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Team:Heidelberg/Templates/Modelling/Ind-Production
From 2013.igem.org
Challenge
A challenge we had to face during the characterization and optimization of indC was to identify the production kinetics of Indigoidine. In order to disentangle the underlying mechanisms of bacterial growth and peptide synthesis, we decided to set up a mathematical model based on coupled ordinary differential equations (ODEs). Calibrated with our experimental time-resolved data, the mathematical model could potentially not only elucidate how Indigoidine production influences growth of bacteria but also provide a more quantitative understanding of the synthesis efficiency of the different T domains and PPTases that were tested.Approach
First, we set up a mind model based on the fact that Indigoidine is produced from Glutamine (Glu) that is cyclized (cGlu)Since we had already established our quantitative Indigoidine production assay (see Tag-Optimization) in a time-dependent manner, we wanted to further exploit these experimental data via quantitative dynamic modeling. The change of bacteria and Indigoidine with time was measured via optical density of the liquid cultures in a 96-well plate of a TECAN reader and can be described in ordinary differential equations (ODEs). Such ODEs contain parameters that characterize e.g. growth or synthesis rates for bacteria or Indigoidine, respectively.
Ordinary Differential Equations (ODEs)
But how to find proper equations for bacterial growth and indigoidine synthesis? From our mind model (Fig. 1), we derived ODEs based on mass-action kinetics$$ \mathrm{d}\mathrm{[Bac]}/\mathrm{d}t = -\frac{\mathrm{[Bac]} \cdot \left(\mathrm{[Bac]} - \mathrm{Bacmax\_native\_svp}\right) \cdot \left(\mathrm{beta\_native\_svp} - \mathrm{[Ind]} \cdot \mathrm{ki\_native\_svp}\right)}{\mathrm{Bacmax\_native\_svp}} $$ $$\mathrm{d}\mathrm{[Glu]}/\mathrm{d}t = - \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn\_native\_svp} $$ $$\mathrm{d}\mathrm{[cGlu]}/\mathrm{d}t = - \mathrm{kdim\_native\_svp} \cdot {\mathrm{[cGlu]}}^2 - \mathrm{kdegg\_native\_svp} \cdot \mathrm{[cGlu]} + \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn\_native\_svp} $$ $$\mathrm{d}\mathrm{[Ind]}/\mathrm{d}t = {\mathrm{[cGlu]}}^2 \cdot \mathrm{kdim\_native\_svp} - \mathrm{[Ind]} \cdot \mathrm{kdegi\_native\_svp} $$
This system contains 4 dynamic variables: Bacteria ($$Bac$$), Glutamine ($$Glu$$), cyclized Glutamine ($$cGlu$$) and Indigoidine ($$Ind$$) that change with time $$t$$. Bacteria and Indigoidine was experimentally measured, we thus call $$Bac$$ and $$Ind$$ observables of our system. The equations are described by 10 parameters:
