The encapsulation of transcription-translation (TX-TL) cell-free machinery inside lipid vesicles (liposomes) is a key element in synthetic cell technology. The PURE system is a TX-TL kit composed of well-characterized parts, whose concentrations are fine tunable, which works according to a modular architecture. For these reasons, the PURE system perfectly fulfils the requirements of synthetic biology and is widely used for constructing synthetic cells. In this work, we present a simplified mathematical model to simulate the PURE system operations. Based on Michaelis-Menten kinetics and differential equations, the model describes protein synthesis dynamics by using 9 chemical species, 6 reactions and 16 kinetic parameters. The model correctly predicts the time course for messenger RNA and protein production and allows quantitative predictions. By means of this model, it is possible to foresee how the PURE system species affect the mechanism of proteins synthesis and therefore help in understanding scenarios where the concentration of the PURE system components has been modified purposely or as a result of stochastic fluctuations (for example after random encapsulation inside vesicles). The model also makes the determination of response coefficients for all species involved in the TX-TL mechanism possible and allows for scrutiny on how chemical energy is consumed by the three PURE system modules (transcription, translation and aminoacylation).
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