Mathematical Model for Fed-Batch Penicillin-G Fermentation
Based on work described by Bajpai and Reuss (1980), the mass balances for fed batch fermentation of P. Chrysogenum to produce Penicillin G are, as told by Patnaik (2001) Birol et. al (2002) :
where X is biomass concentration in culture, S is substrate concentration in culture and Sf is the substrate concentration in feed, P is product (penicillin G) concentration in culture, V is the culture volume, F is the feed rate, µ is the specific growth rate of P.Chrysogenum, π is the specific rate of product formation, m is the maintenance coefficient on substrate, Yx/s is the yield of biomass per unit substrate and Yp/s is the yield of penicillin G per unit substrate (glucose). The specific growth rate equation for the P.Chrysogenum is based on the Monod equation but accommodates the reality that specific growth rate lessens with increasing biomass concentration. There are other factors such as temperature and pH which will be included in the equation later, but the basic growth equation is:
where Ks is the substrate saturation constant and µmax is the max specific growth rate of P.Chrysogenum. The presence of X implies that with increasing concentrations the specific growth rate will be lower. The specific rate of penicillin production can be found by an equation similar to the growth equation but with some changes:
where Kp is the Monod inhibition constant, KI is the inhibition constant for product formation and πmax is the max rate of penicillin formation.
These equations are the most basic unstructured model equations for industrial fed-batch Penicillin-G manufacture. In order to have a more realistic model, parameters such as dissolved oxygen consumption, pH, input variables such as agitation power and aeration, heat generation, acid and base additions, broth evaporation rates etc. need to be included in the complete model. Work done by Gulnur Birol et. al (2002) will be the basis for the predictive model used for this design project. Unstructured models with enough detail are sufficient for process prediction, however they do not account for cell morphology changes or strain optimisations. For this project, unstructured model equations will be the sole basis for the mathematical model.
Expansion of Growth Rate Expression
The growth rate of biomass is, from the Monod equation, proportional to the utilisation of substrate. In the growth rate equation above implies glucose is the only substrate but clearly oxygen is a vital substrate also. So, by including dissolved oxygen consumption into the equation, the first rearrangement of the growth rate equation is obtained:
where CL is dissolved oxygen concentration in the culture and Kox is the oxygen limitation constant. The same is true for the specific rate of penicillin G production:
where Kop is the oxygen limitation constant for product formation.
The effects of pH and temperature must also be incorporated into this equation in order to satisfactorily replicate the growth kinetics in the bioreactor environment, thus implying they also affect substrate utilisation and penicillin production. Their direct effects on the latter two parameters are mitigated from the model due to the complicated nature of the relationships and lack of experimental data.
Effect of pH on Growth Rate
The growth rate is found to be inversely proportional i.e. inhibited, to a function of an acid dissociation model described as:
where K1 and K2 are constants that have values elucidated in previous literature that Girol et. al have researched and [H+] is the concentration of hydrogen ions in the culture broth. The change in hydrogen ion concentration is dependent on biomass growth, and is generally counteracted by the...
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