available electrons per g atom of the compound. The number of equivalents for carbon, hydrogen, oxygen and nitrogen are 4, 1, -2 and -3 respectively. In view of this, for different cell compositions, degree of reductance can be calculated. Examples of the degree of reductance values for a wide range of compounds can be found in [514],

3.6.2 Thermodynamics of Cellular Growth

A complex network of metabolic reactions in microbial growth is involved.

These reactions are either catabolic or anabolic. The former type releases energy while the latter consumes energy. However, some energy is always lost as heat. For this reason, in large-scale processes, it is necessary to remove this heat so that the culture is maintained at its optimum temperature. When there is negligible amount of extracellular product formation under aerobic conditions, the growth reaction (Eq. 3.36) may be rewritten as,

Cell + Source (Carbon, Nitrogen, etc.) + 02 —>

and assuming cell composition is CH1.sO0.5N0.2, Eq. 3.37 becomes

C6H1206 + aNH3 + b02 —> aCHi.sOo.nNo.t + PC02 + -yH20 (3.47)

The total heat evolved (AQ) during growth can be calculated from an enthalpy balance

AQ = (-AHs)(-AS) + (-AHn)(-AN) - (-AHx)(-AX) (3.48)

where, AHs, AHpj, AHx are the heats of combustion of carbon substrate, nitrogen substrate and cells in kcal/g respectively. AS, AN, AX are the amounts of the corresponding materials consumed or produced. The value of the heat of combustion of cells can be estimated using a modification of the Dulong equation [67] using cell composition data that is experimentally determined,

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