# A compartment size-dependent selective threshold limits mutation accumulation in hierarchical tissues

^{a}MTA-ELTE “Lendület” Evolutionary Genomics Research Group, H-1117 Budapest, Hungary;^{b}Department of Biological Physics, Eötvös Loránd University, H-1117 Budapest, Hungary;^{c}MTA-ELTE Statistical and Biological Physics Research Group, Hungarian Academy of Sciences, H-1117 Budapest, Hungary;^{d}Evolutionary Systems Research Group, Centre for Ecological Research, Hungarian Academy of Sciences, 8237 Tihany, Hungary

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Edited by Robert H. Austin, Princeton University, Princeton, NJ, and approved December 12, 2019 (received for review August 1, 2019)

## Significance

Renewed tissues of multicellular organism accumulate mutations that lead to aging and cancer. To mitigate these effects, self-renewing tissues produce cells along differentiation hierarchies, which have been shown to suppress somatic evolution both by limiting the number of cell divisions, and thus reducing mutational load, and by differentiation “washing out” mutations. Our analytical results reveal the existence of a third mechanism: a compartment size-dependent threshold in proliferative advantage, below which mutations cannot persist, but are rapidly expelled from the tissue by differentiation. In sufficiently small compartments, the resulting selective barrier can greatly slow down somatic evolution and reduce the risk of cancer by preventing the accumulation of mutations even if even they confer substantial proliferative advantage.

## Abstract

Cancer is a genetic disease fueled by somatic evolution. Hierarchical tissue organization can slow somatic evolution by two qualitatively different mechanisms: by cell differentiation along the hierarchy “washing out” harmful mutations and by limiting the number of cell divisions required to maintain a tissue. Here we explore the effects of compartment size on somatic evolution in hierarchical tissues by considering cell number regulation that acts on cell division rates such that the number of cells in the tissue has the tendency to return to its desired homeostatic value. Introducing mutants with a proliferative advantage, we demonstrate the existence of a third fundamental mechanism by which hierarchically organized tissues are able to slow down somatic evolution. We show that tissue size regulation leads to the emergence of a threshold proliferative advantage, below which mutants cannot persist. We find that the most significant determinant of the threshold selective advantage is compartment size, with the threshold being higher the smaller the compartment. Our results demonstrate that, in sufficiently small compartments, even mutations that confer substantial proliferative advantage cannot persist, but are expelled from the tissue by differentiation along the hierarchy. The resulting selective barrier can significantly slow down somatic evolution and reduce the risk of cancer by limiting the accumulation of mutations that increase the proliferation of cells.

Tumors develop as genetic and epigenetic alterations spread through a population of premalignant cells, and some cells accumulate changes over time that enable them and their descendants to persist within tissues (1, 2). From an evolutionary perspective, each tumor is an independent realization of a common reproducible evolutionary process involving “adaptive” mutations that are preferentially selected by the tumor environment. This process is clonal, which means that a subset of mutations termed “drivers” confer clonal growth advantage, and they are causally implicated in cancer development.

A large body of work (2⇓⇓–5) has focused on understanding clonal evolution of an initially homogeneous population of identical cells, a subset of which progress toward cancer as they accrue driver mutations. Beerenwinkel et al. (6), for instance, considered the Wright–Fisher process (a homogeneous population of initially identical cells) to explore the basic parameters of this evolutionary process and derive an analytical approximation for the expected waiting time to the cancer phenotype and highlighted the relative importance of selection over both the size of the cell population at risk and the mutation rate.

Self-renewing tissues, which must generate a large number of cells during an individual’s lifetime and in which tumors typically arise, comprise a hierarchy of progressively differentiated cells and, as a result, are not homogeneous populations of identical cells. There is empirical evidence (7⇓–9) and theoretical rationale (10⇓–12) that such hierarchical tissue architecture has a profound effect on neoplastic progression. Theoretical work has demonstrated that hierarchically organized tissues suppress tumor evolution by limiting the accumulation of somatic mutations in two fundamentally different ways, as follows.

As described in a seminal paper by Nowak et al. (11), the linear flow from stem cells to differentiated cells to apoptosis in a spatially explicit, strictly linear organization has the property of canceling out selective differences. Nowak et al. considered a system where only asymmetric cell divisions are allowed, that is, after each cell division, one of the daughter cells differentiates to the next level of the hierarchy, pushing all cells at higher levels farther along the hierarchy (Fig. 1*A*). In this idealized construction, mutations, irrespective of how much they increase division rate, are invariably “washed out” unless they occur in the stem cell at the root of the hierarchy. In a more general setting, where symmetric divisions are allowed, the strength of this washing out effect can be quantified by introducing the self-renewal potential of cells. The self-renewal potential is defined as the logarithm of the ratio between the rate of cell divisions that increase the number of cells at a given level of the hierarchy (division producing two cells at the same level) and the rate of events that result in the reduction at that level (division producing two differentiated cells that move higher up in the hierarchy or cell death). In healthy homeostatic tissues, the self-renewal potential of stem cells is zero (corresponding to equal rates of differentiation and self-renewal), while, for differentiated cells, it is always negative, as these cells have an inherent proliferative disadvantage as a result of which they are eventually washed out of the tissue from cells differentiating from lower levels of the hierarchy. In the following, lower (higher) refers to levels closer to (farther away from) the stem cell compartment.

More recently, Derényi and Szöllősi (12) showed that, in self-renewing tissues, hierarchical organization provides a robust and nearly ideal mechanism to limit the divisional load (the number of divisions along cell lineages) of tissues and, as a result, minimize the accumulation of somatic mutations. The theoretical minimum number of cell divisions can be very closely approached: As long as a sufficient number of progressively slower dividing cell types toward the root of the hierarchy are present, optimal self-sustaining differentiation hierarchies can produce N terminally differentiated cells during the course of an organism’s lifetime from a single precursor with no more than

Here, we examine the effect of compartment size by introducing interaction among cells in the form of cell number regulation, which acts on the cell division rates such that the number of cells at each hierarchical level of the tissue has the tendency to return to its desired homeostatic value. We consider a single (non-stem cell) level of the hierarchy that is renewed from below by cell differentiation. We introduce mutants with a proliferative advantage, that is, mutants with a positive self-renewal potential. As detailed below, using both simulations and an approximation adopted from nonequilibrium statistical physics, we find that, under a wide range of parameters, a third fundamental mechanism exists by which hierarchically organized tissues can slow down somatic evolution and delay the onset of cancer.

## Results

We consider level *B* and *C*).

At homeostasis (i.e., when the number of cells,

The particular choice of how the confining potential is distributed between the cell number increasing and decreasing rate modifiers (to satisfy Eq. **2**) has only marginal effect on the dynamics. Here, for simplicity, we make the symmetric choice,

The self-renewal potential of the cells in a healthy homeostatic tissue is defined as

We introduce mutants with an elevated rate of divisions that increase cell number,

Denoting the number of mutant cells by

The continuous interpolation of this potential is shown in Fig. 2, *Bottom* for different parameters. An effective potential, such as Eq. **9**, can always be defined if the mutant and wild-type transition rates depend only on the number of cells of the given type, and if cell number regulation—which acts as a multiplicative modifier of these rates—depends only on the total number of cells (*SI Appendix*).

Here we are concerned with cell number regulation that can be described by a confining potential with a single minimum, for which Eq. **3** is the parabolic approximation. In this case, the presence of size regulation (i.e.,

The characteristic residence time of a population of mutants that have initially spread corresponds to the mean exit time τ of escape from the quasi-stationary state described above. Following the approach described by Gardiner (17) and Derényi et al. (18), an analytical approximation can be derived for τ of the general form (for details, see *SI Appendix*)

Using the mean exit time for escape from the quasi-stationary state, the probability P that a single mutant persists (i.e., first spreads, and then avoids escape) for the lifetime of the individual can be expressed as*A* *Top*, the above approximation for the escape time τ is highly accurate, and it depends very sharply on the selective advantage of mutants. This results in a well-defined threshold selective advantage (Fig. 3*A*, *Bottom*) below which mutants, even if they avoid early stochastic extinction, will rapidly go extinct, that is, will be washed out by cells differentiating from below. Furthermore, the threshold value depends only weakly on the value of β for reasonably strong cell number regulation, that is, for

Realistic values for the rates **1**. Furthermore, in the context of our model, as is apparent on inspection of Ψ, the dynamics does not depend on the absolute rates but only on the ratio **5**).

Fixing *SI Appendix*, Fig. S4, even for strong self-renewal and correspondingly weak washing out, the threshold spreading factor can be large in small compartments.

## Discussion

In classical population genetics models of finite populations, a mutation is either fixed in the population or lost from it within a finite length of time. A fundamental result of population genetics theory is that, in constant populations, mutations with a given selective advantage will avoid early stochastic extinction and fix with a probability independent of population size and proportional to the selective advantage (14⇓–16). As a corollary, in the context of somatic evolution, Michor et al. (19) demonstrated that the accumulation of oncogene-activating mutations (i.e., mutations that provide a proliferative advantage) that occur at a constant rate per cell division is faster in large than in small compartments. Consequently, as pointed out by Michor et al., the classical theory of finite populations of constant size implies that the organization of self-renewing tissues into many small compartments, such as the stem cell pools in colonic crypts, from which the tissue is derived, protects against cancer initiation (5). Further work by Beerenwinkel et al. (6), using qualitatively similar models with a single compartment without differentiation from below, found that the average waiting time for the appearance of the tumor is strongly affected by the selective advantage, with the average waiting time decreasing roughly inversely proportional to the selective advantage. The mutation rate and the size of the population at risk, in contrast, were found to contribute only logarithmically to the waiting time and hence have a weaker impact.

In hierarchically organized tissues with finite compartment size, the situation is more complicated. A mutant that avoids early stochastic extinction and achieves a sizable seemingly stable population can go extinct as a result of differentiation from below. This results in a qualitatively different and more profound ability of smaller compartment size to limit the accumulation of mutations. Similarly to classical population genetics models, the initial spreading probability of a mutation in a compartment of a hierarchical tissue is proportional to the proliferative advantage *A*, the probability of the mutation to persist in the tissue exhibits a threshold that is strongly dependent on compartment size. For small compartments, even mutants with a very large selective advantage will only persist for a very short time; for example, a mutant with a selective advantage of

An important exception is constituted by tissue-specific stem cell compartments residing at the bottom of the hierarchy, such as the stem cells at the bottom of colonic crypts. As these compartments do not receive an influx of cells from lower levels, their dynamics can be described by the classical population genetics models discussed above, that is, mutations can accumulate more easily.

The derivation of the results presented above relies on the existence of the potential defined in Eq. **9**. In our model, this is ensured by the assumptions that 1) the transition rates for cells of each type depend only on the number of cells of that type and 2) cell number regulation acts as a multiplicative rate modifier and depends only on the total number of cells. There are several biologically relevant violations that must be considered. In real tissues, the first assumption, the independence in the absence regulation, is, in general, violated by mutation of wild-type cells into mutant cells (and vice versa), as this increases the number of mutant cells at a rate dependent on the number of wild-type cells (and vice versa). In the context of most, if not all, somatic tissues, the rate of mutations that confer significant selective advantage is sufficiently low that the waiting time between successive mutations is much longer than the relevant time scale of the dynamics considered here; thus, it has a negligible effect on the persistence time, and, as a result, it does not affect our conclusions. This is even more the case for back-mutations from mutants to the wild type. The second assumption, the postulation of a simple form of cell number regulation that acts as a multiplicative modifier and depends only on the total number of cells, is clearly a simplification. It neglects, for instance, explicit spatial organization and any potential long-term memory, such as hysteresis of the homeostatic compartment size dependent on either intrinsic or extrinsic parameters. Such a simplified form of regulation, however, is consistent with more detailed models of homeostatic tissue size regulation, such as recent work on the stability of regulation (20⇓–22) and its optimality in terms of reducing mutation accumulation (23).

In order to quantitatively discuss the biological relevance of our results, we must consider relevant values of two parameters: compartment size (*B*).

At present, systematic data on the selection advantage of mutations in somatic tissues is not available. Vermeulen et al. (7), however, measured the fixation probability of several known drivers of colorectal cancer in the mouse intestine, finding values between 0.4 (Kras

## Methods

Detailed derivation of the results presented above are provided in *SI Appendix*. All data are contained in the manuscript text and *SI Appendix*.

## Acknowledgments

G.J.S. received funding from the European Research Council under the European Union’s Horizon 2020 research and innovation program under Grant Agreement 714774 and Grant GINOP-2.3.2.-15-2016- 00057.

## Footnotes

↵

^{1}I.D. and G.J.S. contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: ssolo{at}elte.hu.

Author contributions: I.D. and G.J.S. designed research; D.G., I.D., and G.J.S. performed research; and D.G., I.D., and G.J.S. wrote the paper.

The authors declare no competing interest.

This article is a PNAS Direct Submission.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.1913104117/-/DCSupplemental.

- Copyright © 2020 the Author(s). Published by PNAS.

This open access article is distributed under Creative Commons Attribution-NonCommercial-NoDerivatives License 4.0 (CC BY-NC-ND).

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