Model

Overview

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Goal and Motivation

Theoretically, when our functional BCAA strain is grown in vitro, for instance in a broth with some arbitrary BCAA concentration, we expect that the BCAA concentration decreases predictably with respect to time given by some rate law. Our goal is to derive this rate law from our gene circuit, and calculate most of the rate constants empirically or from literature. If we have a reasonable rate law for BCAA concentrations in vitro, we hope that it can be extended to the human gut microbiome. In addition, knowing a rate law will help with calculating rate constants from empirical data. Below we derive our rate law by breaking it down into separate models that we combine (using algebraic substitution) to get the rate law for BCAA disappearance.

Mutlicompartment Modeling

First, let us establish how mass action kinetics work in multi-compartment systems. We know that mass must be conserved, so the number of moles disappearing from one compartment is the number of moles appearing in another compartment, and when accounting for volume:

$$\frac{dA_{1}}{dt} = - \frac{dA_{2}}{dt}$$ $$\frac{dA_{1}}{dt} \cdot \frac{Vol_1}{Vol_1} = - \frac{dA_{2}}{dt} \cdot \frac{Vol_2}{Vol_2}$$ $$\frac{dA_{1}}{dt} \cdot \frac{Vol_1}{Vol_1} = - \frac{dA_{2}}{dt} \cdot \frac{Vol_2}{Vol_2}$$ $$\frac{d[A_{1}]}{dt} \cdot Vol_1 = - \frac{d[A_{2}]}{dt} \cdot Vol_2$$ $$\frac{d[A_{1}]}{dt} = - \frac{d[A_{2}]}{dt} \cdot \frac{Vol_2}{Vol_1}$$

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References

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Team:MIT/Model