On a class of sigmoidal growth models induced by reaction networks

Milen Kolev Borisov, Svetoslav Markov*


We formulate and study a class of sigmoidal growth models induced by catalytic reaction network via the mass action law. The proposed reaction network, called “pre-logistic”, is based on the autocatalytic logistic reaction inducing the familiar logistic growth model. The pre-logistic reaction network consists of two catalytic reactions. We show that the dynamics of the pre-logistic network is close to the dynamics of either the logistic, or the first-order  saturation reaction depending on the ratio of  the rate constants. In particular we show that the autocatalytic logistic reaction can be obtained as a limit case of the pre-logistic reaction network whenever one of the rate constants tends to infinity while the other is kept fixed. Hence the pre-logistic reaction can be considered as a generalization of both the logistic and the saturation reactions and thus can be used in a number of real-world situations when modeling growth processes in life sciences. A generalization of the pre-logistic network involving more intermediate steps is also discussed. The proposed growth models possess clear physic-chemical mechanisms and are suitable for fitting a variety of measurement data. Numerical examples of the proposed growth models using experimental measurement data from the field of microbial cultivation are presented and graphically visualized.

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Anguelov R., Borisov M., Iliev A., Kyurkchiev N., Markov S., On the chemical meaning of some growth models possessing Gompertzian-type property, Math Meth Appl Sci. 2017;1–2 https://doi.org/10.1002/mma.4539

Iliev A., Kyurkchiev N., Markov S., On the Approximation of the step function by some sigmoid functions, Mathematics and Computers in Simulation 2017; 133:223--234.

Lente G. Deterministic Kinetics in Chemistry and Systems Biology, Briefs in Molecular Science, Springer 2016.

Chellaboina V., Bhat P., Haddat M., Bernstein S., Modeling and Analysis of Mass-Action Kinetics, IEEE Control Systems Magazine 2009; 60--78.


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