Multi-region sampling of a hepatocellular tumour and cell-based simulations.

A shows the spatially resolved sequencing data of 285 samples of a hepatocellular carcinoma analyzed by Ling et al. [34]. Each sample is indicated by a small pie chart in which colors indicate specific mutations, and slice sizes indicate the mutation frequencies within each sample. The 23 samples highlighted by a black outline were also subjected to whole-exome sequencing. B and C Results of a cell-based simulation in the surface growth mode (B) and the volume growth mode (C). In each case, 280 evenly spaced samples were taken from the population of 10000 cells of a 2d simulation, see text. The most frequent mutations are shown as in (A), superimposed on the structure of the simulated tumour.

Relative position of mutants under different growth modes.

A The direction angle θ quantifies the direction of a new mutant clone relative to its parent clone. In this illustration, a new mutant clone indicated in blue appears and grows radially outward on a red parental background, resulting in an angle θ near zero. Each pair of cells indicated in red and blue contributes to the distribution of θ, with the statistical weight of each mutant clone adding to one (see SI S2). B The distribution of angles θ for different mutant clones found in the spatially-resolved data of Ling et al. [34]. C and D show the corresponding distribution of angles for numerical simulations. Subfigure C shows simulations of surface growth, resulting in a distribution of direction angles with a pronounced maximum near zero. Under volume growth (D) a nearly flat distribution is seen. For C and D, simulations were run in three dimensions with a maximum population size of 40000 cells grown at division rate b = 1, a rate of cell death d = 0.4 and d = 0.8 for surface and volume growth respectively, and a whole-exome mutation rate μ = 0.3 before taking a two-dimensional cross-section of 280 samples mimicking the sampling procedure in [34]. (The different death rates were chosen to make the extinction probabilities of the populations comparable for the two cases. Changing these rates did not affect the distributions of angles.)

Dispersion of mutations within the tumour.

(A) Cells with a particular mutation can form tight spatial clusters within a tumour (simulated example here: mutant shown in blue), or they can be more widely dispersed (mutant shown in red). We quantify the dispersion of a mutation using the dispersion parameter σ, see text and SI S4. In this illustrative example, the blue mutation has a small dispersion parameter σ = 1.3, the red one has σ = 2.5. (B) Histogram of the dispersion parameters σ across 217 mutations in the whole-exome data of [34]. (C) and (D) show the corresponding histograms for simulations of surface growth and volume growth, respectively. The simulations were run in 3D with populations grown up to 40000 cells before taking 23 evenly spaced samples from a 2D cross-section. Only mutations with a whole-tumour frequency larger than 1/40 were considered, mimicking the limited sequencing resolution in the Ling et al. data. Simulation parameters were division rate b = 1, mutation rate μ = 0.3, and death rates d = 0.4 and d = 0.8 for surface growth and bulk growth, respectively.

Rate of cell death and the mutation rate.

(A) We ask how a limited number of samples identify mutations. We pick a subset of the whole-exome sequenced samples of Ling et al. [34] and plot the fraction of mutations present at least in one of these samples against the number of samples in the subset (red symbols, fractions are relative to the number of mutations present in at least one of the 23 samples. Mutations must be supported by at least 5 reads at a coverage of at least 150). The procedure is repeated (blue symbols) with those mutations removed that occur in some other sample with a higher frequency than the frequencies with which the mutation occurs in the subset of samples. Error bars indicate the range of the 95-percentile. (B) The schematic Muller plot shows how cell death leads to the loss of clones, some of which have extant offspring. In the example shown here, the clone in light red becomes extinct, leaving behind its darker shaded offspring clone with no parental clone. The rate of this loss of parental clones depends on the rate of cell death, and can be used for inference, see text. (C) shows the inferred rate of cell death and the inferred rate of mutation per generation. The violin plots show how the inferred values vary when subsampling different fractions of all mutations and scaling the inferred mutation rate correspondingly, with the dashed lines indicating the mean inferred values. (The fraction of mutations sampled ranges from 0.5 to 0.9, with the results shown separately in SI 9.)

Mutational signature decomposition.

Relative weights of single-base substitution (SBS) mutational signatures [45] in all three tumours were derived with the in-house method SigNet [47] where possible (highlighted pie charts), otherwise with non-negative least squares (Methods). (left) Ling et al., (centre) tumour T1 of Li et al., (right) tumour T2 of Li et al. Top: all mutations, bottom: mutations stratified by their clonality. Signature SBS22 is associated with exposure to aristolochic acid. In the Ling et al. data, this signature is prominent among clonal mutations, but absent in subclonal mutations. SBS5 and SBS40 are endogenous mutational processes. Signatures with non-positive weights were combined into a single category (Unknown).