Extreme value statistics of mutation accumulation in renewing cell populations

The emergence of a predominant phenotype within a cell population is often triggered by the chance accumulation of a sequence of rare genomic DNA mutations within a single cell. For example, tumors may be initiated by a single cell in which multiple mutations cooperate to bypass their natural defense mechanism. The risk of such an event is thus determined by the extremal accumulation of mutations across tissue cells. To address this risk, here we study the statistics of the maximum mutation numbers in a generic, but tested, model of a renewing cell population. By drawing an analogy between the genealogy of a cell population and the theory of branching random walks, we obtain analytical estimates for the probability of exceeding a threshold number of mutations to trigger a proliferative advantage of a cell over its neighbors, and we determine how the statistical distribution of maximum mutation numbers scales with age and cell population size.

Figure 1

The genealogy and its implications. (a) Illustration of the history of mutation accumulation. Vertical lines represent mutational paths; horizontal arrows mark loss/replacement events. Dashed lines are mutational paths that are lost, while bold lines are paths that survive until time $t=T$, constituting the genealogy. If $T$ is large enough, the genealogy possesses a last common ancestor (LCA). (b) Mean difference between maximum mutation number and the population mean, $\langle \mathrm{\Delta }{m}^{*}\rangle =\langle m^{*}\rangle -\langle m_i\rangle$, as a function of $\mu T$, for fixed $N=10000$. Blue pluses are results from Monte Carlo simulations for parameters from human eyelid epidermis stem cells $\lambda =6067\mu$ (corresponding to $\mu =0.27/\left(63\text{years}\right)$ and $\lambda =0.5/\text{week}$ [7, 8]) so that $\langle {\widehat{T}}_{\mathrm{LCA}}\rangle \approx N/\lambda =1.65/\mu$. The red dashed line illustrates the saturation asymptote. Black pluses are corresponding results for $\lambda =0$ and the black dashed line marks $\langle \mathrm{\Delta }{m}^{*}\rangle \simeq {\left(\mu T\right)}^{1/2}{(2lnN)}^{1/2}$.

Figure 2

Mean maximum mutation number ahead of the mean, $\langle \mathrm{\Delta }{m}^{*}\rangle$, as a function of $N$, for $T=10N/\lambda$ such that $T>{\widehat{T}}_{\mathrm{LCA}}$. Shown are the results of Monte Carlo simulations (pluses), and theoretical predictions from the BRW approximation, Eq. (7), for fitted numerical constant ${\mathcal{C}}_{\tilde{m}}^{\mathrm{fit}}$ (blue dashed line) and theoretically estimated value (solid red line), ${\mathcal{C}}_{\tilde{m}}^{\mathrm{th}}=1.79$ for (a) $\mu =\lambda$ (${\mathcal{C}}_{\tilde{m}}^{\mathrm{fit}}=1.43$) and (b) $\mu =0.001\lambda$ (${\mathcal{C}}_{\tilde{m}}^{\mathrm{fit}}=1.63$).

Figure 3

Squared mean maximum mutation number ahead of the population mean, ${\langle \mathrm{\Delta }{m}^{*}\rangle }^2$, as a function of $N$, for fixed $T=1000/\lambda$. Shown are the results of Monte Carlo simulations (pluses) and theory, Eq. (9), with fit parameter ${\mathcal{C}}_{\sigma }^{\mathrm{fit}}$ (dashed line) for (a) $\mu =\lambda$ (${\mathcal{C}}_{\sigma }=1.13$) and (b) $\mu =0.001\lambda$ (${\mathcal{C}}_{\sigma }=1.59$).

Figure 4

Probability $\tilde{P}$ of accumulating a critical number of mutations, $m_c$, in a single cell among $N={10}^5$ cells for $\lambda =6000\mu$ (black points), which, for illustrative purposes, corresponds to an estimated area of skin epidermis of 0.1 ${\mathrm{cm}}^2$ [7, 8] [see also Fig. 1] and for $\lambda =0$ (red points). (a) $\tilde{P}$ as a function of time and threshold $m_c=6$ [22, 55]. The top panel shows the ratio of the latter two cases, ${\tilde{P}}_{\lambda =0}/{\tilde{P}}_{\lambda >0}$. (b) $\tilde{P}$ as a function of the threshold mutation number $m_c$ for fixed time $\mu T=0.27$, corresponding to 63 human years of skin turnover [8].