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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 the functional Indigoidine dimer is produced from two Glutamine (Glu) that are each cyclized (cGlu) , and our observation that Indigoidine (Ind)-producing bacteria (Bac) grow slower than mock controls. Those hypotheses resulted in a general model scheme depicting the interdependency between Indigoidine synthesis and bacterial growth (Fig. 1). With the mathematical model we could then validate whether there is indeed a negative feedback from the Indigoidine production to the growth of bacteria.

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 . However, bacterial growth curves cannot be sufficiently described by mass action, thus we adapted our ODE for the bacterial growth from equation (7) of Kenneth and Kamau, 1993 . Our ODE system is now given by the following 4 equations:

$$ \mathrm{d}\mathrm{[Bac]}/\mathrm{d}t = -\frac{\mathrm{[Bac]} \cdot \left(\mathrm{[Bac]} - \mathrm{Bacmax\_native\_svp}\right) \cdot \left(\mathrm{beta\_native\_svp} - \mathrm{[Ind]} \cdot \mathrm{ki\_native\_svp}\right)}{\mathrm{Bacmax\_native\_svp}} $$ $$\mathrm{d}\mathrm{[Glu]}/\mathrm{d}t = - \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn\_native\_svp} $$ $$\mathrm{d}\mathrm{[cGlu]}/\mathrm{d}t = - \mathrm{kdim\_native\_svp} \cdot {\mathrm{[cGlu]}}^2 - \mathrm{kdegg\_native\_svp} \cdot \mathrm{[cGlu]} + \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn\_native\_svp} $$ $$\mathrm{d}\mathrm{[Ind]}/\mathrm{d}t = {\mathrm{[cGlu]}}^2 \cdot \mathrm{kdim\_native\_svp} - \mathrm{[Ind]} \cdot \mathrm{kdegi\_native\_svp} $$

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 8 parameters:

Bacmax: maximum capacity for bacterial growth

beta:

Framework

Those parameters have to be estimated from experimental data. In order to implement our mathematical model and the wetlab data, we used an open-source software package allowing for comprehensive analysis (D2D Software). With this framework, we were able to calibrate the model Mathematical modelling allows for Identifiability analysis

Results

Conclusion and Outlook

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