Analysis of global control of Escherichia coli carbohydrate uptake
© Kremling et al; licensee BioMed Central Ltd. 2007
Received: 08 May 2007
Accepted: 13 September 2007
Published: 13 September 2007
Global control influences the regulation of many individual subsystems by superimposed regulator proteins. A prominent example is the control of carbohydrate uptake systems by the transcription factor Crp in Escherichia coli. A detailed understanding of the coordination of the control of individual transporters offers possibilities to explore the potential of microorganisms e.g. in biotechnology.
An o.d.e. based mathematical model is presented that maps a physiological parameter – the specific growth rate – to the sensor of the signal transduction unit, here a component of the bacterial phosphotransferase system (PTS), namely EIIA Crr . The model describes the relation between the growth rate and the degree of phosphorylation of EIIA crr for a number of carbohydrates by a distinctive response curve, that differentiates between PTS transported carbohydrates and non-PTS carbohydrates. With only a small number of kinetic parameters, the model is able to describe a broad range of experimental steady-state and dynamical conditions.
The steady-state characteristic presented shows a relationship between the growth rate and the output of the sensor system PTS. The glycolytic flux that is measured by this sensor is a good indicator to represent the nutritional status of the cell.
Mathematical models of cellular systems describing metabolism, signal transduction and gene expression are becoming more and more important for the understanding of the underlying molecular processes. Since the earliest work to elucidate the molecular nature of regulatory structures by J. Monod, the knowledge of the detailed interactions between the components that are responsible for carbohydrate uptake in Escherichia coli is steadily increasing. Although current research on individual uptake systems like glucose still reveals new players that maybe play a role in local control , the knowledge of individual uptake systems is rich and is used as a basis to set up mathematical models to describe and analyze the properties of the control circuits. E.g. for the lactose uptake system in E. coli, it was shown that the autocatalytic action of inducer allolactose is responsible for the existence of multi-stationarity . Such nonlinear properties of sub-networks are often described and assigned to a certain functionality of the system. The understanding of how different stimuli of the same type – in this study carbohydrates – are sensed by the cells and how these different signals are processed is still lacking. Here, we used experimental data published by our group [3, 4] to elucidate and characterize such a global control circuit, that is, a regulatory scheme, that senses a physiological parameter like the specific growth rate and maps it to the degree of phosphorylation of the intracellular component EIIA Crr . EIIA Crr is a component of the phosphoenolpyruvate (PEP): carbohydrate phosphotransferase system (PTS). The PTS is not only a transport system for a number of carbohydrates but also acts as a sensory system. Sensor elements like the PTS can be seen as logic elements that process external stimuli into intracellular signals. High fluxes through the glycolysis, corresponding to high growth rates result in a low degree of phosphorylation of EIIA Crr . At first view, this is surprising, since, assuming a linear reaction chain, high fluxes result in high pool concentrations, based on the (normally) monotone dependency of the reaction rate on the substrate concentration. The PTS together with the glycolysis can now be seen as an element that allows a transformation of high fluxes into a low pool concentration. This is not only due to the existence of two complementary pools like EIIA Crr and its phosphorylated form, but as we will show, depends strongly on the flux distribution at the PEP node. High fluxes through the glycolysis result in low values of the phosphorylated form of EIIA Crr while low fluxes indicate a hunger situation and the global transcription factor cAMP·Crp is activated.
Interestingly, the relationship between growth rate and degree of phosphorylation of EIIA Crr could be seen in various growth situations of the wild type strain growing on single substrates like glucose, lactose, and glycerol and for growth on mixtures of substrates, and of a PtsG deletion mutant strain missing ptsG, a gene that is central for glucose transport.
Carbohydrate uptake by E. coli
The PTS of E. coli consist of two common cytoplasmatic proteins, EI (enzymeI) and HPr (histidine containing protein), as well as of an array of carbohydrate-specific EII (enzymeII) complexes. E.g. for glucose uptake, a phosphoryl group is transferred from phosphoenolpyruvate (PEP) through EI, HPr, EIIA Crr , PtsG (also known as EIICB Glc , that is the membranstanding transport protein) and finally to the substrate. Since all components of the PTS, depending on their phosphorylation status, can interact with various key regulator proteins the output of the PTS is represented by the degree of phosphorylation of the proteins involved in phosphoryl group transfer.
with K1, K2, K3, K L being the respective equilibrium constants from the single reactions shown in Figure 1. If EIIA Crr is bound to lactose permease or glycerol kinase, it acts as an inhibitor that prevents uptake and/or metabolism of the substrate, an effect that is called inducer exclusion.
The intention of this contribution is to develop a model with a small number of state variables and parameters to work out the basic principles for the understanding of the sensor function. Nearly all parameters could be determined from experiments (for material and methods, [see Additional file 1]). The core of the model describes the mapping of the specific growth characteristics represented by the carbohydrate uptake rates to the degree of phosphorylation of the PTS component EIIA Crr . The kinetic properties of the sensor which at the same time is a transport system are characterized and the output of the sensor is mapped to the rate of synthesis of genes that are under control of transcription factor cAMP·Crp. In this way, a closed loop is established that precisely adjusts the respective transport protein to maintain the incoming flux. The results are used to predict the transient behavior during glucose/glucose 6-phosphate diauxic growth and glucose/lactose diauxic growth. Finally, we also show that the approach can be generalized for other main growth substrates like acetate. In the end, a comparison with a corresponding detailed model on catabolite repression  is performed.
Results and discussion
The scheme simplifies the biological knowledge on metabolism and gene expression by lumping together reactions and components. In case A, carbohydrate uptake is represented by reactions r pts_up for a PTS carbohydrate and rn-ptsfor a non-PTS carbohydrate. Glycolysis is simply represented by metabolite Glc 6P. The flux at node PEP is subdivided into r pyk for pyruvate kinase and r pts . The drain from pyruvate (Prv) to other parts of the central metabolism is represented by r pdh . Since other fluxes from or to PEP and pyruvate are rather marginal they are not considered in the model. Proteins EI, HPr, and EIIA Crr of the PTS are represented by only one component X that exists either in the unphosphorylated form X or in the phosphorylated form XP. In case of a PTS carbohydrate, XP is used for transport via r pts_up .
Case B considers gluconeogenetic substrates which feed into TCA or into other central metabolites below the PEP/pyruvate branch. Here, PEP and pyruvate are produced by a number of different reactions, e.g. from the TCA or via Acetyl CoA. Among these, PEP synthase (Pps) is active converting pyruvate directly to PEP. For substrates that enter TCA, two pathways are known that connect TCA and glycolysis: PckA (PEP carboxykinase) connects oxaloacetate and PEP, MaeB/SfcA (malate dehydrogenase) connect malate and pyruvate. These fluxes are represented by h1 r up and h2 r up , respectively, with h1 and h2 are numbers between zero and one, representing a fraction of the uptake rate r up . In a number of subsequent gluconeogenetic reaction steps (r glu ), PEP is then converted to glucose 6-phosphate.
Based on the knowledge presented so far, a simplified model structure is suggested that is able to simulate the different cases proposed above.
where rn-ptsand r pts_up are the systems inputs and are related by the yield coefficients to the specific growth rate. XP is the system output. The following conditions will hold for the defined rates in steady-state:
r pts = r pts_up (6)
r gly = rn-pts+ r pts_up (7)
r pdh = 2 (rn-pts+ r pts_up ) (8)
r pyk = 2 rn-pts+ r pts_up (9)
The kinetics for the rate laws are kept as simple as possible to describe the experimental data. The rate laws are assumed as follows:
r gly = k gly G 6P (10)
r pdh = k pdh Prv (11)
r pts = k pts (PEP(X0 - XP) - K pts Prv XP) (12)
r pyk = k pyk PEP f (PEP, ...), (13)
with X0 is the overall concentration of the PTS protein. The focus of the analysis will be on the branch point at PEP. To elucidate the correct choice of the kinetic rate law for the pyruvate kinase reaction, function f is introduced that represents different model variants. Function f depends on PEP but may also depend on different metabolites in the network.
The steady-state equation for PEP is given in implicit form since it depends on function f. In the following, growth situations on non-PTS and PTS sugars are considered separately.
Since k pyk , in this case, is the maximal reaction rate of r pyk , PEP is an increasing monotone function in dependency on the uptake rate rn-pts. Interestingly, this leads to values for XP that increase for increasing uptake rates. This result is again contradictory to the observed experimental results.
Differences for PTS and non-PTS substrates can be seen in the numerator that is always smaller in case of growth on PTS substrates. Since the denominator is always larger than in the case of non-PTS substrates, the curve of the PTS substrates will always be below the curves for non-PTS substrates.
To describe the available experimental data for growth on PTS and non-PTS substrates (Table 2 in [Additional file 1]), parameters were estimated by a least square approach. A reasonable fit could be obtained with
f = f1(G 6P)·f2(PEP) = G 6P n ·PEP m . (22)
Since the pyruvate kinase in E. coli is a tetramer that needs activation from a glycolytic metabolite (in E. coli PykF is strongly activated by fructose 1,6-bis-phosphate, that is not included in the model, but is represented by glucose 6-phosphate instead), values for n > 1, m ≥ 1 are analyzed. Equation (1) relates the overall PTS constant K pts to individual reactions steps. Since measurements of proteins that influence K pts are not available, K pts represents a mean value for different situations considered in the experiments. For parameter identification 31 data points are considered, values n = 2, m = 1 are fixed and values for K pts and X0 are taken from literature (Table 6 in the [Additional file 1]); so, four parameters are estimated: k gly , k pyk , k pts , and k pdh .
Rate r up is the system input. Rate r bio is the flux from pyruvate to biosynthesis and r glu is the rate of gluconeogenesis:
r bio = k bio P rv (30)
r glu = k glu PEP (31)
For the rate r pps the following simple approach is used:
r pps = k pps Prv g(Prv, ...) (32)
Liao and colleagues observed a drastic increase of the lag phase on acetate in the mutant strain during glucose/acetate diauxic growth. Our simple model predicts, that the degree of phosphorylation is a bit smaller than the values in the wild type strain. This confirms that Pps has nearly no influence on physiological parameters like the growth rate.
With the model developed so far, model predictions can be performed. Two cases are considered: the PEP/pyruvate ratio and growth on different single carbon sources.
Growth on single carbohydrates
Since the value for K3, the equilibrium constant for the phosphoryl transfer HPr to EIIA Crr is approximately 1 [3, 13, 14], values of d EIIA and d HPr are nearly equal. Therefore, in the model, state variable X can be used to represent HPr as well as EIIA Crr .
Transcription efficiency and sensor kinetics
In order to set up a closed loop, further modules have to be characterized. First, the influence of phosphorylated EIIA Crr on transcription efficiency is analyzed, afterwards the kinetics of the PTS transport system is investigated.
Unexpectedly, the Hill coefficient is high (n = 6) indicating a high sensitivity in a narrow range of the input.
Closed loop dynamics and application to diauxic growth
Finally, a model with a closed loop, comprising the core model and individual uptake systems is set up. The model is applied to a complex growth situation, namely growth with a mixture of two substrates. Simulation studies for growth on mixtures of glucose/glucose 6-phosphate and of glucose/lactose are described.
Growth on glucose/glucose 6-phosphate
Glucose 6-phosphate represents an interesting growth substrate. This sugar-phosphate is taken up into the cell via the inorganic phosphate antiporter, UhpT  and can afterwards enter into glycolysis without further modification. A mixture of glucose and glucose 6-phosphate is a very interesting case because expression of both proteins depends on the cAMP·CRP complex. UhpT has been shown to influence cAMP levels in the cell . It was concluded that neither glucose 6-phosphate nor another metabolite of glycolysis was directly involved in this effect but rather the flux through UhpT itself . These results are confirmed by additional studies analyzing the effect of glucose 6-phosphate uptake on the degree of EIIA Crr phosphorylation and the amount of cAMP . In addition, it was shown that high intracellular Glc6P levels lead to the degradation of the ptsG mRNA [6, 7] via the small regulatory RNA, SgrS  and hence to reduced concentrations of PtsG.
with the specific growth rate μ that is calculated with the yield coefficients Yg 6pand Y glc in dependence on the substrate uptake:
μ = Yg 6prn-pts+ Y glc r pts_up . (44)
Parameters k1 and k2 are scaling factors and g T is taken from Equation (37).
In the simulation (Figure 10), only the parameters for the glucose 6-phosphate uptake and the inhibition of the glucose transporter PtsG by the glucose 6-phosphate transporter UhpT are fitted while all other parameters are kept as described in the previous sections. Therefore, the time course of the degree of phosphorylation of EIIA Crr is a prediction based on previous results. The time course of the substrates in the medium hints to an inhibition of glucose uptake during glucose 6-phosphate uptake. After consumption of glucose 6-phosphate, the growth rate slows down which results in a small increase of the degree of phosphorylation. During subsequent growth on glucose, the degree of phosphorylation of EIIA Crr is again very low. For the experiment shown in Figure 10 the course of the glucose transporter was not measured. Therefore, the right plot of Figure 10 shows data from experiments with slightly different initial conditions. To compare the results, the time of the simulation experiment and the time of the wet experiment are scaled. The time course of the glucose transporter indicates that indeed the rate of gene expression is under control and is inhibited during growth on glucose 6-phosphate.
Growth on glucose/lactose
Comparison with a detailed model for catabolite repression
Summary of the state variables of the model
extracellular glucose 6-phosphate
glucose 6-phosphate; represents the metabolites in the upper part of the glycolysis
represents the phosphorylated form of the PTS proteins
(EI, HPr, EIIA Crr )
represents uptake system for glucose 6-phosphate (UhpT)
represents uptake system for glucose (PtsG)
represents uptake system for lactose (LacY)
Several kinetic properties determine the degree of phosphorylation of the PTS protein EIIA Crr . According to this study, the choice of the rate law for the pyruvate kinase is the most important one. While all other kinetic rate laws can be described with simple mass action rate laws, the pyruvate kinase has to be described with a power law kinetics. However, this choice is only true for a certain set of experimental conditions; considering only growth of the wild type on glucose, a simple rate law, as suggested by , is capable to describe experimental data. Based on a systems biology approach that considers different operational modi of the system and a directed stimulation of the system with respect to these modi, the present study shows that the simple rate law is not longer able to describe all experimental data. The core model comprises four reactions for glycolysis, pyruvate kinase, PTS, and drain to monomers. Parameters were determined by fitting experimental data from a wild type strain and a PtsG mutant strain. To deconstruct the results, a robustness analysis was performed that ranks the parameters according to the influence on the degree of phosphorylation of EIIA Crr in dependence on the growth rate. As expected, the biggest influence for both operational modi shows parameter n, that represents the influence of the feed-forward control of glucose 6-phosphate on pyruvate kinase. Furthermore, the overall concentration X0 of enzyme EIIA Crr has a big influence while the concentration of the other enzymes represented by k gly , k pyk , and k pdh is moderate and comparable to the influence of the remaining kinetic parameters k pts and K pts .
The feed-forward loop is a special motive (a regulatory pattern that is more present than others) described in detail for genetic systems . Here, we found that this motive is essential for the transformation of a high incoming flux (high growth rate) into a low PEP/pyruvate ratio. To verify this, the internal metabolites PEP and pyruvate are measured. Since the errors for the procedure of the PEP and pyruvate measurement are rather high , the data shown in Figure 6 should be interpreted rather as a trend and not as quantitative measurements. Although measurements for small growth rates are not available, the PEP/pyruvate ratio could be predicted very well for growth rates in the range between 0.15 1/h and 0.7 1/h.
In engineering science, sensor or measurement systems are designed in such a way that they don't influence the system that is measured. This is called "free of retroactivity". Considering the PTS operational mode in comparison to the non-PTS mode the difference of the curves is due to the transport activity of the PTS. Hence, the sensor PTS is not free of retroactivity; however, for small growth rates, indicating a severe stress situation, the difference between the PTS mode and the non-PTS mode is negligible.
As representative of gluconeogenetic substrates, growth on acetate was considered. The fluxes are adjusted in such a way that a flux distribution published previously, is matched. Measurements of the degree of phosphorylation of EIIA Crr are in good agreement with the predicted values. The results also confirm that the Pps enzyme has only marginal influence on growth on acetate as described by . However, the observation that a Pps mutant strain that grows simultaneously on glucose and acetate shows an extended lag phase could not be explained with model set up in this study.
The transcription efficiency according to Equation (37) revealed that the Hill-coefficient n = 6 is rather high. This might be due to several reasons: although the signal transduction pathway starting from EI and ending with Crp is rather short, several components and processes are involved. First cAMP is generated by the adenylate cyclase (Cya); second cAMP interacts with Crp to activate the transcription factor. Furthermore, transcription of Cya is also under control of Crp leading to a feedback loop. Since the kinetics of the individual steps are not yet characterized, the rather high Hill-coefficient can be seen as an overall measure of the sensitivity of the system. The kinetics determined are used to simulate the two dynamical experiments and a good agreement between the simulation data and the experimental data could be observed. This shows that not only the steady-state behavior can be reproduced well but also the dynamics of the sensor/actuator system.
The simplified scheme is used to analyze the growth behavior and the dynamics of Escherichia coli during growth on glucose/glucose 6-phosphate and on glucose/lactose. The model has to be extended to describe the kinetics of the transporters and the kinetics of gene expression for the relevant transporters. Since experimental data that characterize the K Glc value for glucose can be found in the literature, the respective value K EIIA for the degree of phosphorylation was determined by a simulation experiment with a random bi-bi double substrate kinetics, Equation (38), and experimental data from . Parameters k max and K EIIA are determined by a least-square fit.
Growth on glucose/glucose 6-phosphate reveals the interesting observation that the concentration of the glucose transporter decreased during growth on glucose 6-phosphate. To match the experimental data, an inhibitory effect of the glucose 6-phosphate transporter UhpT on the glucose transporter PtsG was assumed and described with a simple kinetics. Previous studies revealed that the ptsG mRNA is under control by SgrS, a small RNA. It was shown that high levels of intracellular glucose 6-phosphate or fructose 6-phosphate lead to ptsG mRNA degradation [6, 7]. Here, the model can be used to calculate the intracellular levels of glucose 6-phosphate and PEP in model variants with and without control of PtsG. As shown in Figure 11, no difference could be detected, indicating that the interaction between the two transporters is based on the activity of the glucose 6-phosphate transporter as suggested in . Note, that to describe the time course of PtsG in Figure 10, three factors, namely the inhibition of PtsG by UhpT, induction of ptsG and global control of PtsG synthesis by Crp were taken into account and have to be adjusted very precisely.
A comparison with a detailed model for catabolite repression justifies the set up of the new model. Altough validated under different experimental conditons, the detailed model fails to describe growth on PTS carbohydrates on a broad range of the growth rate.
The approach is based on the development of a model with a minimal number of parameters that are necessary to describe the observations. Although some of the parameters have no defined mechanistic interpretation such models will facilitate the procedure of parameter analysis and estimation. The model is capable to simulate a broad range of experimental conditions and is suited for further studies on control systems on E. coli since it can be easily extended to describe other regulatory systems.
For simulation of the algebraic system, solving the o.d.e. system, and parameter estimation MATLAB was used. Files to simulate the system with MATLAB and the experimental data can be found on a website . For the experimental data, see the [Additional file 1] and a further manuscript from our group .
AK and KB are funded by the FORSYS initiative from the German Federal Ministry of Education and Research (BMBF).
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