From Physics to Bioengineering: Microbial Cultivation Process Design and Feeding Rate Control Based on Relative Entropy Using Nuisance Time

Abstract

For historic reasons, industrial knowledge of reproducibility and restrictions imposed by regulations, open-loop feeding control approaches dominate in industrial fed-batch cultivation processes. In this study, a generic gray box biomass modeling procedure uses relative entropy as a key to approach the posterior similarly to how prior distribution approaches the posterior distribution by the multivariate path of Lagrange multipliers, for which a description of a nuisance time is introduced. The ultimate purpose of this study was to develop a numerical semi-global convex optimization procedure that is dedicated to the calculation of feeding rate time profiles during the fed-batch cultivation processes. The proposed numerical semi-global convex optimization of relative entropy is neither restricted to the gray box model nor to the bioengineering application. From the bioengineering application perspective, the proposed bioprocess design technique has benefits for both the regular feed-forward control and the advanced adaptive control systems, in which the model for biomass growth prediction is compulsory. After identification of the gray box model parameters, the options and alternatives in controllable industrial biotechnological processes are described. The main aim of this work is to achieve high reproducibility, controllability, and desired process performance. Glucose concentration measurements, which were used for the development of the model, become unnecessary for the development of the desired microbial cultivation process.

Identification of Functional Bioprocess Model for Recombinant E. Coli Cultivation Process

“…The posterior distribution for the m-th offline sample is: where every sampled prediction m has a constant variance σ 2 X . Prior distribution also has the form of Gaussian distribution [31,32]:…”

“…where X y m is the mth observation value of the biomass concentration and its individual variance is σ 2 X,m . Integration of relative entropy [31] yields:…”

“…where a further procedure neglects coefficient c. In a previous study [31], the uncertainty of prior distribution was set as equal to the squared observed value. However, Reference [3] showed that there are trade-offs between the least squares approach and the squared mean absolute percentage error (MAPE) criterion.…”

“…The tuning coefficient K exp 0 ≤ K exp ≤ 1 with a value of 1 recreates the least squares approach, which has a higher penalty for bigger criterion values. Meanwhile, the value of zero results in the squared MAPE criterion [31], which restricts estimation errors to smaller overall criterion values. Such criteria showed acceptable practical benefits in a generic case of a biomass model fitting procedure.…”

“…The process of identifying E. coli BL21 (DE3) pET28a strain's stoichiometric parameters and protein model fitting coefficients is based on the convex optimization method and the maximization of entropy [31,37]. Figure 2 depicts the workflow of the optimization procedure.…”