When the mathematical model was used to simulate Wnt signaling effects on BC-SCs, it predicted that low Dkk1 concentrations should not affect, or somewhat increase SC proliferation, whereas above a threshold of Dkk1 concentration the BC-SC number would decrease, depending on the dose of the exogenously applied Dkk1 (Fig 2). The gene discussed is DKK1; the disease is breast cancer.