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Conceived and designed the experiments: BW DD TL JMB AT. Performed the experiments: BW AT. Wrote the paper: BW DD TL JMB AT.

The authors have declared that no competing interests exist.

Most tissues in multicellular organisms are maintained by continuous cell renewal processes. However, high turnover of many cells implies a large number of error-prone cell divisions. Hierarchical organized tissue structures with stem cell driven cell differentiation provide one way to prevent the accumulation of mutations, because only few stem cells are long lived. We investigate the deterministic dynamics of cells in such a hierarchical multi compartment model, where each compartment represents a certain stage of cell differentiation. The dynamics of the interacting system is described by ordinary differential equations coupled across compartments. We present analytical solutions for these equations, calculate the corresponding extinction times and compare our results to individual based stochastic simulations. Our general compartment structure can be applied to different tissues, as for example hematopoiesis, the epidermis, or colonic crypts. The solutions provide a description of the average time development of stem cell and non stem cell driven mutants and can be used to illustrate general and specific features of the dynamics of mutant cells in such hierarchically structured populations. We illustrate one possible application of this approach by discussing the origin and dynamics of PIG-A mutant clones that are found in the bloodstream of virtually every healthy adult human. From this it is apparent, that not only the occurrence of a mutant but also the compartment of origin is of importance.

We investigate the average stem cell driven dynamics of cell counts in an abstract multi compartment model. Within this framework one can represent different tissue structures, as for example hematopoiesis, the skin or the colonic crypt. Our analysis is based on an individual cell model in which cells can differentiate, reproduce or die. We give closed solutions to the corresponding system of coupled differential equations, that describe the average dynamics of all cell types. There are three cases of interest: (i) Mutations at the stem cell level, (ii) Mutations in downstream compartments associated with more mature, non stem cell types, (iii) Mutations in downstream compartments with cells acquiring stem cell like properties. The average dynamics shows for (i) and (iii) an increase of mutants towards an equilibrium, in case (ii) the average mutant cell count goes through a maximum, but mutants die out in the long run. We calculate the corresponding extinction times for every compartment. We discuss applications to hematopoietic diseases such as, PIG-A mutant cells or the classic oncogene BCR-ABL. Although the abstract model is a simplified sketch of cell differentiation, it is capable of describing many aspects of a wide variety of such tissues and associated diseases.

Many tissues have a hierarchical multi compartment structure in which each compartment represents a cell type at a certain stage of differentiation. This architecture has been well described for hematopoiesis

a) We consider three possible events during the cell division of a non-stem cell. Cells can differentiate, die, or reproduce. This happens with probabilities

This model does neglect several aspects that may have an impact on the dynamics of the system under consideration, such as biochemical feedback or spatial population structure

One special case of our framework is the model of hematopoiesis discussed in

The individual cell model is based on a finite number of cells that divide and differentiate with certain probabilities. Thus, it is a stochastic process

Let us assume that the number of stem cells

The simulations presented in this paper are individual based stochastic simulations. We implement all elements of the first

The equilibrium of the process is obtained from setting the left hand side of our system of differential equations to zero. Biologically, this corresponds to tissue homeostasis. In this case, we have

Next, we turn to the process of filling empty compartments by a continuous influx from the stem cell pool. Because we do not consider interactions between different cell clones in our differential equations, this corresponds also to the dynamics of a mutation arising in the stem cell pool. Thus, we choose the initial condition

If we choose (i) an exponentially increasing proliferation rate

In

The colored symbols are averages of an individual based stochastic simulation with

Solution (6) describes the deterministic process of filling empty compartments within hierarchical organized tissue structures, as can occur during wound healing, recovery from hematopoietic stem cell transplantation

Next, we turn to mutations occurring downstream of the stem cell compartment. The occurrence of a mutation in a non stem cell compartment is more likely than a mutation in the stem cell pool due to the higher numbers and proliferation rates of non stem cells. The dynamics of such a mutant is not driven by the stem cell pool and thus is not described by the solution form above, equation (6). However, the compartment structure is unchanged and thus the dynamics of such mutants is also described by equations (1a)–(1c), but with altered initial conditions. Assuming there is a mutation in compartment

a) A single mutant occurred in the compartment

a) Average number of mutants in the first 31 compartments of a hematopoiesis model (

Based on equation (8), other mutant dynamics are also possible. If

In the long run the average mutant cell count is given by the dynamics of the slowest decaying exponential function of equation (8). It is often natural to assume that this corresponds to the dynamics in the compartment of the mutant origin

a) The black lines show the average mutant cell count based on equation (11) in the compartments

A special case of interest is a mutation with

Here, we will utilize the model to illustrate the dynamics of a mutation that is seen in virtually every healthy human being. Sensitive flow cytometric analysis of circulating blood cells will identify a small clone that lacks expression of CD55 and CD59 (amongst others)

The model parameters were fixed to represent hematopoeisis following

In Araten et al.

Panels a) to c) show the number of PIG-A mutants per million healthy cells in compartment

In

Moreover the hierarchical structure of hematopoiesis provides an explanation why almost all humans carrying PIG-A mutations do not have symptoms of PNH. Only mutations in the most ‘primitive’ compartments have an impact and only mutations in a HSC will lead to disease. In general, one can predict the dynamics for mutants with very different properties using equation (8). The compartment of the mutant origin can be inferred if one follows the mutant count by taking blood samples at regular intervals.

In this work, we presented closed analytical solutions for the deterministic dynamics of stem cell and non stem cell driven mutants in a multi compartment model of tissues such as hematopoiesis, the skin and the colon. This enables us to describe the dynamics of mutant cells in a general approach. We can predict the time development of a mutant depending on its origin and its specific proliferation properties. The process of cell differentiation is conceptually fairly well understood, but it is of course a challenge to estimate the various parameters in our model for real systems. Fortunately, very often, simplifying assumptions, e.g. exponentially increasing cell proliferation rates, can lead to insights

Let us turn to hematopoiesis to address some of the implications of our model because recent technological developments allow the detection of well known mutations in many otherwise healthy people. Perhaps the best examples are derived from blood disorders, since repeated blood sampling is a minimal invasive procedure and molecular probes for many blood disorders are available. The case of PIG-A mutant cells present in healthy humans has been analyzed extensively in an earlier section. There are several other specific examples

A mutation in the janus like kinase 2 where phenyalanine substitutes valine (JAK2V617F) is a common mutation in patients with chronic myeloid neoplasms. However, one can find this mutation in a substantial fraction of healthy adults (perhaps 0.2–0.4 percent) and with an even higher frequency (0.94 percent) in hospitalized patients who do not have an overt hematologic disorder

Finally, the classic oncogene BCR-ABL

We can also think of other mutations altering cell division properties. For instance, one can consider a mutation occurring in compartment

Our model provides a mathematical abstraction of hierarchically structured tissues and neglects many factors that can have an important impact on the dynamics, as for example spatial population structure or temporal changes of cell division properties, e.g. due to aging or injury. Nonetheless, the most important aspects of such tissue structures are captured by our model. It takes the form of ordinary differential equations that allows analytical solutions in many cases. An alternative would be a numerical solution, but such a solution has to be implemented for specific sets of parameters. We are convinced that our model can readily be applied to various hierarchical tissues and expect that general features of mutant dynamics will be conserved across different tissues.

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