INTRODUCTION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

INSULIN IS AN ESSENTIAL peptide hormone discovered in 1921 that regulates metabolism (26). Interestingly, the ability of insulin to promote glucose uptake into tissues was not demonstrated until 1949 (27). In 1971, specific cell surface insulin receptors were identified (12). The discovery that insulin-stimulated glucose transport involves translocation of glucose transporters (e.g., GLUT4) from an intracellular compartment to the cell surface was made in 1980 (8, 53). Since genes encoding the human insulin receptor and GLUT4 were cloned in 1985 (10, 54) and 1989 (5, 13), respectively, steady progress has been made in identifying components of insulin signal transduction pathways leading from the insulin receptor to translocation of GLUT4 (see Ref. 32 for review).

On binding insulin, the insulin receptor undergoes receptor autophosphorylation and enhanced tyrosine kinase activity. Subsequently, intracellular substrates (e.g., insulin receptor substrate-1, IRS-1) are phosphorylated on tyrosine residues that serve as docking sites for downstream SH2 domain containing proteins, including the p85 regulatory subunit of phosphatidylinositide 3-kinase (PI 3-kinase). The p85 binding to phosphorylated IRS-1 results in activation of the p110 catalytic subunit of PI 3-kinase that catalyzes production of phosphoinositol lipids including phosphatidylinositol 3,4,5-trisphosphates [PI(3,4,5)P₃] that activate the Ser/Thr kinase 3-phosphoinositide-dependent protein kinase (PDK)-1. PDK-1 phosphorylates and activates other downstream kinases, including Akt and protein kinase C (PKC)-, that mediate translocation of GLUT4. PTP1B is a protein tyrosine phosphatase (PTPase) that negatively regulates insulin signaling pathways by dephosphorylating the insulin receptor and IRS-1. Interestingly, IRS-1 and PTP1B upstream from Akt and PKC- have recently been identified as substrates for these downstream kinases, suggesting that feedback mechanisms exist (9, 33, 38, 39). Elements downstream from Akt and PKC- linking insulin signaling pathways with trafficking machinery for GLUT4 are unknown (34). Thus a complete understanding of mechanisms regulating the metabolic actions of insulin has remained elusive.

One reason it has been difficult to comprehend metabolic insulin signaling pathways is that determinants of signal specificity are poorly understood. Many signaling molecules are shared in common among pathways initiated by distinct receptors. Moreover, cross talk and feedback between a multitude of receptor-mediated pathways generate signaling networks rather than linear pathways. Without a theoretical framework, it is difficult to understand how complexities evident from experimental data determine cell behavior. Now that the human genome has been sequenced, it may be possible to generate a vast experimental database for understanding cellular signaling. Alfred Gilman has founded The Alliance for Cellular Signaling (http://www.cellularsignaling.org/) with the goal of integrating relevant experimental data (temporal and spatial relationships of signaling inputs and outputs in a cell) into interacting theoretical models. This comprehensive approach may enable a full understanding of the complexities of cell signaling. In this spirit, we now develop a mathematical model of metabolic insulin signaling pathways that explicitly represents many known insulin signaling components. Our goal is to define a comprehensive model that not only accurately represents known experimental data but will also serve as a useful tool to generate and test hypotheses. This modeling approach may lead to novel insights regarding the molecular mechanisms underlying insulin signal transduction pathways that regulate metabolic actions of insulin.

	MODEL DEVELOPMENT

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

We use our previously validated models of insulin receptor binding kinetics (57), receptor recycling (36), and GLUT4 translocation (35, 37) as subsystems in conjunction with a novel mechanistic representation of postreceptor signaling pathways to generate a complete model with 21 state variables. This complete model is then extended to incorporate feedback pathways, and consequences of feedback are explored. Differential equations derived from the structure of the complete model were solved by use of a fourth order Runge-Kutta numerical integration routine (42), using the WinPP version of XPPAUT (available at http://www.math.pitt.edu/~bard/xpp/xpp.html; see APPENDIX A for complete list of equations, initial conditions, and model parameters; see http://mrb.niddk.nih.gov/sherman for WinPP source files used to run simulations). A sufficiently small step size (0.001 min) was used to ensure accurate numerical integrations for all state variables. First order kinetics were assumed except where noted.

Complete Model without Feedback

Insulin receptor binding subsystem. Our model of insulin receptor binding kinetics (57) (Fig. 1A) was extended here to include additional steps representing insulin receptor autophosphorylation and dephosphorylation (Fig. 1B). On binding the first molecule of insulin, the receptor is rapidly phosphorylated (1), resulting in receptors that may either bind another molecule of insulin or dissociate from the first molecule of insulin. Binding of a second molecule of insulin does not affect the phosphorylation state of the receptor, whereas receptor dephosphorylation occurs when insulin diffuses off of the receptor, leaving a free receptor. In addition, protein tyrosine phosphatases that dephosphorylate the insulin receptor and whose levels can vary under pathological conditions (15) are explicitly represented as a multiplicative factor ([PTP]) that modulates receptor dephosphorylation rate. The differential equations for this subsystem are

(1)

(2)

(3)

(4)

(5)

Definitions for state variables and rate constants are given in legend to Fig. 1, A and B. Note that we do not explicitly include an intermediate state of free receptors that are still phosphorylated because receptor occupancy and phosphorylation are tightly coupled, and we assume that there are virtually no receptors in the unbound phosphorylated state.

View larger version (20K): [in this window] [in a new window]   Fig. 1.   Schematic of insulin receptor binding and life cycle subsystems. A: previously validated model of insulin binding kinetics (57). x1, Free insulin concentration (system input); x2, free receptor concentration; x3, concentration of receptors with 1 molecule of insulin bound; x4, concentration of receptors with 2 molecules of insulin bound; k1 and k1, association and dissociation rate constants, respectively, for the first molecule of insulin to bind the receptor; k2 and k2, association and dissociation rate constants, respectively, for the second molecule of insulin to bind the receptor. B: receptor binding subsystem extended to include receptor autophosphorylation and dephosphorylation. x5, Concentration of once-bound phosphorylated receptors; x4, redefined as concentration of twice-bound phosphorylated receptors; k3, rate constant for receptor autophosphorylation; k3, rate constant for receptor dephosphorylation; [PTP], a multiplicative factor modulating k3 that represents the relative activity of protein tyrosine phosphatases (PTPases) in the cell that dephosphorylate the insulin receptor. C: previously validated model of insulin receptor recycling (36). x6, Concentrations of intracellular receptors; k4, endocytosis rate constant for free receptors; k4, exocytosis rate constant; k4', endocytosis rate constant for bound receptors; k5, zero order rate constant for receptor synthesis; k5, constant for receptor degradation. D: extension of insulin receptor binding and recycling subsystems that includes phosphorylated receptors. x7 and x8, concentration of twice-bound and once-bound intracellular phosphorylated receptors, respectively; k4', exocytosis rate for twice-bound and once-bound intracellular phosphorylated receptors; k6, dephosphorylation rate constant for intracellular receptors that is modulated by the multiplicative factor [PTP].

Insulin receptor recycling subsystem. Our previous model of insulin receptor life cycle explicitly represents synthesis, degradation, exocytosis, and both basal and ligand-induced endocytosis of receptors (36) (Fig. 1C). We now extend this subsystem so that ligand-induced endocytosis is only applied to phosphorylated cell surface receptors (Fig. 1D). Both once- and twice-bound phosphorylated receptors are treated identically with respect to internalization. An additional step representing dephosphorylation of internalized phosphorylated receptors and their incorporation into the intracellular pool is included. State variables representing free surface insulin receptors (x₂) and phosphorylated surface receptors (x₄ and x₅) are shared by both binding and recycling subsystems. Thus differential equations for these coupled subsystems (depicted in Fig. 1D) are

(6)

(7)

(8)

(9)

(10)

(11)

(12)

(13)

Definitions for additional state variables and rate constants are given in legend to Fig. 1, C and D.

Postreceptor signaling subsystem. The postreceptor signaling subsystem developed here (Fig. 2) comprises elements in the metabolic insulin signaling pathway that are well established (32). It is assumed that this is a closed subsystem so that synthesis and degradation of signaling molecules are not explicitly represented. The concentration of phosphorylated surface insulin receptors is the input to this subsystem. Activated insulin receptors phosphorylate IRS-1, which then binds and activates PI 3-kinase. We modeled the dependence of IRS-1 phosphorylation on phosphorylated surface receptors as a linear function. That is, the rate constant for IRS-1 phosphorylation, k₇, is modulated by the fraction of phosphorylated surface receptors (x₄ + x₅)/(IR_p), where IR_p is the concentration of phosphorylated surface receptors achieved after maximal insulin stimulation. The association of phosphorylated IRS-1 with PI 3-kinase is assumed to occur with a stoichiometry of 1:1. Differential equations governing phosphorylation of IRS-1 and subsequent formation of phosphorylated IRS-1/activated PI 3-kinase complex are

(14)

(15)

(16)

(17)

Activated PI 3-kinase converts the substrate phosphatidylinositol 4,5-bisphosphate [PI(4,5)P₂] to the product PI(3,4,5)P₃. This is modeled as a linear function so that k₉, the rate constant for generation of PI(3,4,5)P₃, is dependent on x₁₂, the amount of activated PI 3-kinase (see APPENDIX B for detailed derivation). 5'-Lipid phosphatases such as SHIP2 convert PI(3,4,5)P₃ to phosphatidylinositol 3,4-bisphosphate PI(3,4)P₂] (6), whereas 3'-lipid phosphatases such as PTEN convert PI(3,4,5)P₃ to PI(4,5)P₂ (45). The differential equations describing interconversion between these phosphatidylinositides are

(18)

(19)

(20)

As with [PTP], the lipid phosphatase factors [SHIP] and [PTEN] correspond to the relative phosphatase activity in the cell and are assigned a value of 1 under normal physiological conditions.

View larger version (28K): [in this window] [in a new window]   Fig. 2.   Schematic of postreceptor signaling subsystem. x9, Concentration of unphosphorylated insulin receptor substrate (IRS)-1; x10, concentration of tyrosine-phosphorylated IRS-1; x11, concentration of free phosphatidylinositol 3-kinase (PI 3-kinase; PI3-K); x12, concentration of the phosphorylated IRS-1/activated PI 3-kinase complex; x13, x14, and x15, percentages of various phosphoinositol lipids in the cell; x16 and x17, percentages of unphosphorylated and phosphorylated Akt in the cell, respectively; x18 and x19, percentages of unphosphorylated and phosphorylated protein kinase C (PKC)- in the cell, respectively. The rate constants k7 to k12 and k7 to k12 govern the conversion between state variables as indicated. [PTP] is a multiplicative factor modulating k7 that represents the relative activity of PTPases in the cell that dephosphorylate IRS-1. [PTEN] and [SHIP] are multiplicative factors modulating k9 and k10, respectively, that represent the relative activity of these lipid phosphatases in the cell. Arrows with solid lines indicate first-order reactions. Arrows with dashed lines indicate reactions where the value of a state variable influences the value of the rate constant.

(21)

(22)

(23)

(24)

(25)

GLUT4 translocation subsystem. The final subsystem is our previous model of GLUT4 translocation (35, 37) (Fig. 3). Under basal conditions, GLUT4 recycles between an intracellular compartment and the cell surface. The differential equations for this subsystem are

(26)

(27)

Definitions for additional state variables and rate constants in the GLUT4 translocation subsystem are given in the legend to Fig. 3. On insulin stimulation, there may be a separate pool of intracellular GLUT4 recruited to the cell surface (17, 18, 40). To represent this aspect of GLUT4 trafficking, the insulin-stimulated exocytosis rate (k_13') is increased to its maximum value as a linear function of the metabolic effect produced by phosphorylated Akt and PKC-. By assuming that the basal equilibrium distribution of 4% cell surface GLUT4 and 96% GLUT4 in the intracellular pool transitions on maximal insulin stimulation to a new steady state of 40% cell surface GLUT4 and 60% intracellular GLUT4, the equations governing changes in k₁₃ and k_13' are

(28)

(29)

Thus changes in k_13' are linearly dependent on the output of the signaling subsystem (Effect).

View larger version (11K): [in this window] [in a new window]   Fig. 3.   Schematic of GLUT4 translocation subsystem (35, 37). x20, Percentage of intracellular GLUT4; x21, percentage of GLUT4 at the cell surface; k13, rate constant for GLUT4 internalization; k13, rate constant for translocation of GLUT4 to the cell surface under basal conditions; k13', rate constant for translocation of GLUT4; k14 (zero order) and k14, rate constants for GLUT4 synthesis and degradation, respectively. The "metabolic Effect" from postreceptor signaling subsystem increases k13'.

View larger version (20K): [in this window] [in a new window]   Fig. 4.   Complete model of metabolic insulin signaling pathways obtained by coupling subsystem models for insulin receptor binding, receptor recycling, postreceptor signaling, and GLUT4 translocation.

Complete Model with Feedback

View larger version (19K): [in this window] [in a new window]   Fig. 5.   Complete model of metabolic insulin signaling pathways with feedback. Identical to model shown in Fig. 4 except that new elements comprising positive and negative feedback pathways are indicated by dotted lines. PKC- serine phosphorylates IRS-1 to create a negative feedback pathway, and Akt phosphorylates PTP1B to create a positive feedback pathway.

We also incorporated a negative feedback loop in which serine phosphorylation of IRS-1 by PKC- impairs formation of the phosphorylated IRS-1/activated PI 3-kinase complex (39). To represent this, we assumed that serine phosphorylation of IRS-1 by activated PKC- creates an IRS-1 species unable to associate with and activate PI 3-kinase (represented by the state variable x_10a). The formation of x_10a tends to decrease the level of activated PI 3-kinase in response to insulin stimulation (when compared with a system without negative feedback). Additional differential equations describing interconversion between unphosphorylated IRS-1 (x₉) and serine-phosphorylated IRS-1 (x_10a) are

(30)

(31)

where Eq. 30 is an updated version of Eq. 14, k_7' is the rate constant for serine phosphorylation of IRS-1 by PKC-, and k_7' is the rate constant for serine dephosphorylation. We include a multiplicative factor, [PKC], that modulates k_7' to model the ability of phosphorylated, activated PKC- (x₁₉) to generate serine-phosphorylated IRS-1. [PKC] is defined as a standard Hill equation that is commonly used to represent enzyme kinetics, where [PKC] = V_maxx₁₉(t  )ⁿ/[K + x₁₉ (t  )ⁿ], where V_max is maximal velocity and K_d is dissociation constant. This explicitly incorporates a time lag () into the negative feedback loop. The parameters used in this equation are listed in APPENDIX A.

	RESULTS

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Model Simulations without Feedback

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View larger version (29K): [in this window] [in a new window]   Fig. 6.   Model simulations without feedback. Time courses for unbound receptors (A), once- and twice-bound phosphorylated surface receptors (B), total phosphorylated surface receptors (C), and unphosphorylated and tyrosine-phosphorylated IRS-1 (D) after a step input of 107 M insulin for 15 min.

View larger version (27K): [in this window] [in a new window]   Fig. 7.   Model simulations without feedback. Time courses for activated PI 3-kinase (A), levels of phosphatidylinositol 3,4,5-trisphosphate [PI(3,4,5)P3] and phosphatidylinositol 3,4-bisphosphate PI(3,4)P2] (B), activated PKC- (C), and cell surface GLUT4 (D) after a step input of 107 M insulin for 15 min.

Maximal formation of the phosphorylated IRS-1/activated PI 3-kinase complex in response to 10⁷ M insulin occurred within ~1.5 min (Fig. 7A). This is in good agreement with published data showing that PI3-kinase and tyrosine-phosphorylated IRS-1 molecules associate quickly after insulin stimulation (2, 14). The time course for disappearance of activated PI 3-kinase after the removal of insulin followed the time course for dephosphorylation of IRS-1. PI 3-kinase activated in response to insulin stimulation catalyzed the conversion of PI(4,5)P₂ to PI(3,4,5)P₃, which in turn drove the formation of PI(3,4)P₂ (Fig. 7B). The level of PI(3,4,5)P₃ increased from 0.31 to 3.1% of the total lipid population, and the level of PI(3,4)P₂ increased from 0.29 to 2.9% of the total lipid population as it equilibrated with PI(3,4,5)P₃. On removal of insulin, the phosphatidylinositides returned to basal levels (time to half-maximal levels was ~11 min). The levels of PI(3,4,5)P₃ controlled the formation of phosphorylated, activated PKC- (Fig. 7C). Maximal PKC- activation occurred within 3 min of insulin stimulation. After insulin was removed, the level of activated PKC- declined back to basal levels (time to half-maximal levels was ~11 min). The time course for phosphorylated, activated Akt was identical to that for PKC- (data not shown). Insulin-stimulated activation of PKC- and Akt mediates increased exocytosis of GLUT4 so that 40% of total cellular GLUT4 was at the cell surface and 60% was intracellular after maximal insulin stimulation (Fig. 7D). Our simulations of insulin-stimulated GLUT4 recruitment occurred with a half-time of ~3 min, matching published experimental results (18, 20). When insulin was removed, surface and intracellular GLUT4 levels returned to their basal values (time to half-maximal levels was ~16 min). Thus the overall response of our complete model without feedback to an acute insulin input is in good agreement with both a variety of published experimental data and previously validated subsystem models.

Model Simulations with Feedback

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View larger version (34K): [in this window] [in a new window]   Fig. 8.   Model simulations with feedback. Time courses for unbound receptors (A), once- and twice-bound phosphorylated surface receptors (B), total phosphorylated surface receptors (C), and unphosphorylated, serine-phosphorylated, and tyrosine-phosphorylated IRS-1 (D) after a step input of 107 M insulin for 15 min.

View larger version (27K): [in this window] [in a new window]   Fig. 9.   Model simulations with feedback. Time courses for activated PI 3-kinase (A), levels of PI(3,4,5)P3 and PI(3,4)P2 (B), activated PKC- (C), and cell surface GLUT4 (D) after a step input of 107 M insulin for 15 min.

The transient oscillatory behavior observed for tyrosine-phosphorylated IRS-1 in response to insulin stimulation was also observed for activated PI 3-kinase (Fig. 9A). Activated PI 3-kinase transiently peaked at 5.6 fM within 1.8 min. This was followed by a rapid undershoot and then equilibration at ~1.9 fM by 10 min. On removal of insulin, the concentration of activated PI 3-kinase returned to basal levels (time to half-maximal levels was ~17 min). The time courses for PI(3,4,5)P₃, PI(3,4)P₂, PKC-, and Akt displayed a qualitatively similar dynamic (Fig. 9, B and C). That is, after 1 min of insulin stimulation, the level of PI(3,4,5)P₃ increased from 0.31% of the total lipid population to a peak of ~6.1% followed by an undershoot before equilibration at ~2.5%. The level of PI(3,4)P₂ increased from 0.21% to a peak of ~5.6% before equilibrating at a steady-state level of ~2.4%. On removal of insulin, both PI(3,4,5)P₃ and PI(3,4)P₂ returned to their basal levels (time to half-maximal levels was ~17 min). In response to insulin, the percentage of activated PKC- transiently peaked at ~17.1% after ~1.5 min followed by an undershoot and equilibration at ~7.4% by 11 min. After insulin was removed, the level of activated PKC- declined to basal levels with a time to half-maximal levels of ~17 min. Because we modeled the behavior of Akt identically to PKC-, the time course for phosphorylated, activated Akt mirrored that for PKC- (data not shown). With respect to translocation of GLUT4, the overall shapes of the time courses for cell surface GLUT4 with and without feedback were similar. However, with inclusion of positive and negative feedback loops, described above, the half-time for translocation of GLUT4 to the cell surface in response to insulin was slightly shorter than that observed in the model without feedback (~2.5 min), whereas the time for return to basal levels after insulin removal was longer (time to half-maximal level was ~18 min). Thus the inclusion of positive and negative feedback loops into our model of metabolic insulin signaling generates predictions regarding the dynamics of various signaling components that may be experimentally testable.

Insulin Dose-Response Characteristics

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View larger version (26K): [in this window] [in a new window]   Fig. 10.   Experimentally generated insulin dose-response curves adapted from Ref. 47 were compared with dose-response curves generated from our model with and without feedback for bound receptors (A), phosphorylated receptors at the cell surface (B), PI 3-kinase activity (C), and glucose uptake (D; assumed to be directly proportional to GLUT4 levels at the cell surface).

For activated PI 3-kinase, model simulations without feedback generated an insulin dose-response curve with an ED₅₀ of 0.83 nM. Model simulations with feedback showed a slightly greater sensitivity with an ED₅₀ of 1.43 nM. These simulation results seem reasonable, since downstream components should have greater insulin sensitivity than proximal events if one assumes that signal amplification occurs for downstream elements. Interestingly, the experimentally determined ED₅₀ for activated PI 3-kinase reported by Stagsted et al. (47) is 8 nM. With respect to insulin-stimulated translocation of GLUT4, model simulations without feedback generated a dose-response curve with an ED₅₀ of 0.53 nM, whereas model simulations with feedback generated a dose-response curve with a slightly smaller ED₅₀ of 0.19 nM. In rat adipose cells, the ED₅₀ = 0.17 nM for insulin-stimulated glucose uptake (47). Again, our simulation results are not only consistent with kinetic expectations but also closely match experimental results, including observations that maximal glucose uptake occurs with partial insulin receptor occupancy (47). A general result from these model simulations is that insulin sensitivity increases for components farther downstream in the signaling pathway. In addition, the presence of feedback in our model resulted in slightly increased sensitivity for each signaling component examined compared with simulations without feedback.

Biphasic Activation of PKC-

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View larger version (22K): [in this window] [in a new window]   Fig. 11.   Biphasic activation of PKC- in response to insulin stimulation. Published data on insulin-stimulated activation of PKC- (adapted from Ref. 48, Fig. 2A; ) were compared with simulation results from our model with () and without () feedback using a step input of 108 M insulin for 15 min.

Effects of Increased Levels of PTPases on Insulin-Stimulated Translocation of GLUT4

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View larger version (17K): [in this window] [in a new window]   Fig. 12.   Effects of increasing PTP on time courses for insulin-stimulated translocation of GLUT4. Simulations for cell surface GLUT4 in response to a 15-min step input of 107 M insulin (A), 109 M insulin (B), and 1010 M insulin (C) are shown for [PTP] = 1.0 (open symbols) and [PTP] = 1.5 (solid symbols) using our model without feedback.

	DISCUSSION

TOP ABSTRACT INTRODUCTION MODEL DEVELOPMENT RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

Since the discovery of insulin over 80 years ago, tremendous progress has been made in elucidating the molecular mechanisms of insulin action. However, recent investigations of insulin signaling reveal biological complexities that are still not fully understood. For example, the determinants of specificity for metabolic insulin signaling pathways are largely unknown. Rapid progress in the field of signal transduction and genomics has inspired the foundation of groups such as the National Resource for Cell Analysis and Modeling (NRCAM; pioneers in the Virtual Cell project; see Ref. 41) and The Alliance for Cellular Signaling. These groups strongly argue that a theoretical approach with a comprehensive database is absolutely necessary for a full understanding of cellular signaling behavior.

Model Development

One potential pitfall in developing such a complicated model is that the number of arbitrary free parameter choices may decrease the predictive power of the model. To address this issue, we incorporated previous models as subsystems in our complete model to significantly reduce the number of arbitrary elements. Rate constants and parameters for these subsystems had previously been independently obtained and validated. The majority of remaining model parameters and rate constants characterizing newly modeled steps of the metabolic insulin signaling pathway were based on experimental data in the literature. We also used boundary value conditions to derive fixed relationships among various rate constants and model parameters to further decrease the degrees of freedom in the structure of the model. Finally, as a simplifying measure, we represented the numerous kinetic reactions in our model as first-order reactions coupled by shared common elements. We have attempted to thoroughly validate our complete model by demonstrating that the behavior of each individual state variable in response to a step insulin input closely matches experimental data from a variety of independent sources. As further evidence of the validity of our complete model, our system as a whole generated properties that have been observed experimentally, including the presence of "spare receptors" (e.g., maximal activation of GLUT4 translocation with submaximal insulin receptor occupancy), signal amplification, and increased insulin sensitivity for downstream components in the signaling pathway. Because three of the four model subsystems have previously been validated and the behavior of the overall model agrees closely with published experimental data, we conclude that our postreceptor signaling subsystem is reasonable and robust. Moreover, the rate constants chosen for the signaling subsystem were based on data in the literature. Although a more complete exploration of the postreceptor signaling subsystem is of interest, this is beyond the scope of the current study.

In some cases where mechanisms regulating interactions between signaling elements are not fully understood, we modeled these interactions as linear relationships. The rate of IRS-1 phosphorylation in response to activated insulin receptors, the rate of PI(3,4,5)P₃ generation in response to activated PI 3-kinase, the rate of PKC- and Akt phosphorylation in response to increased levels of PI(3,4,5)P₃, and the rate of exocytosis for GLUT4 in response to phosphorylated PKC- and Akt are all modeled as simple linear functions. In the case of the negative feedback loop where PKC- phosphorylates IRS-1 on serine residues, we modulated the rate constant for serine phosphorylation of IRS-1 using a standard Hill equation to incorporate a reasonable time lag. Importantly, we observed a good overall match between experimental data and model simulations for both insulin dose-response curves and time courses. Thus our coupling assumptions seem reasonable. Nevertheless, these points in the model represent areas that could be further refined in the future when a greater understanding of the molecular mechanisms involved is achieved. Indeed, by modeling more complicated coupling mechanisms, specific simulation results may give rise to experimentally testable predictions. This interplay between theoretical predictions and experimental results may yield important insights into the molecular mechanisms of insulin action.

Model Simulations without Feedback

In our simulations we observed a time lag between the insulin input and subsequent steps in insulin signaling that increased as the signal propagated downstream. This lag was present for simulations of both insulin stimulation and removal. Because the kinetics governing most state variables were modeled as first-order events, recovery to basal conditions of each component would be expected to appear as a concave-up, exponential decay curve. Interestingly, in our simulations of insulin removal (after 15-min insulin stimulation), we observed the presence of a concave-down region immediately before the concave-up exponential decay that became more pronounced for distal components and is clearly evident in the simulations of GLUT4 translocation (Fig. 7D). This qualitative behavior is consistent with the presence of a signaling cascade that controls insulin-stimulated translocation of GLUT4 and has been observed experimentally in rat adipose cells (20).

To further validate our complete model without feedback, we compared published experimental data with simulations of insulin dose-response curves for key state variables (Fig. 10). The usefulness of these comparisons was substantially strengthened by the fact that data on insulin receptor binding, receptor autophosphorylation, PI 3-kinase activation, and glucose transport were obtained from a single experimental preparation of rat adipose cells (47). Qualitatively, the sigmoidal shape of dose-response curves generated by our model simulations (when plotted as a semi-log graph) is consistent with the hyperbolic response characteristic of most receptor-mediated biological events. In addition, we observed increased insulin sensitivity for downstream components of the insulin signaling pathway. That is, ED₅₀ = 3.5 nM for receptor binding, ED₅₀ = 3.5 nM for receptor autophosphorylation, ED₅₀ = 0.83 nM for PI 3-kinase activation, and ED₅₀ = 0.53 nM for GLUT4 translocation. This increased sensitivity of downstream components, consistent with the presence of a signal amplification cascade, is a well-described characteristic for biological actions of insulin. For example, only a fraction of insulin receptors need to be occupied by insulin for maximal glucose uptake to occur in adipose cells (20, 47). Stagsted et al. (47) reported an ED₅₀ of 8 nM for insulin-stimulated PI 3-kinase activation. However, close inspection of their data suggests that the actual value may be closer to 4 nM. Nevertheless, this ED₅₀ is still somewhat larger than the ED₅₀ of 0.83 nM derived from our simulations. However, the measurement of PI 3-kinase activity in anti-phosphotyrosine immunoprecipitates derived from whole cell lysates is difficult to perform in a quantitative manner. Thus it is possible that the small discrepancy between our simulation results and experimental data with respect to activation of PI3-kinase may be explained, in part, by imprecision introduced by experimental variability. Of note, the shape and ED₅₀ of insulin dose-response curves for insulin receptor binding, receptor autophosphorylation, and GLUT4 translocation almost exactly match the corresponding experimentally determined dose-response curves (Fig. 10). Thus the insulin sensitivity of key components of our model is realistic. Moreover, given the fact that our model structure, rate constants, and parameters were derived from a large mixture of many different experimental systems, this remarkable fit to experimental data from a single system across several key state variables suggests that the overall structure of our model is quite robust.

Model Simulations with Feedback

We incorporated a negative feedback loop into our model by modeling the ability of PKC- to phosphorylate IRS-1 on serine residues and impair its ability to bind and activate PI 3-kinase (39). Time courses for insulin receptor binding and phosphorylation generated by models with and without feedback were similar. However, the overall dynamics of postreceptor signaling elements in our model were quite different with inclusion of feedback. In response to insulin stimulation, we observed a transient damped oscillatory behavior before equilibrium was reached in all elements of the postreceptor signaling subsystem. This oscillatory behavior was a direct result of negative feedback by PKC-. Immediately after insulin stimulation, effects of PKC- on upstream components are not apparent because of the time lag