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A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues.

Werner B, Dingli D, Traulsen A - J R Soc Interface (2013)

Bottom Line: Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy.We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients.

View Article: PubMed Central - PubMed

Affiliation: Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, Plön, Germany.

ABSTRACT
Cancers are rarely caused by single mutations, but often develop as a result of the combined effects of multiple mutations. For most cells, the number of possible cell divisions is limited because of various biological constraints, such as progressive telomere shortening, cell senescence cascades or a hierarchically organized tissue structure. Thus, the risk of accumulating cells carrying multiple mutations is low. Nonetheless, many diseases are based on the accumulation of such multiple mutations. We model a general, hierarchically organized tissue by a multi-compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters. We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy. We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients. This result can contribute to the explanation of very diverse mutation profiles observed by whole genome sequencing of many different cancers.

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(a) Number of cells without mutations in compartments 1–5, arising from a single cell in compartment 1. Lines are equation (2.8), symbols are averages with corresponding standard deviations over 103 independent runs of stochastic individual-based computer simulations, and squares are equation (2.13). Parameters are n0 = 1, ɛ = 0.85, γ = 1.26, u = 10−6 and r0 = 1/400. (b) Reproductive capacity of a single founder cell in compartment 1. Shown is the number of cells with 0–4 mutations in the first 31 compartments, acquired from a single cell in compartment 1. Symbols are numerical solutions of (2.8) in the limit of infinite time and lines are equation (2.13). The reproductive capacity increases exponentially for increasing compartment number. Cells carrying multiple mutations are strongly suppressed within a hierarchical tissue structure (see equation (2.14)). (Online version in colour.)
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RSIF20130349F4: (a) Number of cells without mutations in compartments 1–5, arising from a single cell in compartment 1. Lines are equation (2.8), symbols are averages with corresponding standard deviations over 103 independent runs of stochastic individual-based computer simulations, and squares are equation (2.13). Parameters are n0 = 1, ɛ = 0.85, γ = 1.26, u = 10−6 and r0 = 1/400. (b) Reproductive capacity of a single founder cell in compartment 1. Shown is the number of cells with 0–4 mutations in the first 31 compartments, acquired from a single cell in compartment 1. Symbols are numerical solutions of (2.8) in the limit of infinite time and lines are equation (2.13). The reproductive capacity increases exponentially for increasing compartment number. Cells carrying multiple mutations are strongly suppressed within a hierarchical tissue structure (see equation (2.14)). (Online version in colour.)

Mentions: We call a mutation neutral if the reproductive capacity of the mutant and the founder cell is equal. In §2.2, we have shown that the reproductive capacity of a cell depends on its differentiation probability ɛ and its mutation rate u, but interestingly it is independent of the reproduction rate r. Therefore, the clonal lineage and the number and type of mutations that arise from a single founder cell do not depend on the proliferation rates of the founder cell. However, the time to reach those states of course depends on r. Therefore, although two mutations lead to the same outcome, this might occur on distinct time scales, with observable differences in the progression of diseases. Nonetheless, our definition of neutral mutations only requires constant differentiation probabilities and mutation rates relative to the founder cell. This assumption allows us to write and thus the number of parameters is reduced from (k + 1) i + 1 for the general case to i + 1 for the neutral case. This number can be reduced to two parameters, u and ɛ, if a constant differentiation probability for all non-stem cell stages is assumed, ɛi = ɛ. This simplifies the evaluation of the recurrence relation (2.12) significantly. The reproductive capacity of neutral mutations in compartment i carrying k mutations becomes2.13Mutants carrying k mutations are suppressed by a factor uk and thus are rare in the early differentiation stages. The number increases exponentially for downstream compartments, and a significant load of cells carrying few mutations can be observed in the late differentiation stages (figure 4).Figure 4.


A deterministic model for the occurrence and dynamics of multiple mutations in hierarchically organized tissues.

Werner B, Dingli D, Traulsen A - J R Soc Interface (2013)

(a) Number of cells without mutations in compartments 1–5, arising from a single cell in compartment 1. Lines are equation (2.8), symbols are averages with corresponding standard deviations over 103 independent runs of stochastic individual-based computer simulations, and squares are equation (2.13). Parameters are n0 = 1, ɛ = 0.85, γ = 1.26, u = 10−6 and r0 = 1/400. (b) Reproductive capacity of a single founder cell in compartment 1. Shown is the number of cells with 0–4 mutations in the first 31 compartments, acquired from a single cell in compartment 1. Symbols are numerical solutions of (2.8) in the limit of infinite time and lines are equation (2.13). The reproductive capacity increases exponentially for increasing compartment number. Cells carrying multiple mutations are strongly suppressed within a hierarchical tissue structure (see equation (2.14)). (Online version in colour.)
© Copyright Policy - open-access
Related In: Results  -  Collection

License
Show All Figures
getmorefigures.php?uid=PMC4043170&req=5

RSIF20130349F4: (a) Number of cells without mutations in compartments 1–5, arising from a single cell in compartment 1. Lines are equation (2.8), symbols are averages with corresponding standard deviations over 103 independent runs of stochastic individual-based computer simulations, and squares are equation (2.13). Parameters are n0 = 1, ɛ = 0.85, γ = 1.26, u = 10−6 and r0 = 1/400. (b) Reproductive capacity of a single founder cell in compartment 1. Shown is the number of cells with 0–4 mutations in the first 31 compartments, acquired from a single cell in compartment 1. Symbols are numerical solutions of (2.8) in the limit of infinite time and lines are equation (2.13). The reproductive capacity increases exponentially for increasing compartment number. Cells carrying multiple mutations are strongly suppressed within a hierarchical tissue structure (see equation (2.14)). (Online version in colour.)
Mentions: We call a mutation neutral if the reproductive capacity of the mutant and the founder cell is equal. In §2.2, we have shown that the reproductive capacity of a cell depends on its differentiation probability ɛ and its mutation rate u, but interestingly it is independent of the reproduction rate r. Therefore, the clonal lineage and the number and type of mutations that arise from a single founder cell do not depend on the proliferation rates of the founder cell. However, the time to reach those states of course depends on r. Therefore, although two mutations lead to the same outcome, this might occur on distinct time scales, with observable differences in the progression of diseases. Nonetheless, our definition of neutral mutations only requires constant differentiation probabilities and mutation rates relative to the founder cell. This assumption allows us to write and thus the number of parameters is reduced from (k + 1) i + 1 for the general case to i + 1 for the neutral case. This number can be reduced to two parameters, u and ɛ, if a constant differentiation probability for all non-stem cell stages is assumed, ɛi = ɛ. This simplifies the evaluation of the recurrence relation (2.12) significantly. The reproductive capacity of neutral mutations in compartment i carrying k mutations becomes2.13Mutants carrying k mutations are suppressed by a factor uk and thus are rare in the early differentiation stages. The number increases exponentially for downstream compartments, and a significant load of cells carrying few mutations can be observed in the late differentiation stages (figure 4).Figure 4.

Bottom Line: Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters.We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy.We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients.

View Article: PubMed Central - PubMed

Affiliation: Evolutionary Theory Group, Max Planck Institute for Evolutionary Biology, Plön, Germany.

ABSTRACT
Cancers are rarely caused by single mutations, but often develop as a result of the combined effects of multiple mutations. For most cells, the number of possible cell divisions is limited because of various biological constraints, such as progressive telomere shortening, cell senescence cascades or a hierarchically organized tissue structure. Thus, the risk of accumulating cells carrying multiple mutations is low. Nonetheless, many diseases are based on the accumulation of such multiple mutations. We model a general, hierarchically organized tissue by a multi-compartment approach, allowing any number of mutations within a cell. We derive closed solutions for the deterministic clonal dynamics and the reproductive capacity of single clones. Our results hold for the average dynamics in a hierarchical tissue characterized by an arbitrary combination of proliferation parameters. We show that hierarchically organized tissues strongly suppress cells carrying multiple mutations and derive closed solutions for the expected size and diversity of clonal populations founded by a single mutant within the hierarchy. We discuss the example of childhood acute lymphoblastic leukaemia in detail and find good agreement between our predicted results and recently observed clonal diversities in patients. This result can contribute to the explanation of very diverse mutation profiles observed by whole genome sequencing of many different cancers.

Show MeSH
Related in: MedlinePlus