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 3: 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 4: Experimental data and simulation of bacterial growth and indigoidine concentration.A)native_sfp. B) T10_delC.

Figure 5: Identifiability analysis of the fitted parameters.A) native_sfp. B) T10_delC.

Our mathematical model was able to describe most of our experimental data and was in those cases structurally and practically fully identifiable. We thus challenged the model to deduce additional facts on indigoidine synthesis. First, we had a look at the parameters that were estimated to describe our experimental data sets. For each parameter estimate, the lower boundary was set to 1e-5 and the upper boundary was set to 1e+5. Initial guesses for parameters during parameter estimations were generated via latin hypercube sampling to ensure broad coverage of all regions in the high-dimensional parameter space (http://www.jstor.org/stable/2346140). Nevertheless, some parameters exhibited broad variation among the best fits of the different data sets while other parameters were estimated in a narrow range (Fig. XXX).

Figure 7: Box and whiskers plot of the parameters fitted for the invidual data sets. The black line represents the median, the box ranges from the first to the third quartile of the sample. The dashed whiskers denote the median ± 1.5 times the interquartile range. The red line denotes the mean, yellow lines the mean ± standard error of the mean, solid whiskers the mean ± standard deviation.

Interestingly, the bacterial growth rate beta and the maximum growth capacity Bacmax were very consistent. The parameters ksyn, kdim and kdegi contributed directly to indogoidine synthesis as it can be inferred from the above mentioned equations (2), (3) and (4). The synthesis rate ksyn characterizes the cyclization of glutamine. The dimerisation rate kdim reflects how two cyclized glutamine molecules form one indgoidine molecule. As such, ksyn and mdim represent the efficiency of the T domain of indC. The degradation rate of indigoidine, kdegi, should be independent from the domain structure of the indigoidine synthetase as the stability of the peptide is not affected by the T domains or the PPTases that were tested. The rate did not vary widely and the diverse data sets could still be explained by the mathematical model. At first glance, kdegi seemed less important for our experimentally assessed conditions. To further investigate the role of kdegi for indigoidine production kinetics, we systematically varied this parameter and compared simulation results. The synthesis of indigoidine was clearly affected by variation of kdegi (Fig. XXX+1). The [Ind] trajectory collapsed more than 3-fold increase of the degradation rate. On the contrary, improved peptide stability could yield much more indigoidine. For very low degradation rates (i.e. kdegi < 1e-3), synthesis would not even saturate within the observed time frame.

Figure 6: Dependency of indigoidine production on the degradation constant. for native_sfp

Sensitivity analysis for optimized indigoidine production

To perceive the role of kdegi for indigoidine synthesis, we have varied the parameter over a broad range and observed qualitatively distinct model trajectories. For a more quantitative understanding of how indigoidine yield changes with altering parameters of the system, we conducted a sensitivity analysis (REVIEW NEEDED). Therefore, we first defined the variable IntInd, the integrated indigoidine reflecting production yield. The change of the indigoidine concentration with time is determined by the ODE (4) and its integral with time can be depicted as the area under the [Ind] curve, i.e. the amount of indigoidine produced by the bacteria with time. Now we were interested in the change of IntInd with the change of the different parameters of our system. Mathematically, this is expressed as the so called sensitivity $\frac{\partial \mathrm{X}}{\partial \mathrm{p}}, with X as our variable of interest (IntInd) and p as the parameter (beta, Bacmax, ksyn, kdim or kdegi, respectively). This partial derivative provides a measure of how much X changes with the change of p or how sensitive X is against small variations in p. Sensitivities can be scaled with the actual values of X and the respective p to fall in the range from -1 to +1. The higher the absolute value of the sensitivity, the more important is the parameter for the variable. A positive sensitivity means positive impact of the parameter on the variable whereas a sensitivity < 0 indicates reduction of X for subtle decrease in p. We utilised our mathematical model and computed the sensitivities of IntInd against the change of all five kinetic parameters of our system and scaled them accordingly. The analysis confirmed that the kdegi was the only parameter of the system that exhibited a negative impact on the indigoidine production (Tab. 1). Slight increase of all the other parameters also augmented additional yield. 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

The initial concentration of bacteria represented the most crucial parameter for indigoidine production, which was intuitive since the amount of produced indigoidine depends on the amount of bacteria in culture that synthesise the peptide. The parameter kdegi exhibited a high impact on IntInd but did not vary so much among the experimental conditions. One would assume that the half-life of the peptide is not affected by different T domains or PPTases. Possibly, the arrangement of modules influences peptide stability as inter modular communication is a precondition for robust peptide assembly (REFERENCE). This links the peptide synthesis optimisation to our module shuffling project where modules of different hosts were combined for custom peptide synthesis. Indigoidine yield was also sensitive against small variations in beta and Bacmax. Culture conditions could be improved and metabolically more active strains could be chosen to further optimize performance in this respect. The parameters ksyn and kdim also determine peptide production. The question is whether those rates could be altered by engineering the NRPS to advanced performance. To answer this question, we performed a multiple regression analysis (Fig. XXX+2). We regressed the alignment score of the synthetic T domains of the indigoidine synthase with the rates from the respective best fit. The score was calculated (http://www.ebi.ac.uk/Tools/msa/clustalw2/) by pairwise multiple sequence alignment of the amino acid sequences of the synthetic T domains with the native T domain of the indigoidine synthase as a template. The parameter values were obtained from parameter estimations of our mathematical model that has been calibrated with experimental data as described above.

Conclusion and Outlook

constraints and a-priori assumptions