Fig 1.
The main research question and the study design.
Quantifying and interpreting the genomic divergence between paired primary and metastatic tumors (M-P divergence, top right panel) utilizing the ever-growing amount of multi-region/longitudinal cancer sequencing data (top left panel) is becoming increasingly important. It can facilitate a deeper understanding of metastatic progression, and thereby, improve its prevention. Here, we sought to ask what exactly is being captured on the trees of tumor evolution by M-P divergence measured in such datasets. We adopt mathematical modeling (non-spatial multi-type branching processes) to illuminate the key evolutionary determinants of M-P divergence; and computational modeling to simulate the three dimensional cancer clonal evolution, and we investigate how the M-P divergence is regulated by the evolutionary forces under various growth dynamics of the primary tumor (bottom panel). The detailed model setting, including the parameters used in both modeling approaches, can be found in Methods and Table 1.
Fig 2.
Elements of Metastatic-Primary genomic divergence and their potential pattern in tumor evolution.
(A) A sample tree (schematic) based on somatic variant analysis of paired Metastatic (M) and Primary (P) tumor samples depicts the two branches involving the number of metastatic specific (Bm, dashed green line) and primary specific somatic variants (Bp, dashed blue line), respectively. We focus primarily on the variants accumulated in the metastatic seeding cell (green circle) before its dissemination (Bmd), as it is directly related to the primary tumor growth and the genetic determinants of metastatic dissemination. Below the sample tree shown a genealogy tree focusing on the lineage of the seeding cell, where we note the branches that represent Bmd and Bp, respectively. Detectable ancestors in the primary at a frequency of γ (DAγ) and the most recent detectable ancestor at α and γ (MRDAα and MRDAγ) of the seeding cell are also indicated, respectively. Here γ is the variant allele frequency threshold above which substantial presence of a variant may be deemed; α is the frequency threshold of sequencing detectability. (B) If a cell with k variants “cancels” its dissemination (i.e., stays in the primary tumor) and its progeny cell with k + 1 variants migrates instead, the resulting Bmd can drop if the progenitor cell with k variants eventually grows into a detectable subclone in the primary tumor. In this case, it becomes the most recent detectable ancestor of the seeding cell. More generally, Bmd can decrease if an ancestor along the genealogy the the seeding cell becomes detectable due to subclone expansion. (C) We hypothesize that Bmd and Bp are affected by the kinetics of primary tumor growth. Under the growth pattern of progressive subclonal divergence, late seeding would lead to larger Bmd than early seeding as only the early subclones are detectable; Bp stays constant as the detectable portion of the seeding cell lineage is restricted. By contrast, detectable subclones can appear late leading to the pattern of sub-clonal convergence (due to selection or other ecological forces), under which scenario seeding from early or late detectable subclones may lead to similar Bmd, but differs in Bp.
Fig 3.
Single-cell-based spatial simulation of paired metastatic and primary tumors validates the phylogenetic definition of Bmd and emphasizes that Bm can appear sensitive or insensitive to seeding time based on the kinetics of primarty tumor growth.
(A)Bmd, the sub branch length (measured in cell generations) from the metastatic seeding cell to its most recent detectable ancestor (MRDAα) captures the majority of the variance of Bm measured from the virtual tumors (green density curves). By contrast, the tumor size at metastatic dissemination, as a surrogate of dissemination time (gray density curves) does not faithfully explain the measured Bm. (B) Two virtual tumors with distinct kinetics exemplify the conditional dependence of Bm on metastatic dissemination time. Upper panel: the genealogy trajectories (gray lines) of metastatic seeding cells (green dots) along the expansion of the primary tumor in a three dimensional lattice. The Euclidean distance from a cell to the center of the lattice is shown against the tumor size fraction when the corresponding cell disseminates. A cell’s distance to the center is strongly correlated with its mutation burden (S3 Fig), reflecting the spatial constraints imposed in our model. The tumor size fraction is plotted at its cube root scale to reflect the clock of actual time. Ancestor cells that are detectable at a frequency greater than 0.01 are marked as blue dots; Middle panel: Bm is plotted against the tumor size fraction when the seeding cell (green dots) disseminates, the running mean (black dots) and standard deviation (gray bars) is also shown; Lower panel: Bm is shown against the Bmd, the sub-branch length from the seeding cell to its MRDAα. The slope of the linear regression is consistent with the passenger variant rate (0.15) used in the model, CI: 95% confidence interval.
Fig 4.
Characteristic Bmd and Bp patterns.
(A) Example phylogenetic tree of somatic variants under neutral model of growth. Notice that the expected Bmd time series are increasing with seeding time since the number of detectable variants along a general branch are expected to reach a fixed limit, beyond which new variants are almost surely undetectable. Similarly, the expected Bp patterns are decreasing (cf. (8)) up until a point where they stabilize to a fixed small number. (B) Example tree depicting the introduction of a new type of cells. Notice that Bmd at time kseed is radically different when seeding originates from type-0 as opposed to type-1. Assuming that seeding is proportional to the fitness of the types, the expected Bmd time series are expected to drop abruptly for seeding times right after the introduction of the new type. Similarly Bp tracks these drops. (C) The connection between the decay in detectability of the variants along the seeding cells lineage and the distribution of the lengths. Notice that a sudden drop in the detectability probabilities along this path, leads to a spike in the probability of the corresponding length of
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Table 1.
Parameters and settings in Comet, the single cell based spatial computational modeling of evolving tumors.
Fig 5.
Spatial computational model verifies that the growth mode governs the dependence of M-P divergence on seeding time.
(A) Mathematical models illuminate the expected patterns of Bmd and Bp under neutral (left) and selective dynamics (right, corresponding to the second and third columns of panel B), respectively. Detailed example plots of mathematical analyses can be found in Fig 7 and S1 Appendix. The variant genealogy schema of seeding cells is also shown. (B) Virtual tumors with three representative clonal kinetics and the changing patterns of M-P divergence along with the metastatic seeding time. Upper panel: the genealogy trajectories (gray lines) of metastatic seeding cells (green dots) during the expansion of the primary tumor in a three-dimensional lattice. The Euclidean distance from a cell to the center of the lattice is shown against the fraction of the final primary tumor size when the corresponding seeding cell is born and disseminated. A cell’s distance to the center is strongly correlated with its mutation burden (S3 Fig), reflecting the spatial constraints imposed in our model. The tumor size fraction is plotted at its cube root scale to reflect the clock of actual time. Blue dots represent progenitor cells that are detectable at a frequency higher than 0.01. Middle panel: Bm is plotted against the tumor size fraction when the seeding cell disseminates, where black dots and gray bars indicate the running mean and standard deviation of Bm, respectively. Lower panel: change of Bp with different seeding time, blue dots and bars represent the running mean and standard deviation of Bp, respectively. (C) The subclonal expansion occurrence is strongly associated with drops and valleys of Bm in spatial simulations (see Methods).
Fig 6.
Summary of the virtual tumor dynamics and the mapping between seeding time and M-P divergence.
(A) For each simulation (a circle), we calculated the average pair-wise Fst (Fixation index) of ten randomly sampled regions of the virtual primary tumor, to reflect the between-region genetic divergence. We plotted the Fst against the average fraction of variants of the seeding cell that fell above the detection limit in the primary tumor. The size of the circle increases with the selection coefficient. For each selection level, the crossbars show the standard deviation along the two axes. Color scale indicates the proportion of variance of Bm explained by the seeding time (based on 10-quantile of all the values). (B) We group the various simulations into three representative clonal dynamics: progressive diversification (s ≤ 0.05), branched evolution (s = {0.1, 0.2}) and linear evolution (s ≥ 0.5). For each group, we plot the actual seeding time (color hues, using the fraction of the final primary tumor size at metastatic dissemination as a surrogate) on top of the corresponding Bm and Bp values. A smoothed layer shows the general distribution of the seeding time.
Fig 7.
Illustrative Bmd and Bp patterns, in the exponential growth setting, under different seeding scenarios explored in Section 4.1.3.
In this example two advanced types of cells are introduced in the population; type-1 is introduced when there are 600 cells, whereas type-2 is introduced when there are approximately 460K cells. The parametric values are: ρ0 = 0.95, ρ1 = 0.93 ρ2 = 0.83, u = 0.15, fm = 0.02, fp = 0.4; the overall population at detection time is 1.06 × 109 cells; finally, the advantageous types 1 and 2 are present on 87.3% and 77.1% of the cells in the population, respectively, and the number of clonal and subclonal variants of the primary tumor with frequency at least 40% is assumed to be 50. Type-2 is a subtype of Type-1. The variability reported on the subplots in the second and third column, captures the effects of uncertainty in seeding origination under seeding scenarios (b) and (c) (note that such variability is absent in the first column which corresponds to seeding scenario (a) which is deterministic; see also Section 4.1.3). Observe that the introduction of advanced cells results in a drop of the expected Bmd and Bp. This drop is more sudden and emphatic in the case where the probability of dissemination is strongly associated with the genetic properties of the seeding cells (as e.g., in the first two columns of the plot, which showcase the resulting patterns under seeding scenarios (a) and (b), discussed in Section 4.1.3), however, even on the other extreme (when seeding is equiprobable among all living cells at the time of dissemination; i.e., seeding scenario (c), detailed in Section 4.1.3), a gradual change in monotonicity of is also present. Similar patterns arise under different parametric choices (see S7 and S8 Figs).