Fig 2. Selection for stepwise de-differentiation when the effective rate of self renewal is highest for stem cells.

Illustration of the selection gradient (comparative fitness) of the S mutant cell population Δλ_S as a function of division rates and symmetric division probabilities, provided that the stem cell compartment has the largest effective self-renewal rate, i.e. λ₀ = r₁(p₁ − q₁). In both panels, colored lines represent analytical approximations from Eq (8) by using the eigenvalue perturbation method and symbols represent exact numerical solutions, which agree very well with each other. The common parameters are n = 4, κ = 0.1, ρ = 0.001, d = 0.05, r₁ = 0.99, r₃ = 0.3. (a) Expanding case (λ₀ > 0). De-differentiation provides a fitness advantage for all values of p₁ and r₂. Here p₂ = 0.55, p₃ = 0.6 and the range of p₁ (0.55 < p₁ < 1.0) ensures that r₁(p₁ − q₁) is the largest eigenvalue. (b) Homeostasis case (λ₀ = 0). De-differentiation also provides a fitness advantage for all values of p₂ and r₂. Here p₁ = 0.5, p₃ = 0 and the range of p₂ (0 < p₁ < 0.3) ensures that λ₀ = 0 is the largest eigenvalue.