Team:Heidelberg/Templates/Modelling/Ind-Production

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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.

Figure 1: The indigoidine synthetase indC produces a blue pigment and is used for the indigoidine-tag. 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.

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. 2), 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:

\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 constant 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 5 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

In addition, the experimental error for the observables was estimated with 2 error parameters and the initial concentration of the bacteria at t=0 was estimated. Data was otherwise normalized between 0 and 1, thus no scaling and offset parameters were required.

Figure 2: Schematic representation of a mathematical model describing bacterial Indigoidine synthesis. ....

Framework

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 and perform robust parameter estimations as well as identifiability analysis yielding valuable insight into bacterial indigoidine production.

Results

Figure 2: Indigoidine synthesis rates ksyn determined by fitting the model to several data sets. The rates were normalized to the range between 0 and 1, rates for data sets where at least one parameter was not identifiable were set to 0. The synthetic T domain T10 is the most efficient of all tested T domains, whereas T12 is the only one that functions with all PPTases.

Figure 3: Dependency of indigoidine production on the degradation constant. for native_sfp

Sensitivity analysis for native_sfp:

Parameter	$$\left.\mathrm{[Bac]}\right\|_{t=0}$$	kdegi	kdim	beta	ksyn	Bacmax
$$\frac{\partial\int\!\mathrm{Ind}\,\mathrm{d}t}{\partial p}$$	0.784	-0.7024	0.5117	0.3338	0.2919	0.1036