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

The insulin-dependent activation and recycling of the insulin receptor play an essential role in the regulation of the energy metabolism, leading to a special interest for pharmaceutical applications. Thus, the recycling of the insulin receptor has been intensively investigated, experimentally as well as theoretically. We developed a time-resolved, discrete model to describe stochastic dynamics and study the approximation of non-linear dynamics in the context of timed Petri nets. Additionally, using a graph-theoretical approach, we analyzed the structure of the regulatory system and demonstrated the close interrelation of structural network properties with the kinetic behavior. The transition invariants decomposed the model into overlapping subnetworks of various sizes, which represent basic functional modules. Moreover, we computed the quasi-steady states of these subnetworks and demonstrated that they are fundamental to understand the dynamic behavior of the system. The Petri net approach confirms the experimental results of insulin-stimulated degradation of the insulin receptor, which represents a common feature of insulin-resistant, hyperinsulinaemic states.


TI
: k in,p , k out,p (internalization of phosphorylated insulin receptor (IR), translocation back to membrane) 5. TI 5 : k bind , k diss (extracellular binding of insulin, release of insulin) 6. TI 6 : k syn , k deg (synthesis, degradation of receptor) Each transition is member of at least one TI, hence the network is covered by TI (CTI).
S2. Quasi-Steady-State Approximation for TI 1 TI 1 describes a cycle of reactions for the species IR, IRI, and IRIP. The corresponding dynamic system is given by where c = ( ir, iri, irip) T denotes a vector of concentrations. The concentration of free insulin is assumed to be constant, i.e., i = i 0 . Within the QSSA we solved the linear system ∂ c ∂τ = 0 (S2) and obtained the steady state for the 3 concentrations For our choice of kinetic rate constants, the insulin-binding equilibrium constant becomes i c = 3.33 nM. Sedaghat et al. assume a fast process of phosphorylation (i.e., k phos k diss and k phos k dephos,m ). In this case the equation is a reasonable approximation. Since the ratio k dephos,m /k phos is less than 0.1 %, we may neglect iri * , and the formula is sufficiently precise for practical applications.

S3. Quasi-Steady-State Approximation for TI 2
The steady-state concentrations ir * , iri * , and irip * completely ignore the process of translocation of receptor into the cytoplasm and are a justifiable approximation only for a short reaction time compared to the time scale of the translocation process. The process of translocation of the activated IR into the cytoplasm (k in,p ) is member of the subnetwork defined by TI 2 . The ODE system of the subnetwork reads with the vector of concentrations, c = ( ir, iri, irip, ir in , irip in ) T . The steady state is given by i † c is the critical insulin concentration for the internalization of receptor. For a fast phosphorylation process as postulated by Sedaghat et al., (i.e., k phos = 2.500 min −1 ) a simplification of equations (S7,S8) is feasible. We considered nonzero degradation and nonzero synthesis of the receptor, i.e., k syn , k deg , in the steady state (S7). However, the degradation and synthesis are not members of TI 2 but form the trivial TI 6 . For k syn = k deg = 0 (i.e. in the case of no degradation and no synthesis), the steady-state concentration, ir † in , becomes a free parameter and has to be determined by a mass conservation equation for the amount of the receptor in the cell.
For our choice of kinetic constants, we get the numerical value, i † c = 0.535 nM, for the critical insulin concentration of internalization of the IR and the steady state concentrations (S7) become The steady state concentrations, iri † and irip † in , of the transient complexes are below experimental detection limits. The steady state concentration, ir † in , of free intracellular receptor is regulated by synthesis (k syn ) and degradation (k deg ), and hence remains constant for all values of i 0 . In the limit of small concentrations of insulin, i 0 → 0, the function approaches zero for vanishing concentration of external insulin, i.e., lim i 0 →0 f (i 0 ) = 0. For increasing concentrations of insulin, i 0 → ∞, the function f (i 0 ) converges to 1. Since the steady-state concentrations, iri † , irip † and irip † in , are proportional to f (i 0 ), they are zero in the basal state of the cell, i.e., in absence of extracellular insulin, i 0 = 0. In the process of down-regulation by insulin, the concentrations, iri † , irip † , and irip † in , increase proportionally to the function f (i 0 ) until they reach their maximal values for i 0 i † c . The steady-state concentration, ir † , of the surface receptor is proportional to 1 − f (i 0 ), and hence, ir † is maximal in the basal state and drops down to zero for i 0 i † c .

S5. Characteristic Eigenvalue for TI 2
The characteristic eigenvalue of ODE (S6) is given by

S6. Drop of Insulin and the Lambert Function
We have abstained from discussing the development of insulin concentration with time based on the functional regimes of the Lambert function W. It is easy to see that for insulin concentrations well below the critical concentration of i † c = 0.535 nM, the differential equation simplifies to and the insulin concentration drops down exponentially in time In the case of a high concentration of insulin (i.e., for i i † c = 0.535 nM), the cell is maximally down-regulated, and the differential equation is given by Consequently, the consumption of insulin with constant maximal velocity leads to a linear diminishment of insulin: The consumption of insulin by the cell leads to an exponential drop on the time scale of t 4 = 5 h 33 min, if the insulin concentration is below the critical insulin concentration, i † c = 0.535 nM. For insulin given in excess (i.e., for i i † c ), the insulin concentration decreases linearly with a flat-angle slope of 0.535 nM/5 h 33 min.

S7. Phosphorylation Dynamics
Cedersund et. al. [1] have discussed the short-term phosphorylation dynamics of the insulin receptor. They have measured a rapid transient overshoot in tyrosine phosphorylation for human adipocytes after a step increase from 0 to 0.1 µM in insulin concentration and have discussed the implication of such an "overshot" on various model structures. Cedersund et. al. [2] have rejected model structures based on the zeros and complex poles of the linearized transfer function, see also Brännmark et. al. [3]. In terms of the Petri net formalism, the model structure requires a certain substructure to produce an overshot behavior. For Sedaghat et al.'s model [4] such a substructure is defined by transition invariant TI 1 . The Petri net approach explains the overshot by the high concentration of phosphorylated receptor, irip * , of the meta-stable quasi-steady state associated with transition invariant TI 1 . Figure S1 shows the percentage of transient phosphorylated IR versus the concentration of insulin. Figure S1. After a step increase in insulin concentration the concentration of phosphorylated IR approach the value irip * of meta-stable steady state (S3). This transient high value of phosphorylated IR drops to the value ν × irip † of meta-stable steady state (S7) due to endocytosis and dephosphorylation of the internalized IR. Plotted is the percentage of transient phosphorylated irip * − ν × irip † versus the concentration of insulin. For 0.1 µM insulin concentration, Sedaghat et al.'s model estimates an "overshoot" at in round numbers 40% .