I have seen that some equation sets have an explicit way of defining the substrate consumption whereas others have an implicit way. I would like to know how to convert the former into the latter.
For instance, in the attached file, it can be seen that there are two ways to describe the growth of bacteria.
In the first (explicit), the presence of a half-saturation constant suggests that the model is a Michaelis-Menten case. The second is straight a logistic model.
I was wondering if I can convert the first into the second, that is explicit to implicit.
Considering that the Monod term says that
μ = μᵢ[S/(S + K)]
then I can write
Ṅ = μN * logistic term
My question is: Would this modification be mathematically and biologically sound?
If yes, how can I estimate the carrying capacity? For instance, if I know the amount of limiting substance, can I estimate the number of cells that a system can sustain?
Thank you
Perhaps looking at the original papers would suggest to you why there are two models. This may lead to the answer to your original question or an excellent dissertation problem for an interested PhD student. Best wishes, David Booth
You will be best judge I think . There are so many model papers of logistic regression on Michales Menten equation - https://www.youtube.com/watch?v=fv6OAVe2vQc
https://www.researchgate.net/post/Logistic_Regression-Applications
The Monod/Michaelis-Menten type growth model in your example allows to eliminate S, if you know the initial S_0 and the consumption of of substrate per biomass c_i of species i. In this case you can substitute (S_0 - \sum c_i N_i) for S and will end up with a growth model that differs from the logistic model by a denominator and the possibly different coefficients c_i. So, as far as I see it, there is no conversion from one model to the other. However, the two models have some similarities, and for certain ranges of values (e.g. if the substrate concentration is always small compared to K_S and the c_i are equal) the logistic model would 'fit well', if the first model is valid.
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