Fig. 1: (g−)Hitchhikers’ frequency depends on driver’s effect.

We consider a simple population of cancer cells that grows exponentially N(t) = e^rt; for simplicity, we assign one mutation per cell division. At the time of biopsy T, the frequency of a mutation occurring at time t_n would be equal to \(f_{\mathrm{n}}\left( {T,t_{\mathrm{n}}} \right) = \frac{{{\mathrm{e}}^{r(T - {\mathrm{t}}_{\mathrm{n}})}}}{{{\mathrm{e}}^{rT}}} = {\mathrm{e}}^{ - rt_{\mathrm{n}}}\). At time t₁, a mutation occurs that increases the growth rate r of the specific subpopulation by a scalar multiplier k, such that the new population is now expanding as \(N_{\mathrm{F}} = e^{krt_2}\). Thus, at the time of biopsy T = t₁ + t₂, we expect a generational (g−) hitchhiking mutation that occurred at time t_m < t₁ to have a frequency equal to \(f_{\mathrm{g}}\left( {T,t_{\mathrm{m}}} \right) = \frac{{{\mathrm{e}}^{r\left( {T - t_{\mathrm{m}}} \right)} + N_{\mathrm{F}}-{\mathrm{e}}^{rt_2}}}{{N_{{\mathrm{tot}}}}}\), where N_tot is the total number of cells (or mutations) and N_F is the number of cells that contain the fitness mutation that occurred at t₁ and expanded for t₂. Therefore \(N_{\mathrm{F}} = e^{krt_2}\). In a, we show the mutational frequencies at the time of biopsy T for two growth models; one neutral and one with a fitness mutation occurring at time t₁ = t_fg. Hitchhiking mutations “b” (blue), “r” (red), as well as passenger mutations “g” (gray) and “y” (yellow), also occur at different time points. b Under an exponential model with a fitness mutation occurring at time t₁ = t_fg, hitchhikers “b” and “r” show an increased frequency compared to neutral, subject to time and effect of the fitness mutation. Passenger mutations “y” and “g” that occurred before or with the fitness mutation, but on a different cell lineage, end up with lower frequencies. We characterize mutations “b” and “r” as generational (g−) hitchhikers since they mark the population’s generational growth.