# Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System’s Dynamics: The Life Cycle of the Insulin Receptor

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## Abstract

**:**

## 1. Introduction

**Figure 1.**The processes of insulin-dependent activation and recycling of the IR. First, insulin binds to the IR. Afterwards, the IR gets autophosphorylated. Two alternative processes describe the dephosphorylation of the IR: (1) The IR can dephosphorylate on the plasma membrane by the dissociation of insulin; (2) The IR can be internalized into the cytoplasm. There, the insulin is degraded, and the IR is transported back to the plasma membrane. The processes of dephosphorylation on the plasma membrane and in the cytoplasm are both catalyzed by the enzyme PTP1B.

**Figure 2.**The PN describes the topological network structure of insulin dependent activation and recycling. Rectangles represent transitions, i.e., reactions. Places are plotted as circles. Directed, weighted edges connect transitions and places. The places represent chemical species, e.g., insulin, receptors, or complexes, and can carry tokens, which represent discrete amounts of the chemical species. Transitions can consume tokens from the pre-places and generate tokens on the post-places.

## 2. Methods

#### 2.1. Petri Nets

- P is a finite set of places,
- T is a finite set of transitions,
- $F\subseteq (P\times T)\cup (P\times T)$ is a set of edges,
- $W:F\to \mathbb{N}$ is the set of edge weights, and
- ${M}_{0}:P\to {\mathbb{N}}_{0}$ is the initial marking.

#### 2.1.1. Timed Petri Nets

**Figure 3.**An example for a TPN. The TPN consists of three places, place 1, place 2, and a time generation place (TGP), and one transition connected by directed edges with an edge of weight 1. (

**a**) In the initial state of the TPN, place 2 carries no token and place 1 10 tokens, each of them with a time stamp of $@0$. There exists a global clock time (not explicitly depicted) which is in the initial state 0. For the transition, a time delay of @ + 2 is defined, indicating that the time stamp of each token processed by this transition increases by 2. The TGP is a pre- and post-place of the transition, i.e., the token on that place will not be consumed when the transition fires. This token is initialized with a time stamp of $@0$. (${1}^{\prime}\left(\right)$ and ${1}^{\prime}\left[1\right]$ are specific notations of the CPN Tools.); (

**b**) The state of the TPN after four simulation steps. 4 tokens on place 1 were consumed and produced on place 2, carrying the time stamps, $@2,@4,@6$, and $@8$, respectively. The other six tokens remain still on place 1 with a time stamp of zero. The transition is enabled if on place 1 and the TGP exist at least one token with a time stamp less or equals the global clock time. If there is no transition activated anymore, the global clock time will be increased until at least one transition can fire. Here, the global clock time has to be increased by 2 with each firing step. The TGP ensures that the tokens can be distinguished from each other, i.e., only one token is moved from place 1 to place 2 for a given global clock time. After the first firing step, the time stamp of the TGP token and the token on place 2 have a time stamp of 2. The global clock time is still 0. Now, the transition is not anymore enabled, because there is no token with a time stamp less than or equals the global clock time on the TGP. For the next firing, the global clock time has to be increased by 2.

#### 2.1.2. General Properties

#### 2.1.3. Invariant Properties

## 3. Results and Discussion

**Table 1.**List of abbreviations and initial concentrations. Initial concentrations are adopted from Sedaghat et al. [34]. The concentrations are given in units of pM $={10}^{-12}\phantom{\rule{0.277778em}{0ex}}$M.

Abbreviation | Species | Initial Concentration(s) [pM] |
---|---|---|

I | insulin | [I]${}_{0}={10}^{3}$–${10}^{6}$ |

IR | insulin receptor | [IR]${}_{0}=0.9$ |

IRI | I-IR complex | — |

IRIP | phosphorylated IRI | — |

IRIP${}_{in}$ | intracellular IRIP | — |

IR${}_{in}$ | intracellular IR | [IR${}_{in}$]${}_{0}=0.1$ |

Parameter | Process | Value | Units |
---|---|---|---|

${k}_{\text{bind}}$ | binding of insulin | $6\times {10}^{7}$ | M${}^{-1}$ min${}^{-1}$ |

${k}_{\text{diss}}$ | dissociation of insulin | $0.2$ | min${}^{-1}$ |

${k}_{\text{phos}}$ | phosphorylation | $2.500$ | min${}^{-1}$ |

${k}_{\text{dephos,m}}$ | dephosphorylation on membrane | $0.2$ | min${}^{-1}$ |

${k}_{\text{in}}$ | internalization of IR | $3.\overline{3}\times {10}^{-4}$ | min${}^{-1}$ |

${k}_{\text{out}}$ | transport of IR to plasma membrane | $3\times {10}^{-3}$ | min${}^{-1}$ |

${k}_{\text{in,p}}$ | internalization of phosphorylated IR | $2.1\times {10}^{-3}$ | min${}^{-1}$ |

${k}_{\text{out,p}}$ | transport of phosphorylated IR to plasma membrane | $2.1\times {10}^{-4}$ | min${}^{-1}$ |

${k}_{\text{dephos,c}}$ | dephosphorylation in cytoplasm | $0.461$ | min${}^{-1}$ |

${k}_{\text{deg}}$ | degradation | $1.67\times {10}^{-4}$ | min${}^{-1}$ |

${k}_{\text{syn}}$ | synthesis | ${}^{*1}$ $1.67\times {10}^{-17}$ | M min${}^{-1}$ |

${}^{*2}$ $1.00\times {10}^{-16}$ | M min${}^{-1}$ |

#### 3.1. The P/T-PN Model and Its Properties

**Figure 4.**The transition invariant, TI${}_{1}$, is highlighted in the PN. TI${}_{1}$ describes a cycle of the processes: binding of insulin to the IR (${k}_{\text{bind}}$), phosphorylation of the insulin-IR complex (${k}_{\text{phos}}$), and extracellular dissociation of the activated insulin-IR complex (${k}_{\text{dephos,m}}$). The steady state of this subnetwork has an equilibrium constant of ${i}_{c}=3.33$ nM for the binding of insulin to the receptor.

**Figure 5.**The transition invariant, TI${}_{2}$, is highlighted in the PN. The six transitions of TI${}_{2}$ form a chain of six consecutive reactions: buffering of insulin (buffer), binding of insulin to the IR (${k}_{\text{bind}}$), phosphorylation of the insulin-IR complex (${k}_{\text{phos}}$), internalization of the activated insulin-IR complex (${k}_{\text{in,p}}$), dephosphorylation of the internalized insulin-IR complex (${k}_{\text{dephos,c}}$), and translocation of internalized IR back to the membrane (${k}_{\text{out}}$).

#### 3.2. The TPN Model and Its Properties

**Figure 6.**The TPN model of activation and recycling of the IR. It explicitly includes the enzyme, protein-tyrosine phosphatase 1B (PTP1B), see Figure 1 for a sketch of the catalytic function of PTPN1B. The circles represent chemical species, here, mainly the different insulin complexes, see Table 3. The rectangles describe the transitions, see Table 4. For a full list of places and transitions of the TPN, we refer to Table 3 and Table 4, respectively.

Name | Molecule | Initial Number of Tokens |
---|---|---|

I | insulin | 10,000 |

IR | insulin receptor | 90 |

IRI | I–IR complex | 0 |

IRIP | phosphorylated IRI | 0 |

IRIP${}_{intra}$ | intracellular IRIP | 0 |

IR${}_{intra}$ | intracellular IR | 10 |

PTPN1B | protein-tyrosine phosphatase 1B | 1000 |

IRIP PTP1B | IRIP–PTPN1B complex | 0 |

IRIP${}_{intra}$ PTPN1B | IRIP${}_{intra}$–PTPN1B complex | 0 |

Phos | phosphate | 1000 |

**Table 4.**List of transitions of the TPN model of receptor phosphorylation and recycling. The time inscriptions of the transitions are constant delay increments. We adapted the values of the constant delays to each initial insulin concentration of $1\phantom{\rule{0.166667em}{0ex}}\mu $M, $100\phantom{\rule{0.166667em}{0ex}}$nM, 10 nM, and $10\phantom{\rule{0.166667em}{0ex}}$nM separately. This list exemplifies the constant delays for ${i}_{0}=1\phantom{\rule{0.166667em}{0ex}}\mu $M.

Name | Process | Time Inscription |
---|---|---|

bin_1 | binding of insulin | @ + 1 |

dis_1 | dissociation of insulin | @ + 40 |

autophos_1 | phosphorylation of IRI | @ + 1 |

intra_1 | internalization of IR | @ + 200 |

memb_1 | transport of IR${}_{intra}$ to plasma membrane | @ + 85 |

intra_2 | internalization of IRIP | @ + 110 |

memb_2 | transport of IRIP${}_{intra}$ to plasma membrane | @ + 400 |

dephos_1 | dephosphorylation of IRIP by PTPN1B | @ + 1 |

dephos_2 | IRIP binds to PTPN1B | @ + 40 |

dephos_3 | IRIP${}_{intra}$ binds to PTPN1B | @ + 20 |

dephos_4 | dephosphorylation of IRIP${}_{intra}$ by PTPN1B | @ + 1 |

**Table 5.**Adaption of time delays of timed transitions to initial concentrations of insulin, 1 μM, 100 nM, 10 nM, and 1 nM. The values in parentheses are applied after 4500 and $15,000$ clock times for the insulin concentrations, ${i}_{0}=10$ nM and ${i}_{0}=1$ nM, respectively.

Transition | 1 μM | 100 nM | 10 nM | 1 nM |
---|---|---|---|---|

bin_1 | @ + 1 | @ + 7 | @ + 16 (@ + 29) | @ + 20 (@ + 30) |

dis_1 | @ + 40 | @ + 40 | @ + 40 (@ + 39) | @ + 40 (@ + 60) |

memb_1 | @ + 85 | @ + 85 | @ + 86 | @ + 86 |

intra_2 | @ + 110 | @ + 110 | @ + 119 | @ + 119 |

dephos_2 | @ + 40 | @ + 40 | @ + 40 (@ + 52) | @ + 40 (@ + 65) |

**Figure 7.**The left part depicts the concentrations of the IR (dark blue line), the phosphorylated insulin-IR complex IRIP (light blue line), the internalized complex IRIP${}_{intra}$ (purple line), and the intracellular IR${}_{intra}$ (yellow line) versus reaction time for initial insulin concentrations, 10 nM (part A, top), and 1 nM (part B, bottom), respectively. The concentrations are precise numerical simulations of the reaction system Equation (5). The right part shows the number of tokens versus the global clock time for the corresponding TPN. One token equates to a concentration of about 10 fM, and 2000 counts of the global clock time represent one minute reaction time.

#### 3.3. Quasi-Steady-State Approximation

**Figure 8.**The fractions of the steady-state concentrations, $i{r}^{*}/i{r}_{0}$ and $iri{p}^{*}/i{r}_{0}$, are plotted versus the concentration of the external insulin. The vertical line indicates the value of the equilibrium constant, ${i}_{c}=3.33$ nM. For insulin concentration above ${i}_{c}$, more than 50% of the extracellular IR binds an insulin molecule. All the IR becomes saturated for an insulin concentration which is large compared to ${i}_{c}$, i.e., for $i\gg {i}_{c}=3.33$ nM.

#### Time Behavior

**Figure 9.**Precise numerical solutions for the concentration of accessible IR on the membrane (solid line with the label free receptor), the total concentration of IR on the membrane, $i{r}_{memb}=ir+iri+irip$ (solid line with the label receptor (membrane)), and the total concentration of the IR, $i{r}_{tot}=i{r}_{memb}+iri{p}_{in}+i{r}_{in}$ (solid line with the label receptor (total)), are plotted versus the logarithmic time axis. The concentrations are given in percentage of the total concentration, $i{r}_{tot}$, of the IR of the basal cell. The initial insulin concentration is ${i}_{0}=1$ nM. The down-regulation of the cell for insulin given in excess passes through three phases: binding of insulin, internalization of the activated IR, and degradation of the IR inside the cell. The analytical approximation (Equation 6) for the concentration of the IR is drawn as dotted line with the label TI${}_{1}$ and describes the fast binding of insulin to the IR. The dotted line with the label TI${}_{2}$ shows the approximation (8). The analytical solution (Equation 11) for the concentration of the IR is depicted as dotted line with the label, TI${}_{2}$ + TI${}_{6}$, and is indistinguishable from the precise numerical solution (solid line free receptor) until the QSSA breaks down, i.e., until the consumption of insulin becomes measurable. The concentration of insulin is not plotted.

## 4. Conclusions

**Figure 10.**The steady-state concentrations, $i{r}^{\u2020}$ (red line), $iri{p}^{\u2020}$ (green line), $i{r}_{in}^{\u2020}$ (blue line), and $i{r}_{tot}=i{r}^{\u2020}+iri{p}^{\u2020}+i{r}_{in}^{\u2020}$ (violet line) are plotted versus the external insulin concentration ${i}_{0}$. The critical insulin concentration, ${i}_{c}^{\u2020}=0.535$ nM, and the equilibrium constant, ${i}_{c}=3.33$ nM, are indicated by left and right vertical lines, respectively. The steady-state concentration, $i{r}_{in}^{\u2020}$, is regulated by synthesis and degradation of the IR in the cytoplasm and hence, remains constant. The steady-state concentration, $iri{p}^{\u2020}$, is zero in the basal state of the cell, i.e., in absence of extracellular insulin, ${i}_{0}=0$. In the process of down-regulation of the cell, i.e., for increasing insulin level, ${i}_{0}$, the concentration, $iri{p}^{\u2020}$, increases until it reaches its maximal values for ${i}_{0}\gg {i}_{c}^{\u2020}$. The steady-state concentration, $i{r}^{\u2020}$, of the surface IR is maximal in the basal state and drops down to zero for ${i}_{0}\gg {i}_{c}^{\u2020}$.

## Supplementary Materials

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**MDPI and ACS Style**

Scheidel, J.; Lindauer, K.; Ackermann, J.; Koch, I. Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System’s Dynamics: The Life Cycle of the Insulin Receptor. *Metabolites* **2015**, *5*, 766-793.
https://doi.org/10.3390/metabo5040766

**AMA Style**

Scheidel J, Lindauer K, Ackermann J, Koch I. Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System’s Dynamics: The Life Cycle of the Insulin Receptor. *Metabolites*. 2015; 5(4):766-793.
https://doi.org/10.3390/metabo5040766

**Chicago/Turabian Style**

Scheidel, Jennifer, Klaus Lindauer, Jörg Ackermann, and Ina Koch. 2015. "Quasi-Steady-State Analysis based on Structural Modules and Timed Petri Net Predict System’s Dynamics: The Life Cycle of the Insulin Receptor" *Metabolites* 5, no. 4: 766-793.
https://doi.org/10.3390/metabo5040766