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Team:Heidelberg/Templates/Modelling/Ind-Production
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| - | 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) | + | 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) ref[Brachmann, Alexander O, Kirchner, Ferdinand, Kegler, Carsten, Kinski, Sebastian C, Schmitt, Imke, Bode, Helge B: Triggering the production of the cryptic blue pigment indigoidine from Photorhabdus luminescens., J. Biotechnol. 157(1), 96–9, January 2012] (<a class="fancybox fancyFigure" href="/wiki/images/8/83/Heidelberg_IndSynthesis_scheme.png" title="<b>Figure 1: The indigoidine synthetase indC produces a blue pigment and is used for the indigoidine-tag</b>. After activation of L-glutamine by the A domain and its binding to the T domain, it is cyclized by the TE domain and oxidized by the Ox domain. Dimerization is thought to happen by air.">Fig. 1</a>), 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. With the mathematical model we could then validate whether there is indeed a negative feedback from the Indigoidine production to the growth of bacteria. |
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Revision as of 17:02, 28 October 2013
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) ref[Brachmann, Alexander O, Kirchner, Ferdinand, Kegler, Carsten, Kinski, Sebastian C, Schmitt, Imke, Bode, Helge B: Triggering the production of the cryptic blue pigment indigoidine from Photorhabdus luminescens., J. Biotechnol. 157(1), 96–9, January 2012] (Fig. 1), 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. 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\begin{align} \mathrm{d}\mathrm{[Bac]}/\mathrm{d}t &= \frac{\mathrm{[Bac]} \cdot \left(\mathrm{[Bac]} - \mathrm{Bacmax}\right) \cdot \mathrm{beta}}{\mathrm{Bacmax}}\label{bacgrowth}\\ \mathrm{d}\mathrm{[Glu]}/\mathrm{d}t &= - \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn}\label{glu}\\ \mathrm{d}\mathrm{[cGlu]}/\mathrm{d}t &= - \mathrm{kdim} \cdot {\mathrm{[cGlu]}}^2 + \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn}\label{cglu}\\ \mathrm{d}\mathrm{[Ind]}/\mathrm{d}t &= {\mathrm{[cGlu]}}^2 \cdot \mathrm{kdim} - \mathrm{[Ind]} \cdot \mathrm{kdegi}\label{ind} \end{align}
Initially, inhibition of bacterial growth by indigoidine and degradation of cyclic glutamine were also described by the model, using equations \eqref{bacgrowthinhib} and \eqref{cgludeg} instead of \eqref{bacgrowth} and \eqref{cglu}, respectively, however the degradation rate turned out to be non-identifiable for all data sets, converging to the lower bound, whereas the growth inhibition term led to frequent convergence failures, the inhibition konstant being very low in cases where convergence was achieved. These two terms were thus removed from the model. \begin{align} \mathrm{d}\mathrm{[Bac]}/\mathrm{d}t &= \frac{\mathrm{[Bac]} \cdot \left(\mathrm{[Bac]} - \mathrm{Bacmax}\right) \cdot \left(\mathrm{beta} - \mathrm{ki} \cdot \mathrm{[Ind]}\right)}{\mathrm{Bacmax}}\label{bacgrowthinhib}\\ \mathrm{d}\mathrm{[cGlu]}/\mathrm{d}t &= - \mathrm{kdim} \cdot {\mathrm{[cGlu]}}^2 - \mathrm{kdegg} \cdot \mathrm{[cGlu]} + \mathrm{[Bac]} \cdot \mathrm{[Glu]} \cdot \mathrm{ksyn}\label{cgludeg} \end{align}
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 7 kinetic parameters:- Bacmax: maximum capacity for bacterial growth
- beta: maximum attainable growth rate
- ksyn: synthesis rate of cyclized glutamine from glutamine
- kdim: dimerization rate of two cyclized glutamines to an Indigoidine dimer
- kdegi: degradation rate of indigoidine
