Example of the discrete time branching process in Chiron.

<p>Aneuploidy is generated at a specific rate (<i>p</i>) of mitotic mis-segregation, where trisomic (red circles) and monosomic cells (green circles) have certain probabilites (<i>s</i>_t and <i>s</i>_m respectively) of permanent proliferative arrest/death (black horizontal bars) at each mitotic cycle. Two or more aneusomies will result in increased probabilities of arrest/death as exemplified by cells labelled++and +−.</p

Reverse engineering <i>in silico</i> of the LoVo stemline genome by introducing positive selection.

<p>(A) The LoVo stemline with trisomies of chromosomes 5, 7 and 12 was recreated by modelling clonal expansion from a normal diploid (N) state where aneuploid cells were subject to negative selection (<i>S</i>_n) at the magnitude measured in LoVo, with the exception of cells with one or more of the trisomies +5, +7, +12. Such cells were subjected to a variable degree of positive selection (<i>S</i>_p), <i>i.e.</i> a higher probability of undergoing mitotic proliferation than diploid cells. For simplicity, diploid cells were given an average probability of 50% for proliferative survival, resulting in one daughter cell on overage for each mother cell (left). Cells having acquired +5/+7/+12 were set to generate on average >1 daughter cell, with the maximum survival benefit for cells having all three trisomies (right). <i>S</i>_p was set in relation to the proliferation of normal cells with <i>S</i>_p = 100% equalling the generation of 2 daughter cells per mother cell with all three trisomies. Each of +5/+7/+12 were coupled to an identical degree of positive selection i.e. 1/3 <i>S</i>_p for each. Aneusomies were acquired through mis-segregation at the rate measured in LoVo. (B) The prevalence of aneuploid cells reaches a stable, high level in the population (1,000 cells) with 100% positive selection for +5/+7/+12 (red plot), while a parallel proliferation in absence of positive selection retains a diploid genome in the majority of cells (blue plot). (C) In the same simulation of positive selection, the prevalence of cells with each of +5/+7/+12 increase rapidly; chromosome 1, used as an internal control, remains disomic. (D) When the prevalence of aneuploidy has reached a stable state the modal number has shifted to 49 as expected with positive selection for +5/+7/+12, while it remains diploid (E) in the absence of positive selection for trisomies. (F) Using the same parameters as in B–D but introducing <i>S</i>_n = 30% for all aneusomies including +5/+7/+12 prevents expansion of aneuploid clones. Instead a state of mutation/selection balance is reached where trisomic cells never reach a higher prevalence than 4%.</p

Elevated Tolerance to Aneuploidy in Cancer Cells: Estimating the Fitness Effects of Chromosome Number Alterations by <i>In Silico</i> Modelling of Somatic Genome Evolution

<div><p>An unbalanced chromosome number (aneuploidy) is present in most malignant tumours and has been attributed to mitotic mis-segregation of chromosomes. However, recent studies have shown a relatively high rate of chromosomal mis-segregation also in non-neoplastic human cells, while the frequency of aneuploid cells remains low throughout life in most normal tissues. This implies that newly formed aneuploid cells are subject to negative selection in healthy tissues and that attenuation of this selection could contribute to aneuploidy in cancer. To test this, we modelled cellular growth as discrete time branching processes, during which chromosome gains and losses were generated and their host cells subjected to selection pressures of various magnitudes. We then assessed experimentally the frequency of chromosomal mis-segregation as well as the prevalence of aneuploid cells in human non-neoplastic cells and in cancer cells. Integrating these data into our models allowed estimation of the fitness reduction resulting from a single chromosome copy number change to an average of ≈30% in normal cells. In comparison, cancer cells showed an average fitness reduction of only 6% (p = 0.0008), indicative of aneuploidy tolerance. Simulations based on the combined presence of chromosomal mis-segregation and aneuploidy tolerance reproduced distributions of chromosome aberrations in >400 cancer cases with higher fidelity than models based on chromosomal mis-segregation alone. Reverse engineering of aneuploid cancer cell development <i>in silico</i> predicted that aneuploidy intolerance is a stronger limiting factor for clonal expansion of aneuploid cells than chromosomal mis-segregation rate. In conclusion, our findings indicate that not only an elevated chromosomal mis-segregation rate, but also a generalised tolerance to novel chromosomal imbalances contribute to the genomic landscape of human tumours.</p></div

Estimation of negative selection acting on aneuploid cells in normal human fibroblasts.

<p>(A) Chiron-based simulation (20 parallel runs) of a growing cell population with a mis-segregation rate of 4×10⁻⁴ where different degrees of selection (<i>s</i>) against aneuploid cells are imposed. The resulting mean prevalence values of cells aneusomic (non-disomic) for a certain chromosome are given as aneusomy indices (AI) on the y-axis. AI is reduced with higher <i>s</i> values and reaches a dynamic equilibrium around 25 generations even at selection values as low as 10%. (B) Relationship of mean AI for a certain chromosome and negative selection acting on cells with aneusomy for this chromosome. Red lines correspond to estimations of selection made from experimental data for chromosomes 1 and 17 in F1 and F2. Error bars correspond to standard deviations in 20 simulations. (C) Fluorescence <i>in situ</i> hybridisation (FISH) was used to estimate AI in human normal fibroblasts, exemplified by F1 cells disomic (2n) and trisomic (3n) for chromosome 17 (red, q arm probe signal; blue, centromeric signal). (D) Prevalence of monosomic and trisomic cells as well as overall AI estimated by FISH in the two fibroblast lines F1 and F2. (E) Chiron-based estimates of the degree of negative selection acting on cells aneusomic for chromosomes 2 and 17 in F1 and F2, deduced by comparing FISH-data (AI) with modelling of AI as a function of <i>s</i> (<a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070445#pone-0070445-g003" target="_blank">Figure 3B</a>). The max and min values correspond to ±2 standard deviations.</p

Aneusomy indices and aneuploidy tolerance in cancer cells ¹.

1<p>Abbreviations: AI, aneusomy index; s, negative selection level;</p><p><i>p</i>, mis-segregation rate/chromosome/mitosis.</p>2<p>Mode refers to the most common copy number of each chromosome.</p

Modelling the distribution of aneuploidy burden in human cancers.

<p>(A) Reported cytogenetic data from the Mitelman Database of Chromosome Aberrations and Gene Fusions in Cancer show a log-linear relationship between the relative prevalence and the number of numerical aberrations per tumour (Nnapt), with highly similar distributions for Wilms tumour (WT) and colorectal cancer (CRC). (B) Modelling of a certain number of cancer stemlines arising in the same number of patients. Each stemline is assumed to derive from a diploid cell (having 0 numerical aberrations) and is allowed to proliferate for a maximum of 2000 generations (G), when the overall distribution of numerical aberrations is sampled. Stemlines accumulate numerical aberrations at a certain mis-segregation rate (<i>p</i>) and are subject to aneuploidy-dependent selection at a certain degree (<i>s</i>), which may in turn result in termination of the stemline (horizontal dumbbell), corresponding to the end of clonal expansion. Because this may result in regression of tumorigenesis at an early stage, cases where stemlines were thus terminated were removed from sampling. (C) Simulated distribution of tumour cases with a certain number of numerical aberrations as the tumour cohort is sampled at generations 1–2000 in a setting where tumours harbour an elevated mis-segregation rate in the absence of negative selection against aneuploid cells (see main text for details). This will result in a binomial-like distribution already after 100 generations, the modal value of which increases with time, in contrast to the actual distribution in human tumours (compare to 6A).</p

Negative selection acting on aneuploidy in cancer compared to normal cells.

<p>Mean aneusomy-dependent negative selection (<i>s</i>) estimated by Chiron-simulations. The single star denotes p<0.05 and double stars p<0.001 (Student’s <i>t</i>-test) at comparison between each cancer cell line and normal fibroblasts (F1+F2).</p

Major mechanisms of chromosome missegregation.

<p>Based on previous experimental data <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070445#pone.0070445-Gisselsson2" target="_blank">[6]</a> mis-segregation was simulated as resulting from sister-chromatid non-disjunction resulting in one monosomic and one trisomic daughter cell (upper panel) and chromatid lagging resulting in monosomy in one daughter nucleus (lower panel). Each of these events was approximately equal in frequency, in all resulting in an average of 75% aneuploid daughter cells (red and green background) per mis-segregation event.</p

Modelling of aneuploidy rates as a function of negative selection against chromosome changes in cancer cells.

<p>(<b>A–B</b>) A dynamic equilibrium with respect to aneusomy index (AI) is reached before 500 generations even with very low negative selection pressures in cancer cell populations with a low mis-segregation rate (exemplified by 1% for DLD1 having a near-normal mis-segregation rate). Both plots are from single simulation runs. (<b>C–F</b>) <b>I</b>n all four analyzed cancer cell lines simulations predicted a near-linear negative relationship between anusomy index (AI) and the degree of negative selection (<i>s</i>) on a log-log scale. Full lines indicate mean AI values and broken lines the minimum and maximum values for each simulation of AI for a certain <i>s</i>. Grey lines correspond to simulated AI for normal fibroblasts and grey circles indicate AI quantified by FISH for chromosomes 2 and 17 in fibroblasts. Coloured lines correspond to simulated AIs for chromosomes of different modal numbers and coloured circles the FISH-estimated AI values for the analysed chromosomes in the cancer cell lines. Except for one chromosome in LoVo, the negative selection pressures acting on aneusomic cells are lower in the cancer cell lines. The full data calculated from these AI-<i>s</i> estimates are presented in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070445#pone-0070445-t001" target="_blank">Table 1</a> and summarised in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070445#pone-0070445-g005" target="_blank">Figure 5</a>.</p

Distribution of numerical aberrations in human cancers predicted by the presence/absence of chromosomal instability and aneuploidy-dependent selection.

<p>The expected distributions according to different conditions of chromosomal mis-segregation and selection were predicted by simulations as described in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0070445#pone-0070445-g006" target="_blank">Figure 6</a>. In each graph, the reported data for Wilms tumour (WT, black circles) and colorectal cancer (CRC, red circles) are included for comparison. (A) The expected distribution (open blue circles) of numerical changes in a setting with chromosomal instability (CIN) in the absence of aneuploidy-dependent negative selection (<i>s</i>) is binomial-like with a high modal number of aberrations per tumour. All data points from 10 independent simulations were included. (B) A setting with CIN present but with <i>s</i> values similar to normal fibroblasts, results in a distribution with far fewer aneusomies than reported in WT and CRC. The same conditions, but with absence of CIN (C and D) result in similar distributions as when CIN is included, but with fewer abnormalities. (E) Attenuated aneusomy-dependent negative selection (<i>s</i> set as a span corresponding to the magnitude found in cancer cell lines) and with CIN present predicts a distribution of numerical changes highly similar to reported data. (F) In contrast, attenuated negative selection combined with absence of CIN results in distribution skewed towards fewer numerical aberrations than observed in WT and CRC.</p