Fig. 1.

Tug-of-war between drivers and passengers leads to a critical population size. (A) Population size verses time of simulations initiated at various sizes (N⁰ = 500, 1,000, or 2,000). For all simulations presented in this paper, μ = 10⁻⁸, T_d = 1,400, T_p = 10⁷, s_d = 0.1, and s_p = 0.001 (SI Appendix, Table S1), unless specified otherwise. (B) A segment of a trajectory shows periods of rapid growth and gradual decline. New drivers arrive with a frequency f(N), abruptly increasing the population size by an amount ΔN. Passenger accumulation causes a gradual decline with rate v_p. (C) Analytically computed mean velocity of population growth (Upper) and an effective barrier (Lower) as a function of population size N. Velocity is negative below N* and positive above it. (D) The probability of adaptation (cancer) as a function of initial population size N (Left) and a relative initial population size (N/N*; Right) for nine sets of evolutionary parameters. Curves collapse and behave similarly when plotted relative to N*. (E) Same as in D for simulations and theory but for different values of s_d. Higher values of s_d leads to a more gradual transition from nonadaptive to adaptive regime. In our formalism, an increase in s_d results in a larger jump size ΔN and lower potential barrier, allowing more populations to overcome the barrier (Right).