(A) Wnt stimulation induces accumulation of β-catenin. (B) Together with the Tcf/Lef transcription factor, β-catenin activates transcription of specific target genes. (C) We found that the absolute level of β-catenin is sensitive to non-specific variation that may naturally occur in cells. (D) Interestingly, the fold-change in β-catenin induced by Wnt stimulation is buffered against variation in parameters. (E) Together with functional data from Xenopus embryos, we propose that fold-changes in β-catenin, and not absolute levels of β-catenin, is the output of Wnt signaling.

(A) A kinetic model of the Wnt pathway (Lee et al., 2003).
(B-C) Parameter sensitivity analysis. The parameters measured in the Xenopus extracts are used as the “unperturbed setpoint” (the first column). From this setpoint, each parameter was increased by 5-fold, one at a time (corresponds to each column). For each parameter set, we simulated the accumulation of β-catenin induced by a step Wnt stimulation, and recorded the level of β-catenin after Wnt stimulation (B) and fold-change in β-catenin induced by Wnt (C).
(D-E) For example, when Axin1 synthesis rate is increased by 5-fold (D) and the system is then stimulated with Wnt, the Wnt-induced level of β-catenin is 30 nM and the Wnt-induced fold-change in β-catenin is 6-fold. When β-catenin synthesis rate is increased by 5-fold (E), and the system is then stimulated with Wnt, the Wnt-induced level of β-catenin is 1200 nM and the Wnt-induced fold-change in β-catenin is 9-fold. Please see Figure S1 for a more complete sensitivity analysis.

(A-B) Analytical solutions for the level of β-catenin after Wnt stimulation (A) and the Wnt-induced fold-change in β-catenin (B). These surfaces were computed analytically using equations 29-30 derived in the supplement. The black dot denotes the Xenopus parameters (α = 66, γ= 1). (C) The light blue shading indicates the region where the fold-change in β-catenin varies by less than ±10%. Suppose the cells reside within this region (point a), and we gradually inhibit degradation (α), thereby moving from point a to b to c. At each point, we stimulate the cells with Wnt, shown in the right panel. The absolute levels of β-catenin vary immediately, but the fold-change in β-catenin is identical for points a and b, which reside within the blue-shaded region.

In human RKO cells, most β-catenin is cytoplasmic due to absence of cadherin (Breen et al., 1993; Giannini et al., 2000). In each experiment, 2-3 biological replicates were examined, each analyzed using 2-4 dot blots. SD = standard deviation.
(A) A typical Wnt stimulation in RKO cells (i.e., incubation with Wnt3A-conditioned media). Level of β-catenin was quantified using dot blot.
(B) Redrawn from Figure 3B. This figure predicts how the Wnt-induced fold-change in β-catenin would respond to various perturbations, depicted by the arrows.
(C) Definition of the quantities measured in the experiments.
(D) Cells overexpressing Axin1 were stimulated with Wnt. Three independent experiments were performed. 95%CI = 1.96*SD.
(E-F) Cells were pre-treated with different doses of lithium (3h), and then stimulated with Wnt in the presence of lithium. Figure E shows representative profiles of Wnt-induced β-catenin accumulation in the control and treated cells. Figure F shows results from 4 independent experiments. Please see Figure S4 for a detailed experimental protocol, and Figure S6 for similar results using BIO, another GSK3β inhibitor.
(G) Cells overexpressing β-catenin were stimulated with Wnt. Two independent experiments were performed. 95%CI = 1.96*SD.
(H) Cells overexpressing β-catenin were pre-treated with different doses of lithium (3h), and then stimulated with Wnt in the presence of lithium. Three independent experiments were performed.

(A) Model prediction. To facilitate comparison with experimental data, we plot the Wnt-induced fold-change in β-catenin against the basal level of β-catenin. (Please also see Figure S8 to help visualize this.) From the Xenopus setpoint, perturbing degradation (α, green) will vary β-catenin level without initially affecting the Wnt-induced fold-change in β-catenin. However, increasing synthesis (γ, red) will increase both β-catenin level as well as the Wnt-induced fold-change.
(B) RNA was injected to all four cells at 4-cell stage. Lithium treatment was performed at 48-cell stage for 5 min. Dorsoanterior phenotype was scored using the DAI index (Kao and Elinson, 1988). Cadherin-free pool of β-catenin was quantified using western blot on stage 8, around the onset of target gene expression.
(C) Lithium treatment.
(D) Plotting median DAI versus measured level of β-catenin for lithium treatment, GBP RNA injection, and Axin1 RNA injection.
(E and F) β-catenin injection. (F) Plotting median DAI versus measured level of β-catenin for β-catenin RNA injection.
Note: Phenotypic scoring, β-catenin quantitation, and RT-PCR (in Figure 6) were performed in sibling embryos. SD of β-catenin is from two to four western blots. For more raw data and controls, see Table S1 . Sixty percent of the DAI 7/8 embryos developed as Janus twins (Kao and Elinson, 1988). Image in (B) is reprinted from Kao and Elinson, 1988.

(A-B) Quantitative RT-PCR analysis of two direct target genes of the Wnt pathway, siamois and Xnr3, in embryos treated with lithium (A) and injected with β-catenin RNA (B). RT-PCR was performed at stage 10; SD = 2-3 measurements. Cadherin-free pool of β-catenin was quantified at stage 8-9 using quantitative Western blot; SD = 2-4 blots.
(C) Embryos were injected with TopFlash and RL-TK constructs at 4-cells stage, and then treated with lithium. Luciferase activity was assayed at stage 10.
(D) RKO cells stably expressing TopFlash construct was treated with lithium. β-catenin and luciferase profiles were measured in the same experiment.