Figures and data
The distribution of variant allele frequencies changes with the growth phases and by sampling.
(a) In the current population, cells divide symmetrically into two daughter cells or asymmetrically with only one daughter cell kept in the focused population. All other events are mathematically equivalent to and are treated as a part of cell death. (b) The rates of symmetric and asymmetric division change during the population growth and lead to a dynamical distribution of variant allele frequencies. (c) The observed VAF distribution is shifted again during sampling compared to the VAF of the whole population – a fact should be considered when inferring population properties through genetic data.
Bulk sequencing based VAF and mutation rate inferences in healthy esophagus.
(a) Simulated VAF with an initial logarithmic growth phase and subsequent constant population phase at different time points. As the population gets closer to the maximum simulated carrying capacity of N = 104, the VAF moves from a 1/f 2 growing population shape to a 1/f constant population shape. Note, the shift considerably slows at older age. (b) VAF from healthy tissue in the oesophagus of 9 individuals sorted into age brackets. The youngest bracket, 20-39, is closer to the developmental 1/f 2 scaling. The older age brackets are both close to the constant population 1/f scaling, resembling the theoretical expectations. (c) Inferred mutation rates increase linearly with age. (d) Simulations of slowly growing stem cell populations reveal that mutation rates appear to increase with age, although the true underlying per division mutation rate remaining constant.
Inference of evolutionary parameters on simulated stem cell populations.
Simulated populations were run up to age 59, growing exponentially from a single cell to constant size NM = 10′000 at age tM = 5, with mutation rate µ = 1.2 and division rates λ = 5 and p = 0.4. Where sampling is mentioned, the sample size 89 was taken. a) The single-cell mutational burden distribution. The compound Poisson distribution (dashed line) matches the burden distribution when averaging over multiple independently evolved populations (filled curve). b) Distribution of estimated mutation rates from 10’000 individual simulations, obtained from burden distributions of the complete populations (blue) as well as sampled sets of cells (orange). c) VAF spectra measured in the complete population (blue) and a sampled set of cells (orange). A single simulation result is shown (diamonds) alongside the theoretically predicted expected value (Eq. (1)) (dashed line) and the average across 100 simulations (solid line). d) Distribution of NM and p inference results for 100 simulated and sampled populations, through estimation of 
Evolutionary inferences in single cell HSC data
(a) The single-cell mutational burden distribution of the data (bars) and the Compound Poisson distribution obtained from its mean and variance, used to obtain the estimated per division mutation rate 
Comparison of an exponential and logistic growth model.
(a) Population size over time for exponential (dotted) and logistic (dashed) growth functions. At the point of maturity (t = 20) the exponential function reaches the population capacity (NM = 10′000), and the logistic function equals 0.99NM. The time points at which the VAF spectra are measured are indicated by solid vertical lines: t1 = 25 (blue) and t2 = 50 (orange). (b) Comparison of the expected VAF spectra, calculated with Eq. (1), in the exponential (dotted) and logistic (dashed) growth models measured at the time points t1 and t2. For reference, the theoretical predictions for an exponentially growing population without cell death and a constant population in the long time limit are shown as solid purple and green lines respectively.
Stochastic simulations of the Variance/Mean of the mutational burden distribution over time for a per cell division mutation rate of µ = 1.3 and varying stem cell population size N and asymmetric division probability p.Stochastic fluctuations are pronounced for small population size N and low asymmetric division probability p.
Likelihood of the Variance/Mean to be in the interval 3 < θ < 5 for a per division mutation rate of µ = 1.3.
Evolutionary parameters appearing in the model system.
Evolutionary parameters appearing in the analytical derivations of the expected VAF distribution in the Moran and pure-birth models.
Likelihood of the Variance/Mean to be in the interval 3 < θ < 5 for a per division mutation rate of µ = 3.
The standard deviation on the VAF spectrum increases for higher frequencies.
a) The VAF spectrum Vf averaged across 100 simulations of a population evolved according to the model described in section 1.1. The standard deviation from the mean is shown as the band Vf ± sf around the average. b) The standard deviation across all simulations for each frequency f = 1/N, 2/N, …, scaled by the average spectrum Vf.
