The liver and the kidney are the most important insulin degrading tissues 59. However, fat and muscle tissues also contribute to insulin degradation. In the following, we show that the insulin degradation rate of the fat tissue is small compared to the hepatic insulin degradation rate. According to Sedaghat et al. 37, the total receptor concentration in adipocytes is 10^-3 nM and the rate constant of receptor internalization is 3.5·10^-5 s^-1. Insulin receptors in hepatocytes have a minimal internalization rate constant of 2·10^-4 s^-1 34 and a concentration of 40 nM (10⁵ receptors per hepatocyte 44, a hepatocyte is assumed to be a sphere with 20 μm diameter). 78% of the liver volume is occupied by hepatocytes 45. Liver mass is about 5% of body weight 46, the mass of the fat tissue is in the same order of magnitude. We postulate the same insulin binding characteristics to the receptor in both tissues. The product of receptor concentration and the kinetic parameter for receptor internalization in adipocytes is five orders of magnitude lower than in hepatocytes. Therefore, the contribution of the fat tissue to insulin degradation can be neglected. The contribution of the muscle tissue will not be analyzed either. No quantitative data was found and a qualitatively similar behavior in comparison to the liver is expected.

Hepatic insulin receptors have access to insulin molecules in the space of Disse 46. The space of Disse (perisinusoidal space) contains blood plasma and is an extracellular space between liver sinusoids (special blood vessels) and hepatocytes. Insulin molecules from the space of Disse can be bound and are internalized together with the receptor. In the space of Disse, there is also nonspecific insulin binding to hepatocytes. Inside the cell, in the acidic endosomal compartment, insulin dissociates from the receptor and is degraded. The receptor then recycles to the cell surface 1.

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 3738, we provide an extended description, model the in vivo situation and include reversible nonspecific insulin binding.

The kidney's contribution to insulin clearance mainly consists of the filtering of insulin from the blood 9. The filtering function of the kidney is modeled as a degradation rate that, according to experimental data 47, does not saturate and is proportional to insulin concentration in the plasma.

Pancreatic insulin secretion is mainly induced by plasma glucose 2. As we focus on insulin degradation, glucose is not included in the model. We model pancreatic insulin secretion in a highly simplified way as a function of insulin concentration. Due to high robustness to changes in the parameters for insulin secretion (Additional file 3), this simplification does not lead to significant approximation errors. In addition, insulin secretion is irrelevant for stationary analysis at constant insulin concentrations.

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.

In vivo model parameters cannot be measured directly in most cases. Taking parameters from in vitro experiments or models for in vivo processes is a promising alternative. Note that this can be problematic since the in vitro parameter values may not be similar to their in vivo counterparts. However, it is the only possibility if there is not sufficient experimental data and a model structure that guarantees identifiability. Using in vitro parameters or otherwise determined model parameters and linking kinetic models of smaller parts of the overall system together is frequently performed, e.g. by the Silicon Cell project 43. This strategy is structurally supported by modular modeling tools, e.g. ProMoT 48.

In this study, model parameters (Table 1) are taken from previously published in vitro experiments 4749505152535455 and small models of insulin binding 36, receptor internalization 34 and nonspecific hepatic insulin binding 46. The models from literature 343646 were combined and kinetic parameters for the remaining processes were taken from in vitro data.

The parameters from the models of insulin binding 36 and nonspecific insulin binding in the liver 46 were directly taken for our model. The relatively simple models for receptor internalization and recycling at high insulin concentrations and without insulin 34 were combined to describe receptor internalization and recycling at arbitrary insulin concentrations.

The parameters for pancreatic insulin secretion were chosen to guarantee the physiological basal level of insulin (0.07 nM, Gisela Drews, personal communication) and to cut off insulin synthesis at peak concentrations in insulin therapy (0.5 nM 1617).

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 bpm 54) that guarantees a fast distribution of circulating insulin.

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 1). In accordance with experimental results 56, the described processes lead to saturation of hepatic insulin degradation at high insulin concentrations.

Model assumptions that are supported by studies from literature are:

• Insulin binding and dissociation are independent of the phosphorylation state of the receptor. This is directly supported by experimental evidence 42.

• Only receptors with bound insulin show autophosphorylation activity. Autophosphorylation is induced by insulin binding 2 and there is no experimental data quantifying autophosphorylation of receptors without bound insulin.

• Receptor dephosphorylation is independent of insulin binding. Receptor dephosphorylation is performed by protein phosphatases 2. It seems very unlikely that insulin binding to the extracellular α-chain of the receptor induces conformational changes in the intracellular β-chain that are large enough to significantly change the affinity of phosphatases for their phosphorylated substrate sites.

• Insulin dissociation from endosomal receptors is irreversible. Upon internalization, the pH in endosomes decreases rapidly, which promotes insulin dissociation from the receptor 9. Free endosomal insulin is degraded by proteases 9.

• Only receptors without insulin are recycled. Receptor recycling is faster if there is no external insulin 34. This leads to the assumption that an additional step for receptors with bound insulin is necessary before recycling is possible. A very promising candidate for this step is insulin dissociation from the receptor. In this case, a single rate constant for recycling, independent of insulin concentration is sufficient to explain the observation.

• Phosphorylated receptors are internalized faster than unphosphorylated receptors. In the presence of higher insulin concentrations, more insulin receptors are phosphorylated 2. Receptor internalization is faster at high insulin concentrations than without external insulin 34. In addition, there are reports that receptor internalization depends on phosphorylation 9.

• Labeled and unlabeled insulin show the same physiological characteristics. Labeling of the insulin molecules was performed with ¹²⁵I 575859. The size of this modification is small compared to the size of the insulin molecule and should not change its binding characteristics, the effect on receptor phosphorylation, the rate of nonspecific insulin binding or the rate of renal insulin filtration.

• All processes in hepatocytes obey mass action kinetics. The processes that were adopted from other models obey mass action kinetics 343646. Mass action kinetics is a good and frequently used approximation for processes at the molecular level.

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.

• Receptors with one or two bound insulin molecules show the same autophosphorylation activity.

• Receptor recycling is independent of receptor phosphorylation.

In the following, the insulin receptor is denoted as R. The binding of one or two insulin molecules is indicated by a prefix I or I2, respectively. A suffix p indicates receptor phosphorylation, a subscript en indicates internalization to the endosomal compartment. All concentrations of receptor species refer to v_hep, the total volume of hepatocytes.

In general, rates denoted by the standard notation r_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 i_j describe internal processes occurring in endosomes of hepatocytes, and rates denoted by f_j describe flows between the plasma membrane and endosomes of hepatocytes.

Figure 1 shows the reaction scheme of processes in hepatocytes.

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 Ins that is the sum of labeled and unlabeled insulin concentrations. The concentration of labeled insulin is denoted as Ins_* . Unlabeled insulin (Ins – Ins_*) has no separate notation. The total contribution of the liver to insulin degradation is

r_liv = (-r₁ - r₂ - r₃ - r₄)·v_hep/v_p.

The plasma volume is denoted as v_p, the total hepatocyte volume is denoted as v_hep. Strictly speaking, r_liv defines insulin removal from the blood, whereas insulin degradation is performed in hepatic endosomes. However, r_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 r₁ – r₄ describe insulin binding to the insulin receptor at the plasma membrane. The values of the parameters kins, kins1d and kins2d were directly taken from the model of Wanant et al. 36.

r 1 = k i n s ⋅ R ⋅ I n s − k i n s 1 d ⋅ I R r 2 = k i n s ⋅ R p ⋅ I n s − k i n s 1 d ⋅ I R p r 3 = k i n s ⋅ I R ⋅ I n s − k i n s 2 d ⋅ I 2 R r 4 = k i n s ⋅ I R p ⋅ I n s − k i n s 2 d ⋅ I 2 R p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@B64A@

Rates r₅ – r₇ describe receptor phosphorylation at the plasma membrane.

r 5 = k y d ⋅ R p r 6 = k y p ⋅ I R − k y d ⋅ I R p r 7 = k y p ⋅ I 2 R − k y d ⋅ I 2 R p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabmqaaaqaaiabdkhaYnaaBaaaleaacqaI1aqnaeqaaOGaeyypa0Jaem4AaSMaemyEaKNaemizaqMaeyyXICTaemOuaiLaemiCaahabaGaemOCai3aaSbaaSqaaiabiAda2aqabaGccqGH9aqpcqWGRbWAcqWG5bqEcqWGWbaCcqGHflY1cqWGjbqscqWGsbGucqGHsislcqWGRbWAcqWG5bqEcqWGKbazcqGHflY1cqWGjbqscqWGsbGucqWGWbaCaeaacqWGYbGCdaWgaaWcbaGaeG4naCdabeaakiabg2da9iabdUgaRjabdMha5jabdchaWjabgwSixlabdMeajjabikdaYiabdkfasjabgkHiTiabdUgaRjabdMha5jabdsgaKjabgwSixlabdMeajjabikdaYiabdkfasjabdchaWbaaaaa@698E@

Rates i₁ – i₄ describe insulin dissociation from the receptor in endosomes.

i 1 = k i n s 1 d e n ⋅ I R e n i 2 = k i n s 1 d e n ⋅ I R p e n i 3 = k i n s 2 d e n ⋅ I 2 R e n i 4 = k i n s 2 d e n ⋅ I 2 R p e n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@86DC@

Rates i₅ – i₇ describe receptor phosphorylation in endosomes.

i 5 = k y d e n ⋅ R p e n i 6 = k y p ⋅ I R e n − k y d e n ⋅ I R p e n i 7 = k y p ⋅ I 2 R e n − k y d e n ⋅ I 2 R p e n MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@801C@

According to our model assumptions, unphosphorylated receptors without insulin (R and R_en) have no autophosphorylation activity. Therefore, the reactions represented by the rates r₅ and i₅ are irreversible. Rates f₁ – f₆ describe receptor internalization and recycling.

f 1 = i n t k 2 ⋅ R − r e c k 1 ⋅ R e n f 2 = i n t k 2 ⋅ I R f 3 = i n t k 2 ⋅ I 2 R f 4 = i n t k 1 ⋅ R p − r e c k 1 ⋅ R p e n f 5 = i n t k 1 ⋅ I R p f 6 = i n t k 1 ⋅ I 2 R p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xH8viVGI8Gi=hEeeu0xXdbba9frFj0xb9qqpG0dXdb9aspeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@A434@

The value of the parameter intk1 was directly taken from a model of receptor internalization and recycling at high insulin concentrations 34. The values of the parameters intk2 and reck1 are from a model of receptor internalization and recycling without insulin 34.

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

R ˙ = − r 1 + r 5 − f 1 I R . = r 1 − r 3 − r 6 − f 2 I 2 R . = r 3 − r 7 − f 3 R ˙ p = − r 2 − r 5 − f 4 I R ˙ p = r 2 − r 4 + r 6 − f 5 I 2 R p . = r 4 + r 7 − f 6 R ˙ e n = i 1 + i 5 + f 1 I R e n . = − i 1 + i 3 − i 6 + f 2 I 2 R e n . = − i 3 − i 7 + f 3 R ˙ p e n = i 2 − i 5 + f 4 I R ˙ p e n = − i 2 + i 4 + i 6 + f 5 I 2 R p e n . = − i 4 + i 7 + f 6 MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=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@FA7B@

The liver also performs nonspecific insulin binding. This reversible process does not saturate 46 and dampens rapid variations in insulin concentration. The rates r_ub and r_*,ub define nonspecific binding of unlabeled and labeled insulin, respectively.

r u b = ( k 1 u b ⋅ ( I n s − I n s ∗ ︷ unlabeled insulin ) ⋅ v d − k 2 u b ⋅ I n s u b ) / v p r ∗ , u b = ( k 1 u b ⋅ I n s ∗ ⋅ v d − k 2 u b ⋅ I n s ∗ , u b ) / v p MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacPC6xNi=xI8qiVKYPFjYdHaVhbbf9v8qqaqFr0xc9vqFj0dXdbba91qpepeI8k8fiI+fsY=rqGqVepae9pg0db9vqaiVgFr0xfr=xfr=xc9adbaqaaeGaciGaaiaabeqaaeqabiWaaaGcbaqbaeWabiqaaaqaaiabdkhaYnaaBaaaleaacqWG1bqDcqWGIbGyaeqaaOGaeyypa0JaeiikaGIaem4AaSMaeGymaeJaemyDauNaemOyaiMaeyyXICTaeiikaGYaaGraaeaacqWGjbqscqWGUbGBcqWGZbWCcqGHsislcqWGjbqscqWGUbGBcqWGZbWCdaWgaaWcbaGaey4fIOcabeaaaeaacqqG1bqDcqqGUbGBcqqGSbaBcqqGHbqycqqGIbGycqqGLbqzcqqGSbaBcqqGLbqzcqqGKbazcqqGGaaicqqGPbqAcqqGUbGBcqqGZbWCcqqG1bqDcqqGSbaBcqqGPbqAcqqGUbGBaOGaay5n+dGaeiykaKIaeyyXICTaemODay3aaSbaaSqaaiabdsgaKbqabaGccqGHsislcqWGRbWAcqaIYaGmcqWG1bqDcqWGIbGycqGHflY1cqWGjbqscqWGUbGBcqWGZbWCdaWgaaWcbaGaemyDauNaemOyaigabeaakiabcMcaPiabc+caViabdAha2naaBaaaleaacqWGWbaCaeqaaaGcbaGaemOCai3aaSbaaSqaaiabgEHiQiabcYcaSiabdwha1jabdkgaIbqabaGccqGH9aqpcqGGOaakcqWGRbWAcqaIXaqmcqWG1bqDcqWGIbGycqGHflY1cqWGjbqscqWGUbGBcqWGZbWCdaWgaaWcbaGaey4fIOcabeaakiabgwSixlabdAha2naaBaaaleaacqWGKbazaeqaaOGaeyOeI0Iaem4AaSMaeGOmaiJaemyDauNaemOyaiMaeyyXICTaemysaKKaemOBa4Maem4Cam3aaSbaaSqaaiabgEHiQiabcYcaSiabdwha1jabdkgaIbqabaGccqGGPaqkcqGGVaWlcqWG2bGDdaWgaaWcbaGaemiCaahabeaaaaaaaa@A6BC@

The values of the parameters k1ub and k2ub were directly taken from the model of Hammond et al. 46. The volume of the space of Disse, in which nonspecific insulin binding takes place, is denoted as v_d. The concentration of unlabeled insulin is Ins – Ins_* (unit: nM), while Ins_*,ub and Ins_ub are the amounts of substance (unit: nmol) for nonspecifically bound labeled and unlabeled insulin, respectively. The expressions for the forward reactions of the rates r_ub and r_*,ub are multiplied by v_d (unit: l) as Ins and Ins_* (unit: nM) are concentrations, whereas Ins_ub and Ins_*,ub are amounts of substance (unit: nmol). The balances of the amounts of nonspecifically bound labeled and unlabeled insulin are given by