Fachgebiet Systembiotechnologie, Technische Universität München, Garching b. München, Germany

Systems Biology Program, Centro Nacional de Biotecnología-CSIC, Campus Cantoblanco, Madrid, Spain

, Helmholtz Center for Infection Research, Braunschweig, Germany

, Wageningen University & Research centre, Agrotechnology & Food Sciences, Wageningen, Netherlands

, Present address: University Children’s Hospital

Abstract

Background

Signal transduction plays a fundamental role in the understanding of cellular physiology. The bacterial phosphotransferase system (PTS) together with the PEP/pyruvate node in central metabolism represents a signaling unit that acts as a sensory element and measures the activity of the central metabolism. ^{Fru}) and a second branch (PTS^{Ntr}), which communicate with each other by phosphate exchange. Recent experimental results showed a cross talk between the two branches. However, the functional role of the crosstalk remains open.

Results

A mathematical model was set up to describe the available data of the state of phosphorylation of PtsN, one of the PTS proteins, for different environmental conditions and different strain variants. Additionally, data from flux balance analysis was used to determine some of the kinetic parameters of the involved reactions. Based on the calculated and estimated parameters, the flux distribution during growth of the wild type strain on fructose could be determined.

Conclusion

Our calculations show that during growth of the wild type strain on the PTS substrate fructose, the major part of the phosphoryl groups is provided by the second branch of the PTS. This theoretical finding indicates a new role of the second branch of the PTS and will serve as a basis for further experimental studies.

Background

Mathematical modelling of biological processes is a powerful tool towards the thorough understanding of a biological system. In the mathematical simulation, in the first step, experimental data is reproduced and subsequently, the model can be used to predict the behaviour of the system. This type of iterative model-based analysis is a hallmark of systems biology research that, in the future, is expected to be very helpful in enhancing the understanding of cellular systems in a better way. Here, we chose to analyse the PTS^{Ntr} of ^{Ntr} and its cross-talk with the sugar PTS.

On one hand, it can degrade degrade a variety of toxic compounds including methoxylated or hydroxylated aromatic acids. On the other hand,

Central metabolism of

Like many other eubacteria, ^{Ntr}. This was performed in order to get an idea of how the phosphate flow is distributed between the two PTS (PTS^{Fru} and PTS^{Ntr}) and to shed some light on the role of both systems in the process of fructose uptake.

Mathematical modeling of the

Biological background

^{Fru}) and the other is suggested to be involved in signal transduction. In both branches, a high-energy phosphate from PEP is transferred by a number of proteins either to the incoming fructose or to PtsN the last protein in the second branch. As depicted in Figure

The PTS reactions in

**The PTS reactions in ****.** The C branch of the PTS is shown on the left hand side from PEP (reaction _{4}) while the second branch is shown on the right hand side (reactions _{1}−_{3}). The phosphryl group from the final protein in the second branch, PtsN is not metabolized. However, there is evidence that cross talk occurs between the two branches. The C branch transfers the phosphryl group to the second branch directly to PtsN (reaction _{5}).

By ^{Ntr} was recently confirmed to succeed in the direction PEP → PtsP → PtsO → PtsN in analogy to the sugar PTS ^{Ntr} and the PTS^{Fru} communicate with each other by phosphate exchange under specific metabolic conditions ^{Fru} was restricted to conditions in which the PTS^{Fru} was active

Central metabolic reactions of

**Central metabolic reactions of ****.** PEP and pyruvate are hubs that distribute the fluxes to other parts of the network.

Methods

Model for the phosphotransferase system

For the involved PTS proteins shown in Figure

Reactions _{1}
to _{3}
describe the phosphoryl transfer from PEP via PtsP and PtsO to PtsN, reaction _{4}
describes the phosphoryl transfer needed for fructose uptake, and reaction _{5}
describes the cross talk between the two branches. A general form of mass action kinetics for the reaction rate is used. The _{
i
}
read:

where ^{
P
}
the phosphorylated form (e.g. ^{
P
}) and ^{
P
}
are the unphosphorylated and the phosphorylated recipient (e.g. ^{
P
}). Conservation relations can be taken into account for the four proteins:

The system can be described with five equilibrium constants _{
i
}, five velocity reactions _{
i
}
and constant entities representing the overall concentration of the proteins.

Scaling is a well known tool to reduce the number of parameters. Scaling on the overall concentration _{
i,0} of the respective compound and on a chosen time constant leads to a set of equations with seven velocity constants, five equilibrium constants _{
i
} and a scaled uptake rate

The files describes the model equations and the kinetic parameters.

Click here for file

Model for flux balance analysis

The flux distributions were computed by applying Flux Balance Analysis (FBA) on the genome-scale metabolic reconstruction of

Kinetics

Based on the experimental data _{
u
}) and known rates (_{
kn
}). Introducing kinetic rate laws allows the calculation of the respective elasticities _{
ij
}=_{
i
}/_{
j
}. If it is assumed that the two input fluxes are not independent, but are related by a factor (an assumption that is reasonable while taking into account that for higher uptake rates (e.g. _{
a
}) the demands of the cell are changing (e.g. _{
b
})), the following condition holds true for the concentration control coefficients _{
j
}/_{
i
}:

Network with two metabolites and four reactions.

**Network with two metabolites and four reactions.** Given is also the stoichiometric matrix.

with _{
a
} and _{
d
} are the measured rates, and assuming that all remaining reactions _{
b
}(_{
c
}(

where stars mark the respective dependency. The matrix has structural rank one. In case of _{
a
} and _{
b
} are measured, the respective matrix can be inverted in (nearly) all cases since it is of the form

Having the concentration control coefficients at hand, they are used to determine the kinetic properties of the system.

Strains and experimental conditions

Strains, media, and growth conditions

All

Analytical procedures and physiological parameters

Cell growth was monitored spectrophotometrically at 600 nm (OD_{600}) and fructose/ glucose concentrations were determined enzymatically with the fructose/ glucose assay kit (Sigma-Aldrich) according to the supplier’s manual. The following physiological parameters were determined by regression analysis during the exponential growth phase in batch culture, as described elsewhere _{600} were determined from batch cultures of each mutant. Therefore, CDW was measured from at least three parallel 10 ml cell suspensions by harvesting the cells by fast filtration through pre-weight nitrocellulose filters (0.45

Determination of the phosphorylation state of PtsN

The phosphorylation state of PtsN in

Results and discussion

The PTS model describes the available data points

Since metabolic reactions are faster than the growth rate, a pseudo-steady-state can be considered. For the complete model the number of data points (see section Methods) are not sufficient to determine all kinetic parameters directly. However, using the data for the mutant strains allows to decompose the system in a smaller set of equations. Table _{
i
} for the different conditions. The following procedure was applied: The PEP/pyruvate ratio for growth only on CAA was set to 1. As can be seen in Figure

Network fluxes under different conditions (black lines indicate active fluxes).

**Network fluxes under different conditions (black lines indicate active fluxes). ****A** The network for the wild type strain growing on CAA as well as FruB mutant growing on CAA plus fructose. **B** The network of PtsP and PtsO mutants growing on CAA and fructose.

**Strain**

**WT**

**PtsP**

**PtsO**

**FruB**

**(****
r
**

**(****
r
**

**(****
r
**

The conditions that results directly from the mutation are given in the headline. The conditions that results directly from the medium are given in the respective rows.

CAA

_{1}=_{2}=_{3}=0

_{3}=_{4}=_{5}=0

_{1}=_{4}=_{5}=0

_{1}=_{2}=_{3}=0

(_{
up
}=0)

CAA + Fru

–

_{3}=_{5}=0

_{1}=_{5}=0

_{1}=_{2}=_{3}=0

providing an additional constraint for the parameters.

Figure

Comparison of the experimental data with the simulated data.

**Comparison of the experimental data with the simulated data.** Growth conditions as indicated on the x-axis. Black bars represent the simulation results while the grey ones the experimental data. Given is the degree of phosphorylation of PtsN. Kinetic parameters are summarized in the Additional file

The degree of phosphorylation of PtsN of the mutant strains PtsP and PtsO on fructose is characterized by the cross talk between the C and the second branch of the PTS. Although the phosphoryl flux in these mutants is interrupted in the second branch, a certain degree of phosphorylation of PtsN is detected. This can only be explained with a transfer of the phosphoryl group from FruB to PtsN (reaction _{5}). In contrast, for growth on CAA, it is assumed that the

For growth on CAA plus glucose, the values for _{
CAA + Glc
}
and the concentration of FruB represent degrees of freedom and have to be adjusted to describe the data. From the available data on the degree of phosphorylation, FruB is assumed to be present also on growth on glucose, however, it has to be further analyzed whether this is a specific effect of glucose or an artefact of residual presence of fructose in the medium composition.

Model prediciton and verification

To check the performance of the model, the behavior of the model for growth of

Comparison of the experimental data with the predicted simulated data for growth in glucose.

**Comparison of the experimental data with the predicted simulated data for growth in glucose.** Black bars represent the simulation results while grey bars are the experimental data. Given is the degree of phosphorylation of PtsN.

**WT**
^{
a
}

**PtsP**

**PtsO**

**FruB**

**PtsN/PtsN**
_{
0
}

**0.7****
±0.03
**

**0.34****
±0.02
**

**0.44****
±0.08
**

**0.89****
±0.08
**

^{a}Value already published.

Flux distributions at nodes PEP and pyruvate

FBA is widely used to explore the capabilities of genome-scale networks (for a review, see

FBA can be used to estimate flux distributions given a stoichiometric network, uptake and production rates for compounds in the medium and an objective function. Here, the network was analyzed for growth on CAA and growth on CAA plus fructose. The complete flux distribution can be found in the Additional file

The xls-sheet gives all values for the flux distributions based on a FBA analysis for different growth conditions. The xls-sheet is subdivided into two parts.In the first part, the values for the growth conditions are summarized; afterwards the flux for each reaction for the different conditions is provided. For all conditions, three columns are shown: given are the nominal value, and minimal/maximal values based on FVA. Reactions with PEP and pyruvate as substrate (S) or product (P) are highlighted.

Click here for file

Flux map at node PEP pyruvate for growth conditions of the wild type on CAA plus fructose (upper values) and CAA only (lower values).

**Flux map at node PEP pyruvate for growth conditions of the wild type on CAA plus fructose (upper values) and CAA only (lower values).** The ratio of PEP and pyruvate determines the degree of phosphorylation of PtsN.

**Substrate**

**Oxaloacetat.**

**pyruvate to**

**pyruvate**

**PEP to 2PG**

**to pyruvate**

**acetylCoA**

**to PEP**

Given are the results of FBA for the main fluxes with PEP or pyruvate as substrate or product (first row) with the respective minimal and maximal values (second row)The third row gives the ratio (max-min)/nominal value (in per cent).

**CAA** - FBA

6.32

1.83

4.00

3.73

FVA min/max

6.18/6.35

1.68/1.85

3.96/4.01

3.68/3.78

(max-min)/nominal

2.7%

9.3%

1.3%

2.7%

**CAA + Fru** - FBA

4.69

1.47

2.93

2.64

FVA min/max

4.59/4.71

1.36/1.49

2.90/2.97

2.61/2.68

(max-min)/nominal

2.6%

8.8%

2.4%

2.7%

Kinetic properties at PEP and pyruvate node

Having the results from the previous sections at hand, a mathematical model taking into account the kinetics of reactions where PEP and pyruvate are involved was set up. The focus is on the steady-state values of intracellular metabolites PEP and pyruvate in dependence on the input flux. Therefore, fluxes that produce PEP or pyruvate or have PEP or pyruvate as substrate were summed up. This resulted in a network with 5 fluxes and two nodes as shown in Figure _{
c
} and _{
d
}, since in this case matrix _{
a
}, the values for _{
b
} had to be interpolated as shown in Figure _{
b
}
are given as follows:

Left: Reduced flux map at node PEP/pyruvate.

**Left: Reduced flux map at node PEP/pyruvate.** The network is represented with 5 reactions rates and 2 nodes. Right: An increasing input flux _{a}
leads to increasing other fluxes. The increase in flux _{b}
is calculated with the slope

with

The elasticity matrix is in the form:

and the respective concentration control coefficient are:

For the unknown reaction rates _{
c
}
and _{
d
}
power law kinetics are applied:

and the respective elasticities are

From the observation that for increasing input flux _{
a
}
the PEP/pyruvate ratio is also increasing, a necessary condition for the positive slope can be calculated from the following condition:

and therefore

Inserting Equations (18) and (20) result in:

which give first insights for the kinetic parameters. Since for the (relative) simple model, the steady-state values for PEP and pyruvate can be calculated, an estimation of the kinetic parameters is possible. With _{
c
}=_{
d
}, _{
d
}=_{
c
}, and ^{
′
}=1/_{
d
}−1/_{
c
} the PEP/pyruvate ratio is:

Having two measurements available, two values (^{
′
}, corresponding to _{
d
}
and _{
d
}) could be calculated when _{
c
}=1
and _{
c
}=1
(this choice is reasonable since we are mainly interested in a comparison of the two enzymes rather than in absolute values).

Figure

Characteristic curve for the dependency of the PEP/pyruvate ratio on the input flux _{a}.

**Characteristic curve for the dependency of the PEP/pyruvate ratio on the input flux **_{a}**.** The two measured data points are indicated with a line.

Interrelation between the two branches of the PTS

Having both parts of the model available, an estimation of the complete flux distribution, considering fluxes through both parts of the PTS branches, is possible. Table _{5}. This leads to a rather small value for _{4}
and corresponding to a high affinity of PEP to FruB. The high affinity leads to a high phosphorylation of FruB, 99%, hence, the substrate, unphosphorylated FruB, is rare and consequently, the flux _{4} is low. This result is rather unexpected and poses new questions on the function of cross talk.

Complete flux distribution at node PEP and pyruvate for growth on CAA plus fructose.

**Complete flux distribution at node PEP and pyruvate for growth on CAA plus fructose.** Note, that the major part for the phosphorylation of incomming fructose comes from the second branch of the PTS.

Conclusion

The PTS is a key player in the coordination of catabolic reactions in

The model comprises two parts: The first one relates the PEP/pyruvate ratio for different growth conditions of a wild type strain and mutant strains (PtsP, PtsO, FruB) to the degree of phosphorylation of the PTS protein PtsN. Based on the information for the different mutant strains, the complete set of equations could be simplified for every growth condition/strain and the equilibrium constants were calculated or chosen to describe the experimental data at best. Moreover, it was possible to estimate the PEP/pyruvate ratio for growth on CAA plus fructose if the PEP/pyruvate ratio for growth on CAA alone was given. The experimental results are in good agreement with the simulated data; this is reflected by the fact that the mean of the residuals is in the range of 10% of the measured values (the errors in the experimental data are between 5% and 30%). The experimental data revealed differences in the phosphorylation state of PtsN between the wild type and the FruB mutant strain for growth on CAA as well as for the PtsP and PtsO mutant for growth on fructose. These differences could not be reproduced by the model, since in both cases the mathematical equations are the same. However, it was assumed in the model that the PEP/pyruvate ratio is the same for one substrate. So, the differences could be explained with slightly different values of the PEP/pyruvate ratio for growth on the same substrate. The overall equilibrium constant for the PTS was estimated to be _{
eq
}=_{1}
_{2}
_{3}=_{4}
_{5}=0.02. This value is a factor 35 smaller than reported for the _{5}, is 654.6, indicating a very weak connection between the two branches.

Based on the kinetic parameters obtained from the experiments with the wild type _{
CAA + Glc
}=0.05. Based on the data, a prediction of the degree of phosphorylation of PtsN was performed for different mutant strains (PtsP, PtsO, and FruB). The experimental data could be reproduced very well.

Having PEP/pyruvate ratios for growth on CAA and CAA plus fructose available, in a second step, kinetic properties of the system were analyzed based on the respective flux distributions. As expected the flux pattern was different for the two conditions. The main fluxes from or to the both nodes PEP and pyruvate are shown in Figure _{
c
} and _{
d
}
and using the other fluxes as input fluxes, conditions for the hill coefficient for both enzymes could be calculated. Furthermore, setting the two parameters for enzyme PEP synthase (_{
c
}) to 1 the ratio of the parameters of the enzyme for _{
d
}
(representing gluconeogenetic reactions) is calculated. As a result, the hill coefficient for the gluconeogenetic reactions is smaller then the one for PEP synthase (factor 0.30) while the reaction constant

An interesting observation was seen when calculating all reaction rates in case of growth on CAA plus fructose. In this case, the individual reactions are not in equilibrium since fructose as additional input enhances the PTS reaction. Approx. 78% of the required phosphoryl groups for the fructose uptake are provided by the second branch of the PTS by cross talk.

This result allows to speculate on a complete new and unexpected role for the PTS^{Ntr}, at least in ^{Ntr}, they provide a storage system of rapidly available phosphate. This system comes into action when fructose is provided to the cells. Fructose in ^{Ntr} might serve as a “pre”-adaptation to the potential presence of fructose, enabling the cell to rapidly and efficiently metabolize fructose, when it is available. This hypothesis is currently under investigation in our laboratory.

Modeling of signal transduction units together with genome scale stoichiometric models will help for a better understanding of the cellular system. Especially the PTS is an important system that is involved in the coordination of catabolic reactions. The proposed model is good starting point to extend research in direction of the coordination between the carbon and other networks.

Competing interest

The authors declare that they have no competing interests.

Authors’ contributions

AK performed the modeling of the PTS, KPG and MC performed the experiments, JP and VMdS performed the flux balance analysis, AK and VdL designed the study, AK and KPG wrote the manuscript. All authors read and approved the final manuscript.

Acknowledgements

AK was funded in part by the FORSYS initiative from the German Federal Ministry of Education and Research (BMBF).