The roles of APC and Axin derived from experimental and theoretical analysis of the Wnt pathway

Abstract

Wnt signaling plays an important role in both oncogenesis and development. Activation of the Wnt pathway results in stabilization of the transcriptional coactivator beta-catenin. Recent studies have demonstrated that axin, which coordinates beta-catenin degradation, is itself degraded. Although the key molecules required for transducing a Wnt signal have been identified, a quantitative understanding of this pathway has been lacking. We have developed a mathematical model for the canonical Wnt pathway that describes the interactions among the core components: Wnt, Frizzled, Dishevelled, GSK3beta, APC, axin, beta-catenin, and TCF. Using a system of differential equations, the model incorporates the kinetics of protein-protein interactions, protein synthesis/degradation, and phosphorylation/dephosphorylation. We initially defined a reference state of kinetic, thermodynamic, and flux data from experiments using Xenopus extracts. Predictions based on the analysis of the reference state were used iteratively to develop a more refined model from which we analyzed the effects of prolonged and transient Wnt stimulation on beta-catenin and axin turnover. We predict several unusual features of the Wnt pathway, some of which we tested experimentally. An insight from our model, which we confirmed experimentally, is that the two scaffold proteins axin and APC promote the formation of degradation complexes in very different ways. We can also explain the importance of axin degradation in amplifying and sharpening the Wnt signal, and we show that the dependence of axin degradation on APC is an essential part of an unappreciated regulatory loop that prevents the accumulation of beta-catenin at decreased APC concentrations. By applying control analysis to our mathematical model, we demonstrate the modular design, sensitivity, and robustness of the Wnt pathway and derive an explicit expression for tumor suppression and oncogenicity.

Figure 1. Reaction Scheme for Wnt Signaling

The reaction steps of the Wnt pathway are numbered 1 to 19. Protein complexes are denoted by the names of their components, separated by a slash and enclosed in brackets. Phosphorylated components are marked by an asterisk. Single-headed solid arrows characterize reactions taking place only in the indicated direction. Double-headed arrows denote binding equilibria. Blue arrows mark reactions that have only been taken into account when studying the effect of high axin concentrations. Broken arrows represent activation of Dsh by the Wnt ligand (step 1), Dsh-mediated initiation of the release of GSK3β from the destruction complex (step 3), and APC-mediated degradation of axin (step 15). The broken arrows indicate that the components mediate but do not participate stoichiometrically in the reaction scheme. The irreversible reactions 2, 4, 5, 9–11, and 13 are unimolecular, and reactions 6, 7, 8, 16, and 17 are reversible binding steps. The individual reactions and their role in the Wnt pathway are explained in the text.

Figure 2. Kinetics of β-Catenin Degradation: Simulation and Experimental Results

(A) Simulated timecourses of β-catenin degradation. The straight line for t < 0 corresponds to the reference state of β-catenin using the parameters given in the legends of Table 1 and 2. In vitro conditions are simulated by switching off synthesis of β-catenin and axin (ν ₁₂ = 0, ν ₁₄ = 0 for t ≥ 0). Curve a: reference case (no addition of further compounds); curve b: addition of 0.2 nM axin; curve c: addition of 1 μM activated Dsh (deactivation of Dsh was neglected, k ₂ = 0); curve d: inhibition of GSK3β (simulated by setting k ₄ = 0, k ₉ = 0); curve e: addition of 1μM TCF. Addition of compounds (axin, Dsh, TCF) and inhibition of GSK3β was performed at t = 0. (B) Experimental timecourse of β-catenin degradation in Xenopus egg extracts in the presence of buffer (curve a′), axin (curve b′: 10 nM), Dsh (curve c′: 1 μM), Li⁺ (curve d′: 25 mM), or Tcf3 (curve e′: 1 μM).

Figure 3. Preincubation of Dsh in Xenopus Egg Extracts Abolishes the Lag in Dsh Activity

Labeled β-catenin was incubated in Xenopus extracts on ice 30 min prior to (B) or 30 min after (A) the extract had been preincubated with 1 μM Dsh. No degradation of the labeled β-catenin was detected while the reactions were on ice. The reactions were started by shifting to 20°C.

Figure 4. The Effect of Dsh versus Axin or GSK3β on the Half-Life of β-Catenin in Xenopus Extracts

(A and B) Predicted effects of Dsh, axin, and GSK3β on the half-life of β-catenin degradation. The half-lives are calculated from simulated degradation curves. Data are plotted as function of added Dsh (logarithmic scale) for various concentrations of axin (A) and GSK3β (B). (C and D) Measured effects of Dsh, axin, and GSK3β on the half-life of β-catenin degradation. Stimulation of β-catenin degradation by axin occurs throughout the range of Dsh concentrations tested. (C) Axin increases the rate of β-catenin degradation even in the absence of added Dsh. (D) Stimulation of β-catenin degradation by GSK3β is detected only at high concentrations of Dsh. No effects of GSK3β on β-catenin degradation can be detected at less than 30 nM added Dsh. There is a disparity between the concentrations of axin in the experimental and theoretical curves. We assume that this is most likely due to the specific activity of the expressed axin protein.

Figure 5. Effect of the Regulatory Loop for Axin Degradation

The case “with regulatory loop” takes into account that axin degradation is APC-dependent (black curves). Alternatively, the case without this regulatory loop is considered (red curves). For the regulatory loop, the rate law (5) is used assuming that in the reference state the APC activation is half of its maximum (K_M = 98.0 nM). The value of k′₁₅ was chosen such that in the reference state both cases, with and without regulatory loop, yield the same degradation rate of axin (k′₁₅ = 0.33 min⁻¹).

Figure 6. Timecourse of β-Catenin and Axin Concentrations Following a Transient Wnt Stimulation

Transient activation of the pathway is modeled assuming a Wnt stimulus that decays exponentially (Equation [6] with τ_W = 1/λ = 20 min) starting at t ₀ = 0. The straight line for t < 0 corresponds to the steady state before pathway stimulation. The curves are obtained by numerical integration of the differential equation system (see Dataset S1). The various curves for β-catenin and for axin differ in the turnover rate of axin determined by the parameters ν ₁₄ and k ₁₅ (curves a: reference values of these parameters; curves b: increase by a factor of 5; curves c: reduction by a factor of 5). All other parameters are given in the legend of Table 2.

Figure 7. Effects of Increasing Axin Concentration on β-Catenin Degradation

(A) Effect of axin concentration on β-catenin half-life. Curve a: reference case (K ₁₈, K ₁₉ > 1 nM, ordered mechanism); curve b: K ₁₈ = 1 nM, K ₁₉ > 1 nM; curve c: K ₁₈ > 1 nM, K ₁₉ = 1 nM; curve d: K ₁₈ = 1 nM. (B) High concentration of axin inhibits β-catenin degradation in Xenopus egg extracts. Labeled β-catenin was incubated in Xenopus extracts in the absence (0 nM) or presence of moderate (10 nM) and high (300 nM) concentrations of axin. Moderate concentrations of axin greatly accelerate, whereas high concentrations inhibit β-catenin degradation.

Figure 8. Effects of APC Concentrations on β-Catenin Degradation

Effect of APC concentration on β-catenin half-life assuming an ordered (curve a) or nonordered mechanism (curve b: K ₁₇ = 1,200 nM), respectively.

Figure 9. Effects of the Alternative β-Catenin Degradation Pathway and of Axin Degradation at Low Concentrations of APC

(A) The alternative β-catenin degradation pathway (axin independent) can have profound effects on β-catenin levels at low APC concentrations. Variations of β-catenin and axin resulting from changes in APC concentration were calculated from the standard stimulated state. Relative variations were plotted since variation in the share of alternative degradation (1%, 5%, and 10%) results in changes of the standard stimulated state (all parameters are constant). β-Catenin and axin levels for varying contributions of the alternative degradation pathway are as follows: 1.5%, β-catenin 178 nM, axin 0.00728 nM; 5%, β-catenin 151 nM, axin 0.00679 nM; 10%, β-catenin 125 nM, axin 0.00629 nM. (B) Inhibition of axin degradation reduces β-catenin concentration after loss of APC. Plotted is the concentration of a potential proteasome inhibitor I (scaled to its inhibition constant, K _I) necessary to reduce β-catenin concentration to its original level, depending on the concentration of APC.