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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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Schematic of the compartment structure of multiple mutations and the corresponding transition rates. The top compartments contain cells carrying no mutation. The bottom compartments contain cells carrying one mutation. Compartments to the right represent more specialized cell stages and arrows represent transition probabilities, where ɛ denotes the differentiation probability, u denotes the mutation rate of cells and . Initially, no mutated cells are present in the hierarchy. We then determine, how many cells are acquired from the founder compartment (top left) and investigate how many cells with k mutations are on average expected at any stage of the hierarchy. (Online version in colour.)
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RSIF20130349F1: Schematic of the compartment structure of multiple mutations and the corresponding transition rates. The top compartments contain cells carrying no mutation. The bottom compartments contain cells carrying one mutation. Compartments to the right represent more specialized cell stages and arrows represent transition probabilities, where ɛ denotes the differentiation probability, u denotes the mutation rate of cells and . Initially, no mutated cells are present in the hierarchy. We then determine, how many cells are acquired from the founder compartment (top left) and investigate how many cells with k mutations are on average expected at any stage of the hierarchy. (Online version in colour.)

Mentions: The hierarchical tissue organization is typically modelled by a multi-compartment approach [6,10]. Each compartment represents a certain differentiation in the stage of cells. At the root of the hierarchy are stem cells ensuring a continuous influx of cells. A proliferating cell in compartment i divides and the two daughter cells differentiate and migrate into the next downstream compartment (i + 1) with probability ɛ, increasing the downstream compartment by two cells, mutates with probability u or self-renews within its own compartment with probability 1 − ɛ − u. Mutated cells stay in the hierarchy. If a mutated cell proliferates, it differentiates with probability ɛ into the next downstream compartment, it self-renews with probability 1 − ɛ − u, or it mutates with probability u again, leading to a cell with two (or more) mutations. All possible outcomes of a cell proliferation are depicted in figure 1. The directions of the arrows point towards the accessible cell states and the labels give the transition probabilities. We allow arbitrary parameters and introduce as the differentiation probability of cells in compartment i carrying k mutations. Asymmetric cell divisions are not explicitly implemented, as they can be absorbed in the differentiation probabilities on the population level. The fate of a cell's offspring is determined based on the probabilities . Cells proliferate with a rate ri in each compartment i. Usually, cells in upstream compartments proliferate slowly and cell proliferation speeds up in downstream compartments (i.e. ri < ri+1). This general framework is very flexible and different tissue structures can be represented.Figure 1.


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)

Schematic of the compartment structure of multiple mutations and the corresponding transition rates. The top compartments contain cells carrying no mutation. The bottom compartments contain cells carrying one mutation. Compartments to the right represent more specialized cell stages and arrows represent transition probabilities, where ɛ denotes the differentiation probability, u denotes the mutation rate of cells and . Initially, no mutated cells are present in the hierarchy. We then determine, how many cells are acquired from the founder compartment (top left) and investigate how many cells with k mutations are on average expected at any stage of the hierarchy. (Online version in colour.)
© Copyright Policy - open-access
Related In: Results  -  Collection

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

RSIF20130349F1: Schematic of the compartment structure of multiple mutations and the corresponding transition rates. The top compartments contain cells carrying no mutation. The bottom compartments contain cells carrying one mutation. Compartments to the right represent more specialized cell stages and arrows represent transition probabilities, where ɛ denotes the differentiation probability, u denotes the mutation rate of cells and . Initially, no mutated cells are present in the hierarchy. We then determine, how many cells are acquired from the founder compartment (top left) and investigate how many cells with k mutations are on average expected at any stage of the hierarchy. (Online version in colour.)
Mentions: The hierarchical tissue organization is typically modelled by a multi-compartment approach [6,10]. Each compartment represents a certain differentiation in the stage of cells. At the root of the hierarchy are stem cells ensuring a continuous influx of cells. A proliferating cell in compartment i divides and the two daughter cells differentiate and migrate into the next downstream compartment (i + 1) with probability ɛ, increasing the downstream compartment by two cells, mutates with probability u or self-renews within its own compartment with probability 1 − ɛ − u. Mutated cells stay in the hierarchy. If a mutated cell proliferates, it differentiates with probability ɛ into the next downstream compartment, it self-renews with probability 1 − ɛ − u, or it mutates with probability u again, leading to a cell with two (or more) mutations. All possible outcomes of a cell proliferation are depicted in figure 1. The directions of the arrows point towards the accessible cell states and the labels give the transition probabilities. We allow arbitrary parameters and introduce as the differentiation probability of cells in compartment i carrying k mutations. Asymmetric cell divisions are not explicitly implemented, as they can be absorbed in the differentiation probabilities on the population level. The fate of a cell's offspring is determined based on the probabilities . Cells proliferate with a rate ri in each compartment i. Usually, cells in upstream compartments proliferate slowly and cell proliferation speeds up in downstream compartments (i.e. ri < ri+1). This general framework is very flexible and different tissue structures can be represented.Figure 1.

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