# Organ aging and susceptibility to cancer may be related to the geometry of the stem cell niche

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Edited by* José N. Onuchic, University of California at San Diego, La Jolla, CA, and approved September 28, 2011 (received for review April 15, 2011)

## Abstract

Telomere loss at each cell replication limits the proliferative capacity of normal cells, including adult stem cells. Entering replicative senescence protects dividing cells from neoplastic transformation, but also contributes to aging of the tissue. Recent experiments have shown that intestinal mouse stem cells divide symmetrically, at random make decisions to remain stem cells or to differentiate, and gradually lose telomeric DNA. A cell’s decision whether to differentiate or to remain a stem cell depends on the local cellular and chemical environment and thus tissue architecture is expected to play role in cell proliferation dynamics. To take into account the structure of the stem cell niche in determining its proliferative potential and susceptibility to cancer, a theoretical model is introduced and the niche proliferative potential is quantified for different architectures. The niche proliferative potential is quantitatively related to the proliferative potential of the individual stem cells for different structural classes of the stem cell niche. Stem cells at the periphery of a niche are under pressure to divide and to differentiate, as well as to maintain the stem cell niche boundary, and thus the geometry of the stem cell niche is expected to play a role in determining the stem cell division sequence and differentiation. Smaller surface-to-volume ratio is associated with higher susceptibility to cancer, higher tissue renewal capacity, and decreased aging rate. Several testable experimental predictions are discussed, as well the presence of stochastic effects.

Tissue maintenance and repair of different organs in complex organisms is carried out by a small number of stem cells having the capacity to self-renew for a long time and to produce differentiated cells of the necessary type. There is evidence that these stem cells reside in specialized niches formed by their microenvironment (1). Stem cells must simultaneously maintain their niche, by self-replication, as well as maintain the organ in which they reside by producing differentiated cells. The latter process involves the generation of intermediate semidifferentiated cells (transit-amplifying cells), which after a few divisions produce fully differentiated cells (2, 3) as, e.g., in the colonic crypts of the intestine. Both tasks of niche and organ maintenance are accomplished through the ability of adult-tissue stem cells to produce daughter cells that can both differentiate or both continue as stem cells, or one of them can differentiate while the other can remain stem cell (4) (Fig. 1). Both local environmental chemical gradients and internal mechanisms like the orientation of the stem cell with respect to its microenvironment have been implicated in a stem cell’s decision how to divide. An important constraining factor in stem cell proliferation is the homeostatic requirement that their number stays approximately constant, and deregulation in their self-renewal rate has been implicated in cancer (5). Progenitor cells divide only a few times until they become fully committed differentiated cells with no renewal capacity. In fast renewing tissues like the skin, intestines, and hair follicles, stem cells have to go through many cell divisions during the life of the organism, but the possible number of stem cell divisions is limited by the gradual erosion of their chromosome ends containing telomeric DNA (6, 7).

Telomeres are DNA–protein complexes that protect the ends of linear chromosomes from the DNA repair machinery and chromosome fusion (8, 9). At each cell division, part of each telomere is lost, and cells lacking a mechanism to counter this loss gradually reach a point at which an ataxia telangiectasia-mutated, ataxia telangiectasia-mutated and rad3-related, and/or DNA-protein kinase/retinoblastoma response is activated by one or few critically short telomeres, which in the majority of cases leads to cell cycle arrest, apoptosis, or necrosis (10–13). There is substantial evidence that short telomeres limit a cell’s ability to proliferate and that the gradual telomere shortening in normal somatic cells leads to their finite proliferative capacity (10, 14). During early morphogenesis, telomere loss is compensated by the enzyme telomerase (9, 15), a ribonucleoprotein complex that, using an RNA template, can extend telomeric DNA. Later in life, telomerase is down-regulated and adult somatic cells have little or no telomerase activity (16). Exceptions are the germ line and the stem cell compartments. Although adult stem cells show some telomerase activity, the expression levels seem to be insufficient for telomere maintenance and these cells gradually lose telomere repeats, although slower than the more differentiated progenitor cells. Recently it was demonstrated that Lgr5^{+} intestinal stem cells in laboratory mice express telomerase, but not at sufficiently high level to prevent telomere erosion (17). In mouse models, it has been estimated that the intestinal stem cells divide approximately once per day.

Telomere erosion leads to unprotected telomeres that are recognized as double-strand breaks in need of repair. These critically short telomeres induce a p53-dependent DNA damage response, which leads to cell cycle arrest. Organ depletion of proliferative stem cells reduces the regenerative abilities of the tissue and eventually leads to organ failure (18, 19). Although contributing to aging, proliferative senescence also serves as a cancer protective mechanism by limiting the number of mutations that a cell can acquire as well as protecting cells with critically short telomeres from chromosome instability (20). In laboratory mice, short telomeres do not seem to contribute to aging in the first generation because these mice have been selected with very long telomeres, but this is not the case for the wild type (21).

## Results

Several models of cell division kinetics have been proposed that attempt to reconcile the requirement for many adult stem cell divisions with the slow telomere attrition rate in tissue (22–24). Lobachevsky and Radford (24) proposed the existence of two types of stem cells and proposed a model in which stem cells in the intestinal crypt divide in such a way as to maximize their proliferative potential. Here a model is proposed in which the architecture of the stem cell compartment determines the tissue maintenance potential (TMP) of the niche, which is defined here as the maximum number of progenitor cells produced by all stem cells in the niche. The TMP depends on the pattern of stem cell divisions determined by their sequence of symmetric and asymmetric divisions. Assuming that the decisions made by a stem cell to divide and the decision of its daughter cells to remain stem or to differentiate are based only on the *local* (25) environment, it is shown here that the architecture of the stem cell niche may determine the tissue-maintaining potential of this niche. These decisions are determined in part by local concentrations of mitogens and morphogenes, and in vitro studies have related these concentrations to the local tissue curvature (26, 27). Another finding is that stem cell niches with structure supporting larger tissue maintenance potential are associated with higher probability for the accumulation of mutations in a single cell, and thus stem cell niches with high proliferative capacity are more prone to cancer transformation compared to niches with lower proliferative capacity. A class of niches with clustered stem cells can have very large TMP with slow average telomere attrition rate, but can also be susceptible to the appearance of a stem cell with many mutations. This increase of mutations is because the susceptibility of a cell stem cell clone to the accumulation of multiple mutations depends not only on the presence of mutagenic factors and the efficiency of DNA repair, but also on the number of cell divisions that a clone has undergone.

The dynamics of aging of the stem cell niche can be described generically by first-order kinetics. Such kinetic model with a nonlinear feedback that has a steady-state solution for a fixed number of stem cells was introduced in ref. 5. Here this model is generalized to include stem cells with different telomere length and telomere evolution. The feedback now must be a function of all cells in the niche. If *α*_{1} is the combined rate of apoptosis and senescence, *α*_{2} the rate of differentiation, and *α*_{3} the stem cell self-renewal rate, the cell kinetic equations for the number of stem cells *N*(** λ**,

*t*) with telomere length

**at time**

*λ**t*are [1]

The telomere length of each telomere in a stem cell is represented by a vector ** λ** = (

*λ*

_{1},…,

*λ*

_{n}), and

**b**= (

*b*

_{1},

*b*

_{2},…,

*b*

_{n}) is a vector with components representing the telomere attrition rate of the corresponding telomere in the stem cells. The total number of stem cells is and setting

*b*to zero and integrating over the telomere length leads to the equation in the original model. The generalized model represented by the above equation has a propagating solution with amplitude that saturates at a steady-state value. Without the feedback term, the solution can be written analytically and corresponds to the propagation of the initial distribution of cells with different telomeres toward smaller telomere length. The solution is growing or decaying depending on the sign of

*α*

_{3}–

*α*

_{1}and preserves the initial shape of the stem cell distribution if it is zero. The role of the feedback term is to control the number of stem cells at their homeostatic number. The gradient term on the right-hand side of the equation represents the drift of the cells toward cells with shorter telomeres. This drift also represents the aging of the niche. The constants

*k*

_{0}and

*m*

_{0}in the nonlinear feedback term in principle can be related to experimental data. It might be also expected that the number of stem cells would fluctuate due to stochastic effects. These effects can be included in the model through multiplicative or additive noise. Because stem cells can produce different daughter cells, the model in addition to the nonlinear differential equation described above also needs to specify the order and type of stem cell divisions. Different models of stem cell division kinetics are described below.

The main underlying hypothesis in this work is that a stem cell generates a stem cell and a differentiated cell when it is necessary to produce differentiating cells and that it divides symmetrically only when it needs to renew the stem cell niche. In addition, it is assumed that stem cells “sense” their environment only within its neighboring cells and the supporting microenvironment. The last assumption leads to the conclusion that stem cells that are in contact only with stem cells will not divide when all neighboring cells are stem cells (28, 29). From these two hypotheses, it follows that stem cells on the boundary between the stem cell compartment and the differentiating cells will divide asymmetrically. When a cell on the boundary exhausts its proliferative ability and divides symmetrically into two differentiating cells, a stem cell from the interior divides symmetrically, repopulating the stem cell compartment, then the daughter cell on the boundary divides asymmetrically until it exhausts its proliferative capacity, etc. This process can continue until the last stem cell produces two differentiating cells, ending the ability of the corresponding tissue to renew and repair itself. In Fig. 2, various two-dimensional cell arrangements are shown for the same number of stem cells. From this figure, it can be seen that different cell architectures lead to a different ratio between the boundary and internal stem cells. In addition, different packing structures lead to different numbers of first, second, etc. internal layers of cells and from this, it is expected that these different architectures will lead to different proliferative potentials of the corresponding stem cell compartments. The reduced telomere length of the generated progenitor cells pays the price for the increased TMP of the stem cell niche because the telomere reduction is shifted from the stem cell compartment to the progenitor cells. Because the progenitor cells divide only a few times before full differentiation, their shortened telomeres do not affect the aging of the tissue or its maintenance.

To establish why structural arrangement of the stem cells may influence the number of possible cell divisions, we consider a stem cell niche with a small number of stem cells, producing transit-amplifying cells, that after few cell divisions produce fully differentiated cells. During homeostasis, the number of stem cells is approximately constant and the transit-amplifying cells divide fewer number of times, compared to the proliferative potential set by their telomeres. Stem cells have telomere attrition rates that are slower than those of the semidifferentiated cells. In Fig. 3, two different modes of cell proliferation are shown. In Fig. 3*A*, two stem cells are initially produced and then the two stem cells alternate to divide asymmetrically. If we assume that the initial telomere size of the shortest telomere in the original stem cell is *λ*_{0}, the rate of basal loss is *β*, and the critical telomere length is *λ*_{c}, the number of cell divisions at the time of senescence will be 2(*λ*_{0} - *λ*_{c})/*β*, the factor 2 coming from the fact that we have two stem cells. If we assume that the cell cycle time is *δ*, then the time that these two cells can maintain the niche is 2*δ*(*λ*_{0} - *λ*_{c})/*β*. However, the ability of stem cells to divide symmetrically may lead to a very different strategy, as shown in Fig. 3*B*. Here, the first step is the same: A stem cell divides symmetrically to produce two stem cells (this step in both cases is not important and we may start with two stem cells). However, the next steps are very different. One of the stem cells divides asymmetrically until it exhausts its proliferative capacity and terminally divides into two differentiated cells. Up to this point, as in the first case, there are two stem cells at any time except at the last symmetric stem cell division, when we have only one stem cell. This stem cell now divides symmetrically into two stem cells. One of them divides until it exhausts its proliferative potential, after which it divides symmetrically into two differentiated cells. The other stem cell divides symmetrically into two cells. One of these two cells starts to divide asymmetrically. This process can continue until the remaining stem cell is left with only one possible division and then this stem cell divides symmetrically into two differentiated cells. In this case, the number of stem cell divisions is much larger than in the first case, provided that the number of potential stem cell divisions is much larger than three. If we denote by *n* = (*λ*_{0} - *λ*_{c})/*β*, the proliferative potential of the first stem cell (the number of divisions before the shortest telomere reaches critical length), then in the first case there will be 2*n* cell divisions, whereas in the second the number will be *n* + (*n* - 1) + ⋯+1 = *n*(*n* + 1)/2. If the shortest telomere in a cell is 6,000 bp, the basal loss is 20 bp, and the critical telomere length is 500 bp, then *n* = 275 and thus 2*n* = 550, whereas *n*(*n* + 1)/2 = 37,950. If a cell divides on average every 6 d, the time this cell will take to enter senescence using the first model of stem cell dynamics will be approximately 9 y. However, under the second mode of stem cell dynamics, it will be approximately 607 y! The two tissue maintenance models described here correspond to maximum number of asymmetric and symmetric stem cell divisions, respectively. Both of these models must satisfy the constraint of constant number of stem cells in the niche. It can be expected in general that, when the niche consists of internal, bulk, and surface stem cells, the TMP of the niche is *An*(*n* + *B*). The coefficients *A* and *B* depend on the exact order at which each stem cell divides, an example of which is shown in Fig. 4. However, this number is substantially bigger than the TMP for disconnected niches, in which stem cells are isolated or all stem cells border transit-amplifying cells, which is *Cn*, where *C* is another constant. How may the particular architecture determine the TMP coefficients? In the case of disconnected niches, where different stem cell divisions are not correlated, the coefficient *C* is proportional to the number of stem cells. In the case of connected stem cell niches, the relation between the architecture and the coefficients is complicated. Once the stem cells at the boundary are exhausted, symmetric stem cell divisions of the stem cells in the next layer will replenish the boundary stem cell layer. The ratio between the number of stem cells in the outer layer and the next layer will determine the number of symmetric stem cell divisions per cell that each stem cell from the second layer will need to undergo to replenish the first boundary layer. We will call this number *k*_{1}. Once the first layer is replenished, the proliferative potential of the new boundary layer is reduced by *k*_{1} divisions per cell. Repeating this process, a series of numbers is obtained: *k*_{1}, *k*_{2}, *k*_{3}, etc. The average number of possible individual stem cell divisions of the cells in the corresponding layers is *n* - *k*_{1}, *n* - *k*_{2}, *n* - *k*_{3}, etc. The sum of these numbers each multiplied by the number of cells in the corresponding layer gives the total TMP of the niche. Thus the proliferative architecture of the niche can be specified by a sequence of pairs of numbers (*k*_{i},*q*_{i}), where *q*_{i} is the number of cells in the *i*th layer. The ratio between the number of cells in neighboring layers depends on the dimensionality, the curvature, and the packing structure of the niche. In two dimensions, random packing on the average leads to a triangular random lattice and on the average each cell has 6 first neighbors, 12 second neighbors, 18 third neighbors, etc.(30). These numbers in real tissue are approximate and may fluctuate. To a first approximation, it takes three divisions for the central cell to replenish its first neighbors, one division for these neighbors to renew their neighbors, and a decreasing number of divisions per cell as the layers closer to niche periphery are approached (and thus *k*_{i} < 1 for *i* > 2). This decrease is also the case in three dimensions and is a consequence of the general property that the ratio between the area (volume in 3D) of a surface layer and the area (volume in 3D) of the whole niche (and thus the ratio between the number of cells at the surface and the total number of cells in the niche) decreases as the size of the niche increases. The ratio between the number of cells on the surface and the number of cells inside the niche scales as the number of internal cells to the power -1/3 in three dimensions and -1/2 in two dimensions, and therefore the larger the niche, the larger the TMP compared to the TMP of the same number of unconnected cells. For fractal structures, these powers are different and the surface can reduce the TMP even for large niches. The two models of cell proliferation in niches with different architecture are both captured by the nonlinear feedback model described earlier. The difference is that, for disconnected niches, *N*(** λ**,

*t*) is the total number of stem cells, whereas in connected niches this function represents the stem cells on the boundary. In the second case, once the surface stem cells are exhausted, the bulk stem cells replenish the niche.

The increased ability of connected compared to disconnected niches to produce cells has its price. In such niches, each cell can go through many more cell divisions compared to a stem cell in a disconnected niche and thus it can accumulate many more mutations. For example, if we consider two cells for a fixed time period in a disconnected niche, each of the two cells will undergo *n* divisions, whereas in a connected niche one of the two cells will undergo 2*n* divisions (Fig. 3), and the other will not divide at all, leading to a doubling of the probability for the dividing cell to acquire mutations. For more realistic niches with *N*_{bulk} and *N*_{surface} stem cells that need to generate *N* progenitor cells and renew themselves, on average (*N*_{bulk} + *N*_{surface}) number of cells will undergo *N*/(*N*_{bulk} + *N*_{surface}) cell divisions in disconnected niches, whereas *N*_{surface} number of cells will undergo *N*/*N*_{surface} number of cell divisions in connected niches. More detailed calculations based on stochastic dynamics are presented in ref. 31. The increased probability of acquiring mutations according to the multiple-hit theory of cancer will lead to a decrease in the waiting time to cancer in compact niches. Therefore it may be expected that, given the same mutagenic stress, organs with compact niches will be prone to cancer earlier than organs with disconnected niches. For cancers that are initiated through genomic instability caused by critically short telomeres, the effect might be the opposite. Because in disconnected niches more cells would acquire critically short telomeres earlier compared to compact niches, the probability of a cell to bypass DNA damage checkpoints induced by short telomeres will be higher and thus these niches might be more prone to genomic instability. However, the increase in probability is probably insignificant because, e.g., in fibroblast cultures, one in a million cells bypasses senescence induced by short telomeres and even in epithelial cells the number of cells that bypass senescence is one in a 100,000 (32). These numbers are much larger than the number of stem cells in a typical niche.

The stem cell niches in different tissues have different structure. In the intestine, a small number of stem cells are intermixed with the Paneth cell (33) in each crypt and thus are expected to divide randomly. Recently, random stem cell division was observed in a mouse model (34, 35). The stem cell niche in the hair follicle bulge has different architecture. In this case, the putative niche seems to consist of a large number of stem cells grouped in a two-dimensional sheet (36). Disjoint hematopoietic stem cells have also been identified in bone marrow (37).

The results described here can be observed by the different time course of the total telomere size in tissue during the life of an animal or human. In the case of compact niches in which the stem cells are clustered together, the total telomere DNA will oscillate as time progresses (Fig. 5). The source of these oscillations is the periodic replenishing of the surface stem cells. After a gradual decrease in the total telomere DNA during asymmetric stem cell divisions, a point in time is reached when the dividing stem cells reach critical telomere length and terminally divide into two progenitor cells. At this point, they are replaced by stem cells that divide symmetrically a few times to repopulate the niche (Fig. 4) and thus replenish the telomere DNA in the tissue. In Fig. 5, the time course of the total telomere DNA of stem cells, progenitor cells, and differentiated cells in a colonic crypt is shown for the two stem cell division models in two different architectures of the stem cell niche. The different time course of the total telomere DNA in the two models of tissue maintenance can be assessed in animals using Southern blotting techniques or more sophisticated techniques like fluorescent in situ hybridization (38). In the mouse intestine, it was recently observed (34, 35) that all stem cells divide symmetrically at random. Laboratory mice have very long telomeres (> 20 kbp) and if a stem cell loses 20 bp per cell division, 15,000 bp will be lost in 25 mo. However, human telomere length at birth is approximately 10,000 bp and, at such high turnover rate, human stem cells will reach senescence during the first few years of life! Thus it may be expected that human cell turnover is slower than cell turnover in mice.

Stem cell niches contain a small number of stem cells compared to the number of cells in the organ. From a statistical point of view, one might expect that the fluctuations of the mean number of stem cells will be large. In a purely random ensemble, one expects the fluctuations to be proportional to the inverse square root of the number of entities. For example, in the colonic crypt, it has been estimated that the number of stem cells is around six. Thus the fluctuations are expected to be on the order of 7%. In addition, noise is common in living systems (39) and thus stem cells might make “mistakes” in their decisions to differentiate or not. These stochastic effects need to be taken into account in determining the proliferative potential of the stem cell niche. To address the issue of noise, an agent-based model was developed and the cell proliferation dynamics of a model stem cell niche was simulated on a computer. The model consists of a two-dimensional square lattice with 400 cells. The central 36 cells were selected to be stem cells at the beginning of each run. Fixed boundary conditions were used. The initial telomere length was chosen to be 10,000 base pairs and the basal telomere loss at each cell division was chosen to be 100 base pairs. The critical telomere length at which cells stop dividing was chosen to be zero. Cells were assumed to divide every 6 d. At each stem cell division, the daughter cells could randomly choose its fate. The probability for symmetric stem cell division into two stem cells was compensated with the same probability for symmetric stem cell division into two differentiated cells to keep the number of stem cells approximately constant. Stem cells surrounded only by stem cells did not divide. The results of the modeling are presented in Fig. 6. In Fig. 6*A* (*Inset*), the stem cell niche geometry is displayed. The average proliferative potential dependence on the probability for symmetric stem cell division is shown in Fig. 6*A*. The average proliferative potential drops fast from its value for purely deterministic dynamics. For purely deterministic dynamics, the stem cells can divide approximately 200,000 times. This number is approximate because different order in the stem cell divisions is possible and is chosen here at random. The fast drop in the presence of even small noise occurs because the stochastic stem cell decisions can lead less-connected niches to become more disconnected. Because one connected stem cell can repopulate a niche of 36 cells (as the one used in Fig. 6) in less than six cell divisions, moving away from the rest of the stem cells leads to a reduction of the total niche proliferative potential approximately by a factor of the order of the proliferative potential of this cell. Thus the separation of one cell from the niche leads to approximately 200,000/100 = 2,000 potential stem cell divisions of the niche. The lowest number of possible stem cell divisions is 100 and corresponds to a fully disconnected niche (i.e., when all stem cells’ neighbors are only differentiated cells). Surprisingly, the proliferative potential displays a power law dependence on the noise level (Fig. 6*B*). Although the proliferative potential decreases when noise is present, it is still 10–100 times larger for moderate values of the noise compared to the case when the stem cells are distributed randomly in the niche and divide asymmetrically at random.

## Discussion

Here two deterministic models of stem cell proliferation that lead to a very different tissue maintenance potential were described. A simple relation between the tissue maintenance potential of a stem cell niche and the proliferative potential (PP) of the individual stem cells comprising the niche was established. Here the language of symmetric and asymmetric divisions is used, but it is possible that there is only one type of stem cell division and that the daughter cells make a “decision” (perhaps at random) whether to differentiate or to remain stem cells. Here these two descriptions are used interchangeably because, in the presented model, they are equivalent. If the stem cell number in the niche can fluctuate, an increase in the stem cell number will increase the tissue maintenance potential and the increase will be larger the longer the telomeres of the new stem cells are. Another way that a stem cell niche can obtain an enhanced TMP is if the probability for symmetric divisions is higher than the probability for cell division into two stem cells. The constraint that the number of stem cells in the niche is approximately fixed will then lead to an enhanced TMP. This mechanism works because the transit-amplifying cells divide only a few times (approximately six times in the mouse intestinal crypt), much less than their proliferative limit set by their telomeres. If these semidifferentiated cells were dividing until they exhaust their telomeres, the number of cells produced by the two modes of cell division dynamics would be the same. In real tissue, it is not clear that these idealized cases are employed.

During carcinogenesis, the majority of tumors acquire infinite or very large PP (the maximum number of cell divisions a cell can undergo before entering replicative senescence) by reactivating telomerase (40). In some cancers, there is no detectable telomerase and their cells use an alternative lengthening of telomeres (ALT) mechanism for telomere maintenance (41–43). ALT is believed to be recombination-based. In addition, in some cases neither telomerase nor ALT have been found (44–46) and the acquisition of a very large proliferative potential is a mystery. If cancer maintaining cells can explore the mechanism described above (47), it is possible that some cancers might proliferate in the absence of telomerase or ALT. In the simulations presented here, the probabilities for symmetric stem cell division into two stem cells or two differentiated cells were chosen equal to balance the two processes. It is possible that, during carcinogenesis, the process of symmetric division into two stem cells acquires higher probability and disrupts homeostasis in the niche. Indeed, recent studies have shown an increase in the stem cell marker Lgr5 in cancer (48), which could be an indication of this conjecture. An alternative explanation may be that the cancer stem-like cells divide more frequently.

The topology and geometry of the stem cell niche, defining its architecture, may also influence the genotypic and/or phenotypic diversity of the tissue. If stem cells predominantly divide when they are in contact with progenitor cells, it can be expected that the niche surface will drive the diversity of the organ. For a niche with small surface area (length), genetic drift may dominate the genetic diversity of the tissue and a monoclonal tissue may be expected. The architecture of the stem cell niche also may influence the phenotypic diversity and structure of tissue through the spatial and temporal distribution of biochemical gradients in the tissue. The phenotypic diversity in cancer is poorly understood. A recent study (49) suggests that cancer cells can switch between different phenotypes, but the role of the tumor architecture is not well understood. Studies of genetic switches in cell populations indicate that the response of the switches depends on the environment, but little is known how different geometries influence this response (50).

## Materials and Methods

Computer simulations of the agent-based model of the stem cell niche were performed on an Apple computer with an Intel duo core processor using the C++ programming language. The model consists of a 20 × 20 square lattice of cells. Two numbers describe each cell: the length of the shortest telomere and its type (zero if it is differentiated cell and one if it is a stem cell). We assume that all telomeres loose 100 bp per cell division, so the shortest telomere is the one that defines the cell’s proliferative potential. One thousand independent runs were averaged to obtain the average PP of the model stem cell niche. Each run starts with choosing a 6 × 6 array of stem cells in the center of the 20 × 20 lattice of cells. The rest of the cells are differentiated and thus the value of the site type is zero. As time progresses, each differentiated cell divides symmetrically into two differentiated cells and loses 100 bp at each cell division. Each stem cell can divide symmetrically or asymmetrically, as shown in Fig. 2. Each stem cell makes a decision to divide and how to divide based on the types of its nearest neighbor cells. If a cell is surrounded only by stem cells, it does not divide. If any of the stem cell’s nearest neighbors is a differentiated cell, the stem cell will randomly choose one of the differentiated cell neighbors and will divide asymmetrically losing 100 bp of telomeric DNA. Noise in the system is introduced by making the stem cell division decisions stochastic. With a small probability, a stem cell can divide into two stem cells (i.e., the type of the lattice site where the stem cell resides remains one and the chosen type of the lattice site occupied by a differentiated cell changes from zero to one) if at least one of its neighbors is a differentiated cell, even when it is surrounded not only by differentiated cells. If only this process is allowed, the stem cell population will grow without limit. Thus any stem cell (unless surrounded only with stem cells) with a small probability can also divide symmetrically into two differentiated cells, even if its telomere is above the critical value (i.e., the type of lattice site where the stem cell resides changes from one to zero and the type of the chosen lattice site occupied by a differentiated cell remains zero). The standard deviation of the average proliferative potential as well as the average number of stem cells and the corresponding standard deviation also computed as well.

## Acknowledgments

The author thanks Herbert Levine and Kamal Shukla for carefully reading the manuscript and for the useful suggestions. This work was supported by the National Science Foundation.

## Footnotes

- ↵
^{1}E-mail: kblagoev{at}nsf.gov.

Author contributions: K.B.B. designed research, performed research, analyzed data, and wrote the paper.

The author declares no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.

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