Division of Hematology, Mayo Clinic College of Medicine, 200 First Street SW, Rochester, MN 55905, USA

Department of Molecular Medicine, Mayo Clinic College of Medicine, 200 First Street SW, Rochester, MN 55905, USA

Departamento de Matematica e Aplicações, Universidade do Minho, 4710-057 Braga, Portugal

ATP-Group, Centro de Matemática e Aplicações Fundamentais, Complexo Interdisciplinar, 1649-003 Lisboa codex, Portugal

Abstract

Stem cells are the target of mutations that can lead to life threatening diseases. However, stem cell populations tend to be small and therefore clonal expansion of mutant cells is highly sensitive to stochastic fluctuations. The evolutionary dynamics of mutations in these cells is discussed, taking into consideration the impact of such mutations on the reproductive fitness of cells. We show how stochastic effects can explain clinical observations, including extinction of acquired clonal stem cell disorders.

Cancer is a consequence of multicellularity due to the fact that the cellular genome is under continuous attack from a variety of environmental or metabolic genotoxic agents. Moreover, the DNA replication machinery is not perfect

Stochastic dynamics of stem cells

For practical purposes, it is generally accepted that one can consider the number of stem cells contributing to a given tissue (for example, hematopoiesis) as constant (_{M}

The basic principles of the Moran process

**The basic principles of the Moran process**. The Moran process assumes that, over short periods of time, the total population of cells is constant. **(a) **To start with, a cell is selected for reproduction. Selection is dependent on the frequency of the cell in the population and its reproductive fitness (**(b) **The cell divides, and the number of cells increases by one. **(c,d) **Therefore, another cell is selected for export **(c)**, which returns the population to its normal level **(d)**.

Evolutionary dynamics under the Moran process

**Evolutionary dynamics under the Moran process**. **(a) **When normal cells divide, there is a probability **(b) **Cells have a relative reproductive fitness **(c) **The probability that a cell is chosen for reproduction (_{M }_{N }

where

If this process continues for a very long time, the result will ultimately be a state where either all the cells are normal (extinction, Figure

The outcomes of Moran dynamics

**The outcomes of Moran dynamics**. **(a,d) **Assuming that a mutant stem cell is present, stochastic dynamics will predict extinction of the mutant cell **(a) **or fixation **(d)**. **(b,c) **However, fixation may require a long time - hence the clone may persist in a latent state (no disease) **(b) **or could reach a threshold leading to a disease state **(c)**. At any of these steps, stochastic extinction is still possible, although less likely as the burden of mutant cells increases. Once the mutant clone reaches fixation **(d) **this is irreversible. Hence, the only two stable states are extinction or fixation.

In several bone marrow disorders, disease requires that a specific threshold of mutant stem cells is surpassed: at least 20% of the bone marrow cells have to be blasts to diagnose acute leukemia, and 10% clonal plasma cells are required to diagnose multiple myeloma

where _{0 }is the initial number of mutant cells and _{1 }is the number of mutant cells needed to reach the threshold

Probability distribution functions to reach the diagnostic threshold

**Probability distribution functions to reach the diagnostic threshold**. Stochastic simulations of Moran dynamics, recording the probability that the mutant clone reaches the diagnostic threshold at a given time after the occurrence of the mutation (diagnosis is here defined as at least 20% of the cells being mutated) as a function of the fitness advantage (

The Moran model assumes homogenous mixing of populations; that is, the spatial distribution of the population is not considered. This means that one cell in a specific place reproduces and a cell elsewhere is chosen for death, a scenario perhaps easiest to accept in small populations of stem cells, such as in an individual colonic crypt

Application of stochastic dynamics to disease modeling: clonal expansion

Reproduction of mutant cells leads to their expansion into a clone. How frequently the cells are chosen for replication depends on their relative frequency in the population and on their relative reproductive fitness (Figure

Expression of

Stochastic dynamics of latency

It is not uncommon to find mutations in disease-associated genes in healthy adults. For example, almost every human has a

Stochastic extinction

The stochastic dynamics associated with the Moran process predicts the possibility of extinction of any mutant clone, irrespective of its relative fitness. This phenomenon, however, becomes quite likely when the mutation either gives a relative fitness disadvantage or is neutral. It is important to remember that even for mutations that increase the reproductive fitness of the cell, extinction is still possible. One can computationally show that even for a mutation that increases the relative fitness of cells by a factor of two, which, in evolutionary terms, constitutes a very high fitness advantage, the probability of extinction of the mutant cell is still about 50%

However, stochastic extinction is not just an

We wish to remind the reader that, in principle, other potential explanations, alternative to stochastic extinction, could exist for the disappearance of mutant clones. For example, our model does not consider the potential impact of immune surveillance and elimination of tumor cells by the immune response. There is some evidence for this phenomenon, especially in the context of allogeneic stem cell transplantation. However, elimination of a mutant clone due to an autologous immune attack is also possible. These scenarios are not mutually exclusive and perhaps either occurs in a subset of patients.

Conclusions

Stochastic behavior is an intrinsic aspect of life - both at the cellular and organismal level. Acquisition of mutations and the evolution of clones are processes that are highly sensitive to stochastic effects since the population under consideration is often small. Mathematically, it is as if cells play dice - many times the impact of such a gamble is inconsequential, at other times it leads to the loss of a particular cell lineage. Unfortunately, there are times when acquired mutations lead to clonal expansion and disease. Depending on the fitness of the mutant clone, stochastic extinction is possible, especially when the population is small and the relative fitness advantage minimal compared to normal cells but also even when the population is not small or the fitness advantage is significant. It is in these circumstances that chance may play a role in cure versus disease. Reduction in disease burden to low levels could, in principle, lead to clonal extinction, especially if the fitness advantage of the mutant cells can be reduced by therapy.