Fermentation Technologies
Monod Equation - Fermentation Contextual Analysis

Every undertaking of a venture in biotechnology involves the use of the Monod Equation in calculating the growth of cells or microbes. This equation is then related to the substrate consumption rate equation for the purposes of the fermentation reactor design or analysis. The Monod equation is empirical having been developed to describe experimental data; nonetheless, it works fairly well. The Monod equation is represented as

                      dX/dt = K_mXS/(K_e + S)                           (1)

where r (or dX/dt) is the rate of microbe growth, X is the quantity of the microbes, S is the concentration of the substrates, and K_m and K_e are constants. Interestingly, there has been little effort too to derive it from first principles of biochemical science. Yet it is instructive to relate the equation to the characteristics of ethanol fermentation.

The fermentation reaction has been determined to require other reactants besides substrates to support cellular functions including growth in order for the fermentation to proceed significantly. These substances further also vary for every  fermentative microbial reaction. For the oxidative assimilation of glucose by yeast there are about eleven different substances that had to be added to support the reaction. These are carbon, oxygen, nitrogen and hydrogen; to leaser extent quantities of phosphorus, sulfur, potassium, and magnesium must also be provided for the synthesis of minor components; and minerals (i.e. Mn, Co, Cu, Zn) and organic factors (amino acids, nucleic acids, and vitamins) are required in trace amounts. The manner by which the microbes use all these supplements have not been fully determined as yet.

However, it is known that some of these reactants are for supporting the parallel anabolic reactions of the microbial metabolic reactions, while others simply are co-catalysts of the fermentation reactions. More specifically, with respect to the oxidation of glucose by yeast the roles of Magnesium and Potassium are effectively well-documented through the Embden Meyerhoff Pathway, which shows the potassium ion K⁺ and magnesium ion Mg²⁺ enable the stabilization of the ATP in the chain reversible reaction pathway. Therefore these ions are critical to the oxidative assimilation of glucose and consequentially to the growth and survival of the microbe. So,  new cells that are formed from cell division are not likely to survive without potassium and magnesium ions in the broth for the new cells to absorb for the oxidation of glucose.

While this empirical analysis offers some insight into the role of the additional reactants and catalysts, it also raises other issues. First and foremost, the Monod equation does not provide a means for pre-calculating the quantity of the additional reactants needed as feed to support the fermentation reactions under conditions of cellular growth. Consider that as per the Embden Meyerhoff Pathway, only one ion  each of  Magnesium and Potassium ions participates per pathway or glycolysis reaction.  So, if several sets of enzymes are undertaking the reactions simultaneously, which is to say that each microbe is using several sets of the Potassium and

In effect then the concentration of K or Mg in the feed must be related to the population differential increase ΔP over the reaction time ( or reactor residence time) Θ:

                          ΔP = (ΔP/Δt) x Θ                              (2)

where (ΔP/Δt) is the average growth rate per unit time of the microbe while in the reacting state, which must be determined during the experimental studies. The population increase differential however has to be converted into the number of new microbes as the microbial Biochemical Pathway of glycolysis is based on single cell domain. Clearly the number of new of microbes N is readily obtained from the population differential increase, ΔP, divided by the weight of a single microbe, the germ-weight, g_w, leading to

                        N = ΔP/g_w                                          (3)

Now given Es is the number of enzymes which are  simultaneously participating  in the pathway, then the number of ions/atoms of the reactants substances such as magnesium Mg or potassium K would have as a minimum, M_i:

                      M_i = N E_s/litrefeed                              (4)

Of course this value has to be adjusted for Symporters intake rate R_s as the substances are dispersed in the mash, with the effective quantity: 

                        M_ia = N x Es  + ∫R_sdτ                        (5)

with the 'τ' being integrated over the reaction time, Θ, The values M_i and M_ia are converted to moles and then to grams-weight if needed. Obviously then not all the  potassium in the feed are absorbed by the microbe, some are discharged with the effluent, which must now be extracted as not to become pollutants with the discharge.

Analyzing the Monod equation to elicit the factors accountable for the pathway-roles of Magnesium and Potassium ions, presents as best candidates the reaction constants K_m or  K_e. Modification of these reaction-specific constants  to reflect the observation yield the representations:

K_m = m_of(Cp) 
K_e= K₀(a + b/g(Cp) )

where a and b are constants and Cp is the concentration of Potassium in the broth, and with the stipulation as Cp goes to zero the function f(Cp)

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must go to zero faster than K_e does due to the function g(Cp). The fact that f(Cp) is used in adjusting the K_m is simply to denote the complexity of that relation, in anticipation of such complexity based on development drawing from magnetic poles distribution concepts.

The analysis of the Monod equation reveals another curious aspect to the design needs of fermentation reactors. As has been noted, one of the steps of the biochemical pathway is the transporting of the substrate molecules into the cell-interior by the Plasma Membrane Transporters of the microbe membrane. However, the Monod equation presents the microbial glucose oxidative process as uniformly homogenized reactions, beginning with the glucose molecule being absorbed into the Plasma membrane Transporters pores followed by the transporting process. Naturally, this situation would impede the absorption of other substances that impact cellular growth functions and thereby making such substance a limiting reactant of the operational pathway; and therefore raising the issue of the substrate being limiting reaction.

This simple analysis reviewing the use of the Monod equation in the design of fermentation reactors has raised cyclic complexities which reveals that there exists the need for using the actual reaction rate expression derived from the Biochemical Pathway for the design to be as descriptive of operating conditions as possible. Consequentially, the Biochemical Pathway for every microbial fermentation reaction of industrial biotechnology processes needs to be discovered as part of the requirement of efficacious design of equipment for the biotechnology operation. reactions.

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