Fig. 1: Mathematical model of drug-induced tolerance.

a In the mathematical model, cells transition between two states, a drug-sensitive (type-0) state and a drug-tolerant (type-1) state. Sensitive cells divide at rate b₀, die at rate d₀ and transition to tolerance at rate μ. Tolerant cells divide at rate b₁, die at rate d₁ and transition to sensitivity at rate ν. The sensitive cell death rate d₀ and the transition rates μ and ν depend on the drug dose c. b The sensitive cell death rate follows a Michaelis-Menten equation of the form \({d}_{0}(c)={d}_{0}+({d}_{\max }-{d}_{0})c/(c+1)\), where d₀ is the death rate in the absence of drug and \({d}_{\max }\) is the saturation death rate under an arbitrarily large drug dose. The dose c is normalized to the EC₅₀ dose for d₀, meaning that the drug has half the maximal effect at dose c = 1. c The transition rate μ(c) from drug-sensitivity to drug-tolerance is assumed either a linearly increasing function of c or to be uniformly elevated in the presence of drug. These two forms of drug-induced tolerance were observed in a recent study by Russo et al.¹⁷ and they also emerge as limiting cases of more general Michaelis-Menten dynamics. d The transition rate ν(c) from drug-tolerance to drug-sensitivity is similarly assumed either a linearly decreasing function of c or uniformly inhibited in the presence of drug.