Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, 39106 Magdeburg, Germany

Abstract

Background

Analyzing the dynamics of insulin concentration in the blood is necessary for a comprehensive understanding of the effects of insulin

Results

We present a dynamic mathematical model of insulin concentration in the blood and of insulin receptor activation in hepatocytes. The model describes renal and hepatic insulin degradation, pancreatic insulin secretion and nonspecific insulin binding in the liver. Hepatic insulin receptor activation by insulin binding, receptor internalization and autophosphorylation is explicitly included in the model. We present a detailed mathematical analysis of insulin degradation and insulin clearance. Stationary model analysis shows that degradation rates, relative contributions of the different tissues to total insulin degradation and insulin clearance highly depend on the insulin concentration.

Conclusion

This study provides a detailed dynamic model of insulin concentration in the blood and of insulin receptor activation in hepatocytes. Experimental data sets from literature are used for the model validation. We show that essential dynamic and stationary characteristics of insulin degradation are nonlinear and depend on the actual insulin concentration.

Background

Insulin regulates important physiological processes like cellular glucose uptake

Insulin dynamics

A prerequisite for fully understanding the effects of insulin

Long acting insulins tend to form dimers or hexamers in the subcutaneous tissue, whereas fast acting insulin analogues have a decreased ability to form oligomers

There are also efforts to predict glucose concentration and to automate insulin dosage for individuals with impaired glucose levels

In the last few decades, many different kinetics for insulin removal from the blood were proposed. The most frequently used kinetics are linear first order kinetics, Michaelis-Menten kinetics or a combination of both

Insulin receptor dynamics

There are several models in literature that describe insulin receptor dynamics

Sedaghat et al. combined models of insulin binding

Hori et al. described receptor phosphorylation, internalization and recycling in Fao hepatoma cells

Thus, there are many couplings between different processes in all detailed receptor models

Insulin dynamics and insulin receptor dynamics

We present a literature-based mathematical model of insulin dynamics and hepatic insulin receptor activation in rats. Compared to other models

Model validation is performed with experimental data sets from literature. We emphasize that the data sets used for the model validation are not used for parameter estimation. This corresponds to a strict separation of model construction and model validation which is frequently applied

We perform a detailed stationary analysis of the contributions of the liver and the kidney to insulin degradation and insulin clearance as well as of the activation state of hepatic insulin receptors under varying insulin concentrations.

Results and Discussion

The model

The model consists of ordinary differential equations (ODEs) and describes the dynamic behavior of radioactively labeled and unlabeled insulin in the blood and the physiological state of hepatic insulin receptors. It can also be used for the injection of only labeled or only unlabeled insulin. Distinction between labeled and unlabeled insulin is necessary as unlabeled insulin is synthesized in the pancreas whereas labeled insulin is not. Therefore, in experiments with labeled insulin, the fraction of labeled insulin changes over time.

Almost all state variables in the model represent concentrations and are given in _{ub }and _{*,ub }that represent amounts of substances and are given in ^{-1}. The rates describing insulin receptor dynamics (_{j}, _{j }and _{j}, _{hep}. All other rates refer to blood plasma volume _{p}. The executable model is given in MATLAB format in Additional file

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Important tissues and processes

The liver and the kidney are the most important insulin degrading tissues ^{-3 }^{-5 }^{-1}. Insulin receptors in hepatocytes have a minimal internalization rate constant of 2·10^{-4 }^{-1 }^{5 }receptors per hepatocyte

Hepatic insulin receptors have access to insulin molecules in the space of Disse

Our model explicitly describes dynamic insulin receptor activation in hepatocytes of the liver. Processes considered are insulin binding to the receptor, receptor autophosphorylation, internalization and recycling. Compared to other models of the insulin receptor which also include these processes

The kidney's contribution to insulin clearance mainly consists of the filtering of insulin from the blood

Pancreatic insulin secretion is mainly induced by plasma glucose

Robustness and parameter estimation. This file describes the examination of robustness to changes in parameter values and the estimation of the model parameters.

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Altogether, our model describes the following processes: intravenous injection of radioactively labeled and unlabeled insulin, pancreatic insulin secretion, hepatic and renal insulin degradation, hepatic insulin receptor activation and nonspecific insulin binding by the liver.

Parameterization of the model

In this study, model parameters (Table

Model parameters and initial conditions

Parameter

Value

Source

Meaning of the parameter

10^{-3 }^{-1 }^{-1}

[36]

insulin binding to the receptor

4·10^{-4 }^{-1}

[36]

insulin dissociation from the receptor (I1, PM)

4·10^{-2 }^{-1}

[36]

insulin dissociation from the receptor (I2, PM)

1.925·10^{-3 }^{-1}

[49]

insulin dissociation from the receptor (I1, EN)

3.85·10^{-3 }^{-1}

[50]

insulin dissociation from the receptor (I2, EN)

3.85·10^{-3 }^{-1}

[51]

receptor dephosphorylation (PM)

7.22·10^{-3 }^{-1}

[52]

receptor dephosphorylation (EN)

0.0231 ^{-1}

[52]

autophosphorylation of the receptor (I1 and I2)

5.5·10^{-4 }^{-1}

[34]

internalization of phosphorylated receptors

2·10^{-4 }^{-1}

[34]

internalization of unphosphorylated receptors

1.533·10^{-3 }^{-1}

[34]

recycling of receptors without insulin

0.35 ^{-1}

[46]

nonspecific insulin binding in the liver

0.2 ^{-1}

[46]

dissociation of nonspecifically bound insulin

0.0016976 ^{-1}

calc.

pancreatic insulin secretion

0.5

ass.

concentration of half-maximal insulin secretion

_{
liver
}

0.05·_{body}

[46]

mass of the liver

_{
p
}

0.03375·10^{-3 }^{-1}·_{body}

[54]

plasma volume

_{
liver
}

1.051·10^{3 }g·^{-1}

[53]

density of the liver

_{
hep
}

(_{liver}/_{liver})·0.78

[45]

total hepatocyte volume

_{
d
}

0.272·10^{-3 }^{-1}·_{hep}·_{liver}

[46]

volume of the space of Disse

_{
kidney
}

2·0.85 _{body/}(230

[55]

mass of the kidney

0.0225·10^{-3 }^{-1}·_{kidney}

[47]

clearance of the kidney

Note that _{body }(body weight in _{in }(injection time in _{in }and _{*,in }(amounts of injected unlabeled and labeled insulin in _{en }= 4.88528, _{en }= 0.145537, _{en }= 0.000121295, _{en }= 0.122602, _{en }= 0.492464, _{en }= 0.000433466, _{ub }= 1.29948·10^{-6}·_{body}, _{*,ub }= 0. The unit of _{ub }and _{*,ub }is

The parameters from the models of insulin binding

The parameters for pancreatic insulin secretion were chosen to guarantee the physiological basal level of insulin (0.07

This study uses the rat as model organism because much more parameters are known for rats than for humans. The model validation is performed using experimental data sets for rats.

All volumes are assumed to be constant. In addition, all tissues are assumed to contact the same total insulin concentration, which is the sum of labeled and unlabeled insulin. The physiological justification of this assumption is the high heart rate of rats (320 – 480

The liver

Insulin degradation in hepatocytes is modeled in a very detailed way. The described processes are successive binding of two insulin molecules to the insulin receptor, receptor phosphorylation and receptor internalization (Figure

Insulin receptor activation in hepatocytes

**Insulin receptor activation in hepatocytes**. The receptor is denoted as

Model assumptions that are supported by studies from literature are:

•

•

•

•

•

•

• ^{125}I

•

For the following assumptions there is no experimental data in literature supporting them. These assumptions were made to keep the number of parameters as low as possible.

•

•

In the following, the insulin receptor is denoted as _{hep}, the total volume of hepatocytes.

In general, rates denoted by the standard notation _{j }describe processes at the plasma membrane of hepatocytes or outside the hepatocytes (nonspecific insulin binding, pancreatic insulin secretion and renal insulin removal). Rates denoted by _{j }describe _{j }describe

Figure

The hepatocyte part of the model does not distinguish between labeled and unlabeled insulin, which reduces the number of necessary ODEs. Hepatocytes have contact to the total insulin concentration _{* }. Unlabeled insulin (_{*}) has no separate notation. The total contribution of the liver to insulin degradation is

_{liv }= (-_{1 }- _{2 }- _{3 }- _{4})·_{hep}/_{p}.

The plasma volume is denoted as _{p}, the total hepatocyte volume is denoted as _{hep}. Strictly speaking, _{liv }defines insulin removal from the blood, whereas insulin degradation is performed in hepatic endosomes. However, _{liv }is the contribution of the liver to insulin dynamics. In the stationary case, the values of the rates for insulin removal and insulin degradation are identical.

Rates _{1 }– _{4 }describe insulin binding to the insulin receptor at the plasma membrane. The values of the parameters

Rates _{5 }– _{7 }describe receptor phosphorylation at the plasma membrane.

Rates _{1 }– _{4 }describe insulin dissociation from the receptor in endosomes.

Rates _{5 }– _{7 }describe receptor phosphorylation in endosomes.

According to our model assumptions, unphosphorylated receptors without insulin (_{en}) have no autophosphorylation activity. Therefore, the reactions represented by the rates _{5 }and _{5 }are irreversible. Rates _{1 }– _{6 }describe receptor internalization and recycling.

The value of the parameter

Altogether, the described processes result in the following balance equations for hepatic insulin receptor species.

The liver also performs nonspecific insulin binding. This reversible process does not saturate _{ub }and _{*,ub }define nonspecific binding of unlabeled and labeled insulin, respectively.

The values of the parameters _{d}. The concentration of unlabeled insulin is _{* }(unit: _{*,ub }and _{ub }are the amounts of substance (unit: _{ub }and _{*,ub }are multiplied by _{d }(unit: _{* }(unit: _{ub }and _{*,ub }are amounts of substance (unit:

In order to obtain the unit ^{-1 }for all rates, we divide by _{p }within the rates _{ub }and _{*,ub }and multiply the rates by _{p }in the ODEs for _{ub }and _{*,ub}, emphasizing the need for _{p}.

Note that symbols with an asterisk indicate radioactively labeled insulin species. Species with insulin whose symbols do not contain an asterisk can contain labeled or unlabeled insulin, except for _{ub}, which only represents unlabeled nonspecifically bound insulin.

The kidney

The kidney performs insulin degradation by filtering insulin from the blood _{kid }is proportional to insulin concentration

_{kid }= -_{p}

Insulin clearance is defined as the quotient of the degradation rate and the insulin concentration

There are also reports that receptor-mediated transport in man contributes about one third to total renal insulin removal

Insulin injection and secretion

Pancreatic insulin secretion is induced by plasma glucose _{pan }is modeled as a function of insulin concentration and turned off at high insulin concentrations. This corresponds to the implicit assumption that glucose dynamics are faster than insulin dynamics. Peak concentrations in insulin therapy are about 60 – 80 ^{-1 }

Stationary model analysis. In this file, the stationary model equations are analyzed.

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Stationary model analysis in portable document format. This file is equivalent to Additional file 4 but results from a conversion to PDF.

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Intravenous injection of labeled and unlabeled insulin (_{*,in }and _{in}) is performed during injection time _{in }with a constant injection rate that is sharply, but smoothly cut off with Hill coefficient 50. This is an arbitrarily chosen parameter that realizes a switching procedure. Real step functions may cause numerical problems which can be avoided in this way.

The amounts of injected unlabeled and labeled insulin are _{in }and _{*,in }(unit: _{in }or _{*,in }then equal zero.

Note that _{in }and _{*,in }are not defined for _{in }= 0 _{in }= 0 _{in }and _{*,in }then have to be set to zero.

Note that 0.07

Insulin concentration in plasma

The balances of the concentrations of labeled (_{*}) and total insulin (

Note that _{liv }and _{kid }refer to total insulin. Therefore, only their fractions _{*}_{*}.

Dynamic model validation

Insulin dynamics

The dynamic insulin degradation behavior of the model was compared to experimentally determined time courses of insulin concentration in plasma after an intravenous insulin injection. Experimental data sets with extremely high ^{-6 }^{-4 }

Dynamic model validation: physiological insulin concentrations

**Dynamic model validation: physiological insulin concentrations**. Simulation of the concentration of radioactively labeled insulin in plasma after the injection of a very low amount of radioactively labeled insulin is shown and compared to experimental data [59].

Dynamic model validation: extremely high insulin concentrations

**Dynamic model validation: extremely high insulin concentrations**. Simulation of plasma insulin concentration after the injection of a large amount of insulin is shown and compared to experimental data [58]. Note that the model does not match the experimental data set. This results from the presence of unmodeled effects at highly supraphysiological insulin concentrations and limitations in the detection quality of the experiment. Therefore, the model is not valid at these extremely high insulin concentrations.

Simulated insulin concentrations for low amounts of injected insulin

Simulation results for the injection of unphysiologically high amounts of insulin

Note that an insulin concentration of 1600

The effects of pinocytosis and additional nonspecific insulin binding at high insulin concentrations are not quantified in literature and not included in the model. Neglecting these processes (and maybe others that are important at high insulin concentrations) leads to an incorrect model structure for high insulin concentrations. Therefore, the model is not valid at extremely high insulin concentrations.

Hepatic insulin receptor internalization

Simulation results for hepatic insulin receptor internalization at 100

Dynamic model validation: receptor internalization

**Dynamic model validation: receptor internalization**. Simulation results for receptor internalization at 100

It should be stated that the model parameters for receptor internalization and recycling are from the same source as the experimental data sets in Figure

Note that the experimental data sets for receptor internalization result from experiments with Fao cells that are tumor cells of hepatic origin. Though simulations match the experimental data sets almost quantitatively, this can only be regarded as a qualitative model validation for hepatocytes.

Stationary model validation

Simulation results for stationary insulin receptor activation and insulin binding were compared to experimental data sets

Stationary model validation: insulin binding and receptor phosphorylation

**Stationary model validation: insulin binding and receptor phosphorylation**. **Left**: Cell-associated radioactively labeled insulin is shown as a function of the stationary insulin concentration and compared to experimental data (Figure 4 A in [57]). Almost no labeled insulin should bind to receptors at maximal concentrations of unlabeled insulin. Therefore, the value for the highest concentration of unlabeled insulin was treated as background and subtracted from all values. **Right**: The fraction of phosphorylated receptors is shown as a function of the stationary insulin concentration and compared to experimental data for receptor activation (Figure 4 B in [57]). We regard receptor phosphorylation as a good indicator for receptor activity.

Klein et al. also determined the stationary dependency of receptor activity on the insulin concentration (Figure 4B in

Changes in _{in }or in the parameters for the pancreas have no effect on the results of stationary model validation, as the system is analyzed at constant insulin concentrations in the stationary case.

Altogether, the model is able to match the experimental data sets for receptor activity and insulin binding very well. Note that the model parameters were not estimated to get these results.

Model analysis

Insulin degradation

The fractions of insulin that are degraded by the liver and the kidney were investigated in several studies. Values for the relative contribution of the liver to insulin degradation in man range from below 50% to 70%, and those for the kidney from 30% to above 50%

Renal insulin degradation does not saturate

Renal and hepatic insulin degradation

**Renal and hepatic insulin degradation**. **Left**: Stationary insulin degradation rates of the liver (red) and the kidney (blue) and the total insulin degradation rate (black) are shown as functions of insulin concentration. **Right**: Stationary relative contributions of the liver (red) and the kidney (blue) to total insulin degradation depend on the insulin concentration. Note that these fractions are slightly lower in reality. Other tissues, in particular fat and muscle, also contribute to insulin degradation but are not analyzed here. The fractions in this plot refer to the sum of the degradation rates of liver and kidney.

In stationary model analysis, the relative contribution of the liver to overall insulin degradation ranges between 81% for insulin concentration tending to zero and 0% for insulin concentration tending to infinity. The relative contribution of the kidney ranges between 19% and 100% (Additional file

Note that changes in _{in }or in the parameters for the pancreas do not affect the results of stationary model analysis, as the system is analyzed at constant insulin concentrations. The rate of nonspecific insulin binding equals zero in the unique stationary case. Therefore, it also has no influence on stationary insulin degradation. The stationary analysis of degradation rates and relative contributions to insulin degradation is also independent of the parameter _{body }(Additional file

Quantitative results regarding relative contributions to insulin degradation are sensitive to changes in the parameter _{kid}) by 10%. However, changes in

Altogether, relative contributions of the tissues to insulin degradation depend on the insulin concentration. At low insulin concentrations, hepatic insulin degradation is predominant, whereas at high insulin concentrations overall insulin degradation is mainly performed by the kidney. Therefore, different results for the relative contributions of the liver and the kidney to insulin degradation are expected for different experimental settings.

Insulin clearance

The quotient of insulin degradation rate and insulin concentration is denoted as insulin clearance

The physiological range of insulin clearance in man (70 ^{-1 }

Insulin clearance strongly depends on the insulin concentration (Figure ^{-1 }in hepatic insulin clearance (-_{liv}·_{p}·^{-1}) strongly dominates the effect of the saturating degradation rate _{liv }(compare Figures

Renal and hepatic insulin clearance

**Renal and hepatic insulin clearance**. Insulin clearance is defined as the quotient of insulin degradation rate and insulin concentration. Total stationary insulin clearance (black) is a function of insulin concentration because hepatic insulin clearance (liver, red) depends on the insulin concentration, whereas renal insulin clearance (kidney, blue) is independent of insulin concentration. A body weight of _{body }= 200

As an example, in a rat whose body weight is 200 ^{-1 }for insulin concentration tending to zero and 2.0 ^{-1 }for insulin concentration tending to infinity (Additional file

Insulin clearance is often used to characterize the state of insulin metabolism. However, its value for analysis of processes that are dominated by saturable components, in particular hepatic insulin degradation, is very limited. A strong dependence on the insulin concentration hampers precise analysis, especially if one cannot guarantee that the insulin concentration is constant during the experiment. This problem does not occur when analyzing first order processes such as renal insulin degradation, where insulin clearance is independent of insulin concentration (Figure

Altogether, the strong dependency of insulin clearance on the insulin concentration is able to explain the wide range of reported values.

Parameter estimation

As shown above, insulin degradation at high insulin concentrations is mainly performed by the kidney. Nonspecific insulin binding dampens rapid variations in insulin concentration at all insulin concentrations (Additional file

In order to investigate whether the model structure can reproduce the experimental data set for high amounts of injected insulin

Thus, taking the experimental data set for high amounts of injected insulin

Sensitivity analysis

We showed that the parameters for the kidney and nonspecific insulin binding are most important at high insulin concentrations (Figure

On the other hand, the values of

The values from literature of

The parameters for the pancreas (

Changes in the parameter _{in }(for which no value is given) have practically no influence on the simulation results for high amounts of injected insulin _{in}. If one assumes that

Altogether, simulation results for insulin dynamics are sensitive to changes in _{in}. However, they are robust to changes in the other unknown parameters and in the parameters that are most important at high insulin concentrations.

Comparison with other models

Comparison of our model predictions for insulin dynamics with those of other models

Estimating the parameters of our model to match experimental data sets for humans as accurately as the models of human insulin dynamics should be possible. The reason for this is that our model considers more processes and therefore has more degrees of freedom. However, the model parameters are not identifiable if only experimental data sets for insulin concentration are used. Therefore, the estimated parameter values are not physiologically relevant.

We investigate whether the analysis of the other models of insulin receptor activation ^{-12 }^{-17 }^{-1}.

Stationary analysis of the model of Sedaghat et al. The stationary model equations of Sedaghat et al.

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Insulin degradation of adipocytes (i.e. with initial conditions taken from the model of Sedaghat et al.) is five orders of magnitude lower than hepatic insulin degradation (see above). At physiological insulin concentrations, insulin degradation is mainly performed by the liver (Figure

The situation is different when considering a modified model of Hori et al.

Stationary analysis of a modified model of Hori et al. The stationary model equations of a modified model of Hori et al.

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As the receptor concentrations in the models of Hori et al.

Note that our model is not able to quantitatively reproduce the dynamic phosphorylation characteristics in endosomes that were measured by Backer et al. in Fao cells

Note that the modified model of Hori et al. corresponds to a reduced model of our receptor model where some parameters and the total receptor concentration are identical.

Both receptor models from literature

Therapeutic insulin concentrations

The aim of insulin therapy is to achieve sufficient glucose uptake with minimal amounts of insulin

At about 10

Half-maximal insulin receptor phosphorylation in rat adipocytes is at 7 ± 1

Relatively large amounts of insulin or insulin analogues are injected or infused in postprandial glucose control. This mimics the physiological response of healthy individuals ^{-1 }

Therefore, the theoretical upper bound for reasonable therapeutic insulin concentrations in rats (about 3

Altogether, mathematical analysis and experimental results indicate that peak concentrations in insulin therapy are below the upper bound where a higher insulin concentration does not result in a stronger physiological effect.

Conclusion

We present a detailed dynamic model that describes

The vast majority of statements about insulin degradation and insulin clearance in the literature is given without explicitly defining the corresponding insulin concentration, and the reported values widely vary. Mathematical analysis shows that relative contributions of the liver and the kidney to total insulin degradation highly depend on the insulin concentration. At low insulin concentrations, insulin is mainly degraded by the liver, whereas renal insulin degradation is predominant at high insulin concentrations. This explains variations in reported values of relative contributions to insulin degradation.

Mathematical analysis also shows that insulin clearance strongly depends on the insulin concentration, which explains variations in reported values. Due to the concentration dependence of insulin clearance, its value for characterizing insulin metabolism is very limited.

The analysis of relative contributions to insulin degradation and the dose-response characteristics of insulin receptor activation and glucose uptake imply the existence of an upper bound for reasonable therapeutic insulin concentrations. Higher insulin concentrations do not result in higher glucose uptake and additional insulin is degraded without having therapeutic effect. However, the upper bound for reasonable therapeutic insulin concentrations is above peak concentrations in insulin therapy.

The detailed model presented here can be used as a starting point for modeling and analysis of the signaling cascades emerging from the hepatic insulin receptor (e.g. MAP kinase cascade and PI3K pathway). This will significantly contribute to understanding the effect of insulin on hepatocytes

Methods

Model parameters and initial conditions

^{-1}. ^{-1}. ^{-1}.

We compared weights of livers and bodies given in literature ^{5 }insulin receptors per hepatocyte ^{-1 }(Gisela Drews, personal communication). As the molecular weight of insulin is 5.7 ^{-1 }corresponds to 0.07

Stationary model equations (all derivatives set to zero) were solved for the state variables under the basal insulin concentration as a constraint to get initial conditions that correspond to the basal insulin concentration (Additional files

Dynamic model validation

Kruse et al. _{body }= 238 _{*,in }= 100 ^{-6 }_{in }= 0 _{body }= 190 _{in }= 47.5 _{*,in }= 0

Backer et al. investigated receptor internalization in Fao hepatoma cells

Dynamic simulation was performed with the software package MATLAB (The MathWorks). The executable

Stationary analysis

The liver is a dynamic system. Its physiological state and its contribution to insulin degradation depend on the insulin concentration. However, the rate of hepatic insulin degradation is not uniquely defined by the present insulin concentration. The hepatic insulin degradation rate strongly depends on the amount of free receptors at the plasma membrane. Assume that the system is not in steady state. In this case, for the model, insulin concentrations from all points in time since the system left the steady state affect the physiological state of the liver and in particular the amount of free insulin receptors at the plasma membrane.

The steady state of our model is uniquely defined by the steady state insulin concentration. Note that the stationary case corresponds to the steady state of the system. In the stationary case, all characteristics of the system (e.g. insulin degradation rates, insulin clearance, receptor activation or insulin binding) can be expressed as functions of the stationary insulin concentration. Thus, stationary analysis gives insights into the system that are not biased by dynamic effects.

All stationary computations were performed with the software package

Abbreviations

ass: assumption; bpm, beats per minute; calc: calculated; ml: milliliter (10^{-3 }^{-9 }^{-1}); nmol: nano mol (10^{-9 }

Authors' contributions

MK developed the model, performed the analysis and drafted the manuscript. EDG initiated and supervised the study.

Acknowledgements

We thank Johannes Witt (Institute for System Dynamics, Universität Stuttgart), Michael Ederer, Holger Conzelmann (Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg) and Duncan Payne (Department of Computer Science, Shefield University) for inspiring discussions and proofreading the manuscript. We acknowledge Gisela Drews (Pharmazeutisches Institut, Eberhard Karls Universität Tübingen) and Michael Heinrich (Center for Biotechnology and Biomedicine, Universität Leipzig) for valuable hints. This work was supported by the Network Systems Biology HepatoSys, which is funded by the German Federal Ministry of Education and Research (Bundesministerium für Bildung und Forschung, BMBF).