Determinants of stem cell enrichment in healthy tissues and tumors: implications for non-genetic drug resistance

Drug resistance is a major challenge for cancer therapy. While resistance mutations are often the focus of investigation, non-genetic resistance mechanisms are also important. One such mechanism is the presence of relatively high fractions of cancer stem cells (CSCs), which have reduced susceptibility to chemotherapy, radiation, and targeted treatments compared to more differentiated cells. The reasons for high CSC fractions (CSC enrichment) are not well understood. Previous experimental and mathematical modeling work identified a particular feedback loop in tumors that can promote CSC enrichment. Here, we use mathematical models of hierarchically structured cell populations to build on this work and to provide a comprehensive analysis of how different feedback regulatory processes that might partially operate in tumors can influence the stem cell fractions during somatic evolution of healthy tissue or during tumor growth. We find that depending on the particular feedback loops that are present, CSC fractions can increase or decrease. We define characteristics of the feedback mechanisms that are required for CSC enrichment to occur, and show how the magnitude of enrichment is determined by parameters. In particular, enrichment requires a reduction in division rates or an increase in death rates with higher population sizes, and the feedback mediators that achieve this can be secreted by either CSCs or by more differentiated cells. The extent of enrichment is determined by the death rate of CSCs, the probability of CSC self-renewal, and by the strength of feedback on cell divisions. Defining these characteristics can guide experimental approaches that aim to screen for and identify feedback mediators that can promote CSC enrichment in specific cancers, which in turn can help understand and overcome the phenomenon of CSC-based therapy resistance.


Introduction
Healthy tissue is maintained by a small population of tissue stem cells, which self-renew and give rise to transit amplifying cells and differentiated cells that make up the majority of the cell population [1]. Homeostasis is ensured by complex regulatory mechanisms including negative feedback loops [2,3]. Stem cells are thought to be an important target for oncogenic transformation [4], which results from the breaking of homeostatic mechanisms. As the tumor grows, it maintains basic features of the underlying tissue hierarchy [5]. Thus, the bulk of the tumor is made up mostly of more differentiated tumor cells, but maintenance of the tumor is thought to depend on a relatively small fraction of cancer stem cells (CSCs), characterized by the presence of specific markers. The exact proportion of cancer stem cells, however, can vary from tumor to tumor, and, within a tumor, over the time course of growth. CSCs tend to be present in small proportions during earlier phases of tumor growth, and tend to make up larger fractions as the disease progresses, [6,7], a process that can be called CSC enrichment. In addition, therapy can have a significant impact on the CSC fraction [8,9]. Cytotoxic chemotherapeutic agents and radiation kill proliferating DC tumor cells more effectively than quiescent CSCs, and the same has been reported for targeted treatments with small molecule inhibitors [10][11][12]. Following treatment cessation, re-generation of the differentiated tumor cell population from the activated CSCs can restore the tumor composition with a majority of differentiated cells, although there is indication that the fraction of CSCs can in some cases remain permanently elevated post-compared to pre-treatment [13].
Since CSC are less susceptible to therapy than more differentiated cells, the fraction of CSCs in the tumor can be an important determinant of the response to cancer therapy. In particular, a relatively high CSC fraction at the time of treatment can be a cause of non-genetic drug resistance, where a loss of response is observed despite the absence of any known drug-resistant mutants.
Thus, in bladder cancer, it has been argued that CSC enrichment might account for the progressive loss of treatment response with repeated cycles of chemotherapy [8]. In chronic myeloid leukemia, high fractions of undifferentiated cells, especially during advanced stages of progression, might contribute to the phenomenon of primary resistance against tyrosine kinase inhibitors [14]. In mouse mammary tumors, CSC enrichment has been shown to contribute to chemotherapy resistance [9]. Hence, understanding the conditions under which stem cell enrichment, and hence high fractions of CSCs, are achieved is paramount to finding strategies to overcome this form of resistance.
The factors that determine the fraction of stem cells in healthy tissues and tumors are not well understood, and neither are the factors that contribute to CSC enrichment during tumor growth, or to sustained CSC enrichment following treatment cycles. Recent experimental and mathematical work implicates the presence of feedback regulatory loops in the process of stem cell enrichment in tumors. For example, a PGE2-mediated wound-healing type response has been documented in bladder cancer that can result in CSC repopulation during chemotherapy [8,15]. According to this mechanism, chemotherapy-induced death of dif-ferentiated cells results in the release of positive feedback signals that drive CSCs into a state of proliferation. Mathematical modeling work has shown, however, that CSC enrichment cannot be sustained after treatment cessation by the wound-healing response alone, but might further require the presence of negative feedback loops within the tumor [13]. It was shown mathematically that the presence of negative feedback from differentiated cells onto the rate of CSC division promotes sustained CSC enrichment post therapy, while this is not observed in the absence of negative feedback [13]. The model suggests that enriched CSC populations are only maintained post-therapy if feedback mechanisms also act during untreated tumor growth.
The above-mentioned positive feedback wound-healing mechanism during chemotherapy is to our knowledge the only regulatory feedback loop that has been identified experimentally in tumors. Analysis of untreated tumor growth dynamics, however, has suggested that negative feedback processes might play an important role as well [16,17]. Experimentally, negative feedback has been reported, e.g., in the olfactory epithelium, where GDF11 and Activin βB negatively regulate cell division rates in progenitor and stem cells [2,18]. Another prominent example of a feedback implicated in carcinogenesis is the dual tumorpromoter/suppressor role of TGFβ signaling [19], whereby in early-stage tumors, it promotes cell cycle arrest and apoptosis, and at advanced stages, the TGFβ pathway in fact enhances tumor progression by promoting cancer cell motility, invasions, and "stemness" [20][21][22] . Therefore, further investigation is warranted into the role of feedback loops for stem cell enrichment, which is the subject of our study.
In our previous work, we only focused on one particular negative feedback loop and showed that it can contribute to stem cell enrichment, i.e. negative feedback from differentiated cells onto the rate of stem cell division. There are, however, a large number of possible negative and positive feedback loops that can potentially influence the proportions of stem cells and differentiated cells in healthy and tumor tissue. These include negative or positive control of rate functions (such as division rate and death rate) and differentiation probability by stem cells, differentiated cells, or both, see [23][24][25]. The question thus arises, which of these feedback mechanisms can drive stem cell enrichment, and which cannot.
Finding an answer to this question is the subject of our investigation and might help to find treatment methods that can sensitize tumors to chemotherapies and targeted therapies.
We focus on two basic scenarios. First, we assume that partial breakage of feedback regulation results in temporary cell growth towards a new equilibrium, characterized by an overall larger number of cells. This could correspond to a single step in step-wise tumor progression. We investigate the conditions required for the stem cell fraction to be larger at the new compared to the old equilibrium. In particular we study how remaining feedback loops determine the stem cell fraction. Second, we consider unbounded tumor growth and investigate how different feedback mechanisms that remain in a growing tumor cell population can determine whether or not CSC enrichment occurs during growth. Much of this work is done using ordinary differential equations. In the context of unbounded tumor growth, the effect of spatial growth patterns on stem cell enrichment is explored using a stochastic agent-based model and an analytical approximation.

The basic mathematical modeling approach
An ordinary differential equation model has been used to describe tissue hierarchy dynamics in a healthy tissue [2,26], and the models presented here build on these approaches. While cell lineages consist of stem cells, transit amplifying cells, and terminally differentiated cells, our models make a simplification and take into account only stem cells (which encompass all the proliferating cells) and It has been shown that introduction of negative feedback loops can result in more realistic behavior, where a stable equilibrium is attained for P>0.5 [2].
This was shown in the context of two specific feedback loops, and subsequently generalized to comprehensively list all possible (positive and negative) feedback loops compatible with stability [23][24][25] . Here, we also use a general model to assume different kinds of feedback on the rate of cell division, L, the rate of cell death, D, and the probability of self-renewal, P. We also add the possibility that stem cells die with a rate δ (which can also be subject to feedback). In the context of our model, feedback is equivalent to a dependence of rates and probabili-ties on the population sizes, x and/or y. Hence, the model is given by the following ODEs: The division rate L, the death rates, D and δ, and the probability of self-renewal, P, are now functions of either the number of stem cells, x, or the number of differentiated cells, y, or both.
Evolution can result in the generation of mutant cell populations that are characterized by a higher self renewal probability, given by P 2 . Hence, we now have two stem and differentiated cell populations denoted by subscripts 1 and 2 for wild type and mutant types, respectively. The equations are thus given by where x=x 1 +x 2 and y=y 1 +y 2 . The two cell populations are in competition with each other, mediated by the feedback factors that are shared between the two populations. Table 1 summarizes all the variables used in this paper (both in this section and in the later sections).

Cell growth towards a new equilibrium.
The first important scenario happens when the mutant cell population gains a selective advantage, outcompetes the original, healthy cell population, and grows towards a new and higher equilibrium level. This is achieved by assuming that P 2 >P, along with other conditions on the rate functions that are specified in Section 1 of the Supplement. We examine the conditions under which the stem cell fraction at the new equilibrium is increased compared to that at the original equilibrium. In terms of the model notation, we are interested in the quantity v(x,y) = x / y, i.e. the ratio of stem to differentiated cells. We investigate how different combinations of feedback loops that remain in the mutant cell population impact stem cell enrichment.
In the following we assume that the division and death rates in the above models are monotonic functions of the number of stem and/or differentiated cells. We further assume that in the absence of mutants, the system is at equilibrium, characterized by the pair (x , y) , which satisfies: At this equilibrium, the fraction of stem to differentiated cells is given by We assume that the mutant cell population can invade from low numbers and displace the original cell population. This occurs if P 2 (x,y) > P(x,y) (this is a sufficient condition). Assuming that a new equilibrium is reached, it is characterized

and the new fraction of stem cells is given by
We examine under what conditions ν * >ν , i.e. when the stem cell ratio at the newly obtained mutant equilibrium exceeds that at the original equilibrium.
We observe two qualitatively distinct outcomes, see We will illustrate these points by using some specific examples. The first example is of SC enrichment. Consider a system where δ=0, the self renewal probabilities P and P 2 are given by decreasing functions of y (see solid lines in Fig 1(a)), and the division rate L is also a function of y, solid line in Fig 1(b). The cell dynamics and control loops corresponding to this system are schematically shown in Fig   2(a). In this and other such diagrams, blue arrows correspond to cellular processes (characterized by kinetic rates, such as L and D), and cell fate decisions, which are probabilities (P or P 2 ). For example, the horizontal arrow connecting a circle marked "SC" with the circle marked "?" denotes a SC division. The question mark reminds the reader that a cell fate decision needs to be made, whether to self-renew (a blue arrow going back to SC, marked with probability (1-P)), or to differentiate (a horizontal arrow toward a circle marked with DC, probability P).
This notation allows us to show precisely which populations control which processes. In Fig 2(a), the red negative arrows originating in the DC circle represent a negative dependence of both functions L and P on y (the number of DCs).
In this first example, SC enrichment is predicted to occur. Fig 2(b) shows the cell dynamics, once a mutant is introduced at 100 time units. While the wild type population goes extinct, the mutants rise to a new equilibrium characterized by a significantly higher ratio x/y compared to the original equilibrium.
The second example is SC depletion. It is given by a system with rate functions given by Fig 1(c,d), with δ=0 and the division rate positively controlled by the SC population. The controls are schematically shown in Fig 2(c). The resulting dynamics are presented in Fig 2(d), where the proportion of SCs at the new, mutant equilibrium is smaller than the original proportion. Note however that the dependence of x/y on time is non-monotonic and a temporary phase of SC enrichment is experienced before the ratio x/y lowers to its long-term level.
These two examples illustrated in  Fig 2(a). First, we consider the death rate of the stem cells, δ. The amount of enrichment is smaller for nonzero stem cell death rates, compared to the case of δ=0. In Fig 3(b) we can see that in the presence of SC death, the increase in the enrichment parameter, x/y, is more modest than that of Fig 2( The second modification is a smaller self-renewal probability, P 2 , of the mutant cell population, Fig. 3(c). The self-renewal probability P 2 of the mutant cells is that given by the dashed line in Fig 1(a). Again, this results in a more modest increase in the stem cell fraction compared to Fig 2( The third modification is a less pronounced feedback on the division rate. The result is that the SC enrichment becomes smaller. In

Stem cell enrichment in non-equilibrium situations
Next, we study the scenarios where the mutant population grows from low numbers and does not reach a new equilibrium, but instead, continues to grow indefinitely. This would correspond to tumor growth. The definition of enrichment in this context and the relevant methodology will be somewhat different in this case. We will consider the mutant population alone, and study the growth of x 2 and y 2 , in order to find the dynamics of the quantity ν=x 2 /y 2 . We will say that stem cell enrichment occurs if the quantity ν increases during population growth, either infinitely, or temporarily. For simplicity we will assume that the cancer stem cell (CSC) population does not die (δ=0). We further assume that the probability of self-renewal of mutants, P 2 , is a monotonic function of the population size (stem cells or differentiated cells) that satisfies P 2 > 1/2, and that in the limit of large populations, it approaches a limiting value, P > 1/2. When we consider the mutant dynamics, we will drop the subscript 2, and simply study the equations: If we assume that both D and L depend on a single population, that is, L=L(x), D=D(x), or L=L(y),D=D(y), the following approximations can be derived (see Section 4 of the Supplement). As the cell population expands, the DC population behaves as and ratio of stem to differentiated cells, ν, in the limit of large times is given by: Equation ( To summarize, the following three types of SC fraction dynamics are predicted: (1) If L is subject to negative feedback and decays to zero and/or D is subject to positive feedback and increases without bound, then we have unlimited SC enrichment.
(2) If L is subject to negative feedback but never decreases to zero and/or D is subject to positive feedback and increases within bounds, then we However, if the feedback on D is (effectively) greater such that L/D is decreasing and bounded above 0, then this would correspond to type (2) and result in limited SC enrichment.

SC enrichment in spatially structured populations
According to the above results, stem cell enrichment during growth requires certain feedback mechanism to be present in the tumor cell population, such that the ratio L/D is reduced as the population size is increased. Typically such feedback can occur through signaling factors that are secreted from stem or differentiated cells. Another way to achieve a similar result can be spatially restricted reproduction of cells. In such scenarios, cells experience range expansion in two dimensions, or grow as expanding sphere-like structures in 3D. Inside the expanding population, divisions must be balanced with deaths because free space is limited.
In a way this works similarly to control loops affecting division and/or death rates, which were discussed earlier in the paper; in the case of spatially restricted growth, control is essentially competition for space, which leads to slower divisions/ higher death as the density increases.
To explore this, we first considered a two-dimensional stochastic agent- is randomly picked as a target for one of the daughter cells. If the chosen spot is already filled, the division event is aborted, otherwise it proceeds. If division proceeds, both daughter cells will be stem cells with a probability P 0 (self-renewal).
With probability 1-P 0 , both daughter cells will be differentiated cells. If the sam-pled spot contains a differentiated cell, death occurs with a probability D 0 . No explicit feedback processes were included in the model. The simulation was started with 9 stem cells (and no differentiated cells). Assuming that stem cells do not die, the resulting average growth curve is shown in Figure 7(a). While initially, the differentiated cells grow to be more abundant than the stem cells, the stem cell population enriches over time and eventually becomes dominant as the cell population grows. Figure 7(b) shows the same kind of simulation, but assuming that stem cells die with a rate that is smaller than the death rate of differentiated cells.
Consistent with the results obtained for explicit feedback mechanisms, we find that the degree of stem cell enrichment is reduced in the presence of stem cell death (higher rates of stem cell death lead to less enrichment).
A simple mean-field model that takes account of the space limitations inside an expanding population can explain these results (see Section 5 of the Supplement). Indeed, the system in the interim of the expanding globe reaches a dynamic equilibrium state where the density of SCs and DCs is dictated by the balance of division and death rates.

Discussion and Conclusions
The process of SC enrichment can be an important determinant of non-genetic drug resistance in cancer therapy, since CSCs are less susceptible to a variety of drugs, including chemotherapy and targeted treatment approaches [10][11][12]. Understanding the mechanisms that contribute to CSC enrichment is therefore a crucial step for the design of treatments that seek to sensitize tumors to drugs.
SC enrichment as a cause for treatment failure has been suggested in bladder cancer [15], and might also be a reason for the occurrence of "primary resistance" of chronic myeloid leukemia (CML) to tyrosine kinase inhibitors [14].
Previous work suggested that regulatory feedback loops might contribute to CSC enrichment [13]. Feedback mechanisms have been shown to be active in tumors [8], and quantitative analysis of tumor growth data further suggests feedback regulation to influence tumor growth dynamics [16]. These results generalize previous work on the determinants of stem cell enrichment, which was motivated by the experimental observation that differenti-ated cell death during chemotherapy of bladder cancer results in a phase of stem cell repopulation, mediated through a PGE 2 -induced would healing response [8].
Mathematical analysis suggested that in order to maintain this stem cell enrichment beyond the treatment phase, and thus to account for elevated stem cell fractions and a reduced response upon initiation of a new phase of chemotherapy, feedback mechanisms must continue to operate during untreated growth [13]. and/or death rate increases. This has the same effect on the dynamics as the presence of explicit feedback loops. Whether this represents a physiologically important mechanism that drives CSC enrichment remains to be explored further.
According to our models, stem cell enrichment is most pronounced in a 2dimensional setting, and considerably less pronounced in a 3-dimensional setting, which applies to many solid tumors. In addition, even if spatially restricted cell divisions occur in a tumor, the presence of cell migration can destroy the reported effect (because cell migration essentially leads to a higher degree of cell mixing).
All modeling approaches contain simplifying assumptions, and the models presented here are no exception. To gain analytical insights, we reduced the complexity of the lineage differentiation pathway to include only stem cells and differentiated cells, ignoring intermediate transit amplifying cell populations with limited self-renewal capacity. Our previous work included models that explicitly took into account transit amplifying cells [13], and the relationship between the presence of negative feedback on stem cell division and the occurrence of stem cell enrichment remained qualitatively the same [42]. Other modeling approaches have treated the cell differentiation pathway as a continuous process using partial differential equations, rather than considering discrete cell sub-populations     The left panels (a, b) illustrate SC enrichment, and the right panels (c,d) SC depletion. In the top panels, the dynamics and the control loops are presented schematically. SCs (the leftmost circle) divide at rate L, such that the decision of whether to self renew or to differentiate (denoted by a question mark) is governed by probability P. Control loops are depicted by red arrows (positive or negative) directed from the population mediating the control to the rate/probability that is being controlled. In the bottom panels, the time series are shown, where functions for wild type cells, x 1 (t) and y 1 (t), are plotted in black, and functions for mutant cells, x 2 (t) and y 2 (t), are plotted in red. SCs are depicted by solid and DCs by dashed lines. Initially, the system is at the original equilibrium, (x , y,0,0).
At t=100, mutant stem cells are introduced at a low level, resulting in the extinction of wild type cells, and convergence to a new equilibrium, (0,0,x * ,y * ). In the insets, the SC fraction, x/y, is depicted as a function of time. (a,b) Negative control by DCs, resulting in SC enrichment: the functions L, P, and P 2 are given by the solid lines in Fig 1(a,b). (c,d) Positive control by SCs, resulting in SC depletion: the functions L, P, and P 2 are specified in Fig 1(c,d). given by the dashed line in Fig 1(a). (d) Shallower L: same parameters as in Fig   2(b), except the division rate L is given by the dashed line in Fig 1(b). We observe that the SC enrichment is (b-d) is less pronounced compared to Fig 2(b). Positive control by SCs; (d) Positive control by SCs. Unlike in Fig 2(a,c), the mutant self-renewal probability is assumed to be constant (and thus the control loop is completely severed in mutants), leading to unlimited growth. The 2 top panels correspond to Fig 5(a,b); the 2 bottom panels correspond to Fig 6(a,b). Notations are as in Fig 2(b,d). The purple line in each panel plots the approximation for y (formula (3)), and in the insets the approximation for x/y (formula (4)).
(a) Negative control by DCs: parameters are as in Fig 2( Negative control by SCs: similar to (a), except the rate functions depend on SCs: L(x)=2*5  Notations are as in Fig 2(b,d). The purple line in each panel plots the approximation for y (formula (3)), and in the insets the approximation for x/y (formula (4)).
(a) Positive control by SCs: corresponds to parameters of Fig 2(