A stochastic model of adult neurogenesis coupling cell cycle progression and differentiation

Journal of Theoretical Biology 475 (2019) 60–72

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A stochastic model of adult neurogenesis coupling cell cycle progression and differentiation Anna Stopka a,b,∗, Marcelo Boareto a,b a b

Department of Biosystems Science and Engineering, ETH Zürich, Switzerland Swiss Institute of Bioinformatics, Switzerland

a r t i c l e

i n f o

Article history: Received 12 September 2018 Revised 16 May 2019 Accepted 22 May 2019 Available online 22 May 2019

a b s t r a c t Long-term tissue homeostasis requires a precise balance between stem cell self-renewal and the generation of differentiated progeny. Recently, it has been shown that in the adult murine brain, neural stem cells (NSCs) divide mostly symmetrically. This finding suggests that the required balance for tissue homeostasis is accomplished at the population level. However, it remains unclear how this balance is enabled. Furthermore, there is experimental evidence that proneural differentiation factors not only promote differentiation, but also cell cycle progression, suggesting a link between the two processes in NSCs. To study the effect of such a link on NSC dynamics, we developed a stochastic model in which stem cells have an intrinsic probability to progress through cell cycle and to differentiate. Our results show that increasing heterogeneity in differentiation probabilities leads to a decreased probability of long-term tissue homeostasis, and that this effect can be compensated when cell cycle progression and differentiation are positively coupled. Using single-cell RNA-Seq profiling of adult NSCs, we found a positive correlation in the expression levels of cell cycle and differentiation markers. Our findings suggest that a coupling between cell cycle progression and differentiation on the cellular level is part of the process that maintains tissue homeostasis in the adult brain. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction It is now widely accepted that a precise balance of stem cell self-renewal and differentiation is required to maintain long term tissue homeostasis. There are two strategies to achieve this task: asymmetry at the cellular or at the population level (Simons and Clevers, 2011; Watt and Hogan, 20 0 0; Morrison and Kimble, 20 06). In mammals, the applied homeostatic strategies are different for every regenerating tissue including skin, intestine, lung, blood, bone marrow, heart, testis, uterus and mammary gland (Pellettieri and Alvarado, 2007; Blanpain et al., 2007; Anversa et al., 2006). For instance, approximately 85% of the stem cells in the skin divide asymmetrically (Lechler and Fuchs, 2005), whereas stem cells in the intestine divide mostly symmetrically (Snippert et al., 2010). The mammalian brain also regenerates (Ming and Song, 2011), whereas it remains unclear which homeostatic strategy is applied. During embryo neurogenesis, primary neural progenitors in the ventricular zone (VZ) divide mostly asymmetrically (Noctor et al., 2004). For a long time, it has therefore been thought that the strategy to maintain tissue homeostasis in the adult brain would ∗

Corresponding author. E-mail address: [email protected] (A. Stopka).

https://doi.org/10.1016/j.jtbi.2019.05.014 0022-5193/© 2019 Elsevier Ltd. All rights reserved.

rely on asymmetry at the cellular level (Silva-Vargas et al., 2018). In a recent study, Obernier et al. have shown that stem cells in the adult subventricular zone (SVZ) divide mostly symmetrically (Obernier et al., 2018). This implies that for adult neurogenesis the strategy to maintain homeostasis is population asymmetry rather than asymmetry at the cellular level. Although the sustainment of adult organs via population asymmetry is a well-established concept and is found in other tissues (Simons and Clevers, 2011), there are still many open questions. For example, how can a stable stem cell population be achieved when stem cells divide exclusively symmetrically? Homeostasis via population asymmetry could, for instance, be achieved by limited access to a stem cell niche in combination with a short-range cell fate determining signal (Watt and Hogan, 20 0 0; Suh et al., 2010). Conversely, if the stem cell population asymmetry is based on an internal regulation that is either intrinsic or coupled to an extrinsic signaling, it is unknown what mechanisms could govern the process. Is a molecular regulator enabling a balanced stochastic cell fate decision for a stem cell population? There is evidence that cell cycle and differentiation are controlled by the same regulator or pathway in various systems of plants, invertebrates and vertebrates (Myster and Duronio, 20 0 0; Jakoby and Schnittger, 2004; Dalton, 2015). For example, a group of

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transcription factors in higher eukaryotes, the E2F transcription factors, was suggested to activate the expression of genes driving cell cycle progression, but also of genes encoding differentiation factors (Dimova et al., 2003). Proneural genes encode a class of basic helix-loop-helix (bHLH) transcription factors that are known to drive neural differentiation. However, some were also connected to the regulation of cell cycle progression. For example, the expression of Ngn2 in oscillatory waves was suggested to maintain neural progenitors in embryonic mice, while a constant expression level coincides with differentiation (Hindley and Philpott, 2012). Furthermore, an experimental study revealed a molecular link between cell cycle progression and differentiation of neural progenitors in developing mice through the bHLH factor Ascl1: it was suggested that Ascl1 activity can switch from promoting cell proliferation to triggering cell cycle arrest and differentiation (Castro et al., 2011). Moreover, the transcription factor Ato in Drosophila, regulating the expression of Ase, the non-mammalian homolog of Ascl1, plays a role in both proliferation and differentiation of neural progenitors (Mora et al., 2018). These findings indicate that the regulation of the cell cycle progression and differentiation of embryonic neural stem cells (NSCs) are coupled through the expression of proneural genes. At present it remains unclear, whether the regulation of cell cycle progression and differentiation of adult NSCs is coupled by a proneural gene, and, in addition, whether such a coupled regulation would have an effect on tissue homeostasis in the brain, where NSCs divide mostly symmetrically. Over the years, a number of insightful studies on renewal dynamics of stem cells have been conducted, ranging from models on stem cell dynamics in general (Stiehl and Marciniak-Czochra, ´ 2011; Sun and Komarova, 2012; Wu et al., 2013; Pazdziorek, 2014; Sun and Komarova, 2015; Yang et al., 2015; Greulich and Simons, 2016; Alvarado et al., 2018) to brain specific models (Gohlke et al., 2018; Aimone and Wiskott, 2008; Ashbourn et al., 2012; Shahriyari and Komarova, 2013; Barton et al., 2014; Ziebell et al., 2014; Stumpf et al., 2017; Picco et al., 2018; Ziebell et al., 2018). The latter were used to explore specific characteristics of neurogenesis, such as the role of fate determining signaling factors during development (Barton et al., 2014), or the migration of neuroblast cells to the olfactory bulb in adults (Ashbourn et al., 2012). Other models explored the conditions for homeostasis without further specifying the studied regenerating tissue (Sun and Komarova, ´ 2012; Pazdziorek, 2014; Sun and Komarova, 2015; Yang et al., 2015; Greulich and Simons, 2016; Alvarado et al., 2018). For instance, the dynamics of mutant accumulation in homeostatic tissues over time have been explored, showing that type and number of mutants depend on the division-tree length (Alvarado et al., 2018). In particular, the role of feedback signals controlling the balance between stem cell self-renewal and differentiation was investigated. This has shown that homeostasis based on stochastic processes needs negative regulatory feedback loops to be maintained (Sun and Komarova, 2015). Furthermore, depending on the applied feedback network, symmetric stem cell divisions can both stabilize or destabilize the homeostatic state of a population (Yang et al., 2015). However, it remains unclear how a coupled regulation of cell cycle progression and differentiation by a proneural gene influences the maintenance of a NSC population. In order to study how a coupling between cell cycle progression and differentiation on the single cell level affects homeostasis, we developed a model in which stem cells have an intrinsic probability to progress through cell cycle and to differentiate. In our model, stem cells divide symmetrically, representing for instance adult mouse NSCs (Obernier et al., 2018). We chose a stochastic approach as it allows us to study how fluctuations at the cellular level can affect population dynamics. With our model, we investigated the dynamics of stem cell populations depending on the coupling of

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cell cycle activity and cell differentiation. We evaluated how robust populations are maintained and how many pre-differentiated cells are produced. Furthermore, we analyzed published single-cell RNA-Seq data of adult mouse NSCs (Llorens-Bobadilla et al., 2015) for a qualitative comparison to our simulation results. This paper is organized as follows. In Section 2 we define our model and discuss the biological background underlying our assumptions. In Section 3 we present the results of our simulations (Sections 3.1 and 3.2) and the RNA-Seq profiling analysis (Section 3.3), offering biological interpretation of the results. In Section 4 we discuss our findings. 2. Methods 2.1. Modeling a stem cell We developed a stochastic model of stem cell dynamics in which the fate of a stem cell is determined by its probability to progress through cell cycle within a time frame t and its probability to differentiate. In our model, we assign each stem cell a factor  , where  corresponds to:



probability to progress through cell cycle within t, : (1 −  ) : probability to not progress through cell cycle within t,

(1) with  ∈ [0, 1]. The cell cycle activity  is inverse proportional to the cell cycle time τ cc and given by  = τcct . Further, each stem cell has a



probability to differentiate, and a α: (1 − α ) : probability to maintain stemness,

(2)

with α ∈ [0, 1]. Recent findings revealed that the divisions of neural stem cells (NSCs) in adult mice are mostly symmetrical (Obernier et al., 2018). Our model mimics the dynamics of adult NSCs by allowing stem cells to divide only symmetrically, either into two stem cells (symmetric proliferative division) or into two pre-differentiated cells (symmetric differentiative division). If a stem cell does not progress through cell cycle, it remains unchanged while keeping its typical stem cell properties. The dynamics of a stem cell are described by a discrete-time Markov chain, where each iteration has a set of 3 states: symmetric proliferative division, symmetric differentiative division and no division. A graphical representation of one iteration is shown in Fig. 1a. The probability of differentiation and selfrenewal are given by α and  (1 − α ), respectively. 2.2. Modeling a stem cell population Several factors can lead to inhomogeneous populations, e.g. the spatial distribution of stem cells or biomolecular signaling. Our model allows for extrinsic and intrinsic sources of heterogeneity by assigning an individual value pair of cell cycle activity and differentiation probability ( k , α k ) to every stem cell. In case of a symmetric proliferative division, the daughter cell does not inherit the properties of its mother cell, but gets assigned a newly generated pair of cell cycle activity and differentiation probability. Thereby we account for stochasticity in the cell fate decision process. This is in line with an experimental study showing independent cell cycle times between mother and daughter cells in the rat retina (Gomes et al., 2011). We investigated stem cell populations with uncoupled and coupled cell cycle and differentiation probabilities. For populations with uncoupled probabilities, we assumed that the cell cycle and differentiation of a stem cell are controlled independently. In case

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Fig. 1. (a) Schematic representation of one iteration step. The probability of differentiation and self-renewal are given by α and  (1 − α ), respectively. (b) Representative histogram of simulated cell cycle activities { } and corresponding histogram of cell cycle times {τ cc }, with normal and log-normal fit, respectively. .

of a coupling, we presumed cell cycle and differentiation to be regulated in a correlated manner by proneural genes (Bertrand et al., 2002; Ohnuma and Harris, 2003), either positively or negatively. The probabilities ( k , α k ) for every stem cell of a population were generated according to a multivariate normal distribution X ∼ N2 (k, ) with vector k = (μ μα )T and covariance matrix



=

σ2 c σ σα

c σ σα

σα2



.

(3)

The mean values μ , μα and standard deviations σ  , σ α determine the normal distribution of cell cycle activities and differentiation probabilities, respectively. The correlation coefficient c ∈ [−1, 1] defines the type of coupling between cell cycle activity and differentiation: positive (c > 0), negative (c < 0) or no (c = 0) coupling. For our simulations, we assumed average cell cycle times of τ cc ࣃ 20h, which is close to measured cell cycle times for stem cells in the ventricular-subventricular zone (V-SVZ) of the adult mouse brain (Ponti et al., 2013). We chose a small time step (t = 1h) such that t  τ cc . To represent variable cell cycle times {τ cc } within a stem cell population, we set σ  = 0. In the rat retina, the measured cell cycle time distribution of retinal progenitor cells has been fitted by a log-normal distribution (Gomes et al., 2011). Cell cycle times in NIH 3T3 mouse embryonic fibroblast cells were found to be distributed similarly and could be reproduced with a stochastic model (Yates et al., 2017). We chose σ = 0.02 for the cell cycle activities { } and showed by fitting the associated distributions of the cell cycle times {τ cc } with a log-normal distribution, that they resembled the measured cell cycle time distributions. Fig. 1b shows a representative histogram of simulated cell cycle activities { } and the corresponding histogram of cell cycle times {τ cc }, with normal and log-normal fit, respectively. In our simulations, about 90% of all stem cells have a cell cycle time of (20 ± 10)h, while the remaining ones are slowly cycling stem cells that accumulate over time. To study the dynamics of a stem cell population in a tissue, we summed over all stem cells at each iteration step to track the stem cell population size. The expected net change of the stem cell population B after one time step is given by

B = Bt+1 − Bt =

Bt 

k · (1 − 2αk ),

(4)

k=1

where Bt is the number of stem cells before the iteration step. For homeostasis, we assumed the stem cell population to be constant in size:

k (1 − 2αk )k,t = 0.

(5)

For the uncoupled case, we solve k k · 1 − 2αk k = 0 and the non-trivial solution is αk k = 0.5. For all relevant parameters in our study, the differences in the solution for αk k = μα between the uncoupled and coupled case, are negligible: we show that we can approximate μα = 0.5 for coupled cell cycle activity and differentiation (Appendix A.1). This implies that in the homeostatic case (μα = 0.5 and μ > 0), stem cells are on average as likely to maintain stemness as to differentiate. We analyzed the average stem cell population dynamics for the non-homeostatic cases (μα = 0.5 and μ > 0) and found that a deviation from the homeostatic conditions leads to overgrowth or shrinkage of the stem cell population (Appendix A.2). To evaluate and compare our simulations, we analyzed the number of differentiated daughter cells and the probability to maintain a stem cell population, which we define as robustness. We defined the differentiation rate D as the number of predifferentiated cells C that were generated on average per time step, normalized by the initial stem cell population size B0 :

D=

Ctend , tend · B0

(6)

with tend being the simulation time. We simulated the stem cell population dynamics for different tend and we set B0 = 50. For every tend , we simulated n = 300 populations and evaluated how many stem cell populations n˜ could maintain their size, i.e. were neither lost (Btend = 0) nor did overgrow (Btend ≥ 2B0 ) within the given time period tend . We calculated the robustness R as follows:

R=

n˜ . n

(7)

2.3. Analysis of single-cell RNA-Seq data We analyzed published single-cell RNA-Seq data of mouse NSCs from the adult brain (Llorens-Bobadilla et al., 2015) for a qualitative comparison to our simulation results. Llorens-Bobadilla et al. acutely isolated NSCs from the lateral subventricular zone (SVZ) and chose an Illumina platform, which offers a low sequencing error rate (Chu and Corey, 2018). The RNA-Seq libraries were prepared using the Smart-seq2 technology (Sagasser et al., 2013). From the processed RNA-Seq data, we extracted cell cycle and differentiation marker expression levels, which we related to the cell cycle activity  and differentiation probability α of our model. The goal of this analysis was, to extract information on the coupling between cell cycle activity and differentiation of adult NSCs. For our analysis, we plotted the expression levels of neural cell cycle markers (Mki67, Mcm2, Ccnd1 (Llorens-Bobadilla et al., 2015; Zhang and Jiao, 2015; Sherr and Roberts, 2004)) against neural differentiation markers (Ascl1, Neurog2 (Ngn2), Neurod2 (Ponti et al., 2013; Kele et al., 2004; Messmer et al., 2012))

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on a log2 scale, excluding data points with zero expression for both marker types. We use log-transformed expression values to generate a continuous spectrum of gene expression levels accounting for up- and downregulated genes in a similar manner (Quackenbush, 2002). We then computed the Pearson correlation coefficient (PCC) (Rogers and Nicewander, 1988), which measures the linear relationship between two datasets, to evaluate the type of correlation between the cell cycle and differentiation markers. The corresponding p-value indicates the probability of an uncorrelated data pair. Further, we performed a bootstrapping analysis (Henderson, 2005): we randomly resampled the gene expression data 10 0 0 times and evaluated the PCC value for each data set. By calculating mean and standard deviation of the obtained PCC values, we could evaluate the accuracy of the determined correlation. 3. Results 3.1. Uncoupled cell cycle progression and differentiation First, we studied the dynamics of stem cell populations with uncoupled cell cycle activity and differentiation. To do so, we generated probabilities as described in Section 2.2 with c = 0 and σ α ∈ [0.0, 0.12]. The upper limit is defined by the fact that less than 5% of the stem cell populations were robust in the long term (tend = 800h) for σ α > 0.12. Two representative distributions are shown in Fig. 2a and further representative distributions are shown in Fig. A.2. We simulated the dynamics of two stem cell population types with distinct levels of differentiation heterogene-

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ity (σα = {0.02, 0.08}) for different simulation times tend ∈ [100h, 800h]. For both population types the average differentiation rates D were not significantly changing for different simulation times (p-values > 0.005, Table A.1 in Appendix A.4). A population with higher differentiation heterogeneity tended to generate more differentiated progeny (Fig. 2b). This result originates from the dynamics of stem cells that are more likely to maintain stemness: these cells cause a stem cell population to grow and as the number of differentiated progeny scales with the stem cell population size, heterogeneous populations tend to have higher differentiation rates than homogeneous populations. However, the differences are not significantly different in the long term (p-values > 0.005, Table A.1 in Appendix A.4). We observed that for both levels of differentiation heterogeneity, the robustness R of a population decreased with increasing simulation times (Fig. 2b). Although the robustness of populations with lower differentiation heterogeneity decreased significantly slower for tend > 300h (p-values < 0.005, Table A.1 in Appendix A.4), there is no life-long homeostasis for a population with uncoupled cell cycle and differentiation rates. A homogeneous population is more robust on the long-term, probably because less fluctuations at the cellular level can imbalance population dynamics. Long-term simulations (tend = 800h one month) supported the finding that the level of heterogeneity of the differentiation probabilities has a strong impact on homeostasis: the higher the differentiation heterogeneity level, the lower the probability to maintain a stem cell population (Fig. 3a). Furthermore, the

Fig. 2. (a) Two representative stem cell populations with uncoupled cell cycle and differentiation and distinct differentiation heterogeneity levels: σα = 0.02 (light blue) and σα = 0.08 (dark blue). The same colors are used in the histograms at the top and on the right hand side showing cell cycle activity and differentiation probability distributions. Where the two distributions overlap, the bars are mid-blue. (b) Average differentiation rates D (top) and average robustness R (bottom) of two stem cell population types with uncoupled cell cycle and differentiation with distinct levels of differentiation heterogeneity (σα = {0.02, 0.08}) for different simulation times tend . The level of differentiation heterogeneity affects the robustness of homeostasis R significantly, while the production of differentiated progeny D is not affected. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 3. Long-term (tend = 800h) robustness R and differentiation rates D of stem cell populations with uncoupled cell cycle and differentiation for various levels of heterogeneity of differentiation probabilities σ α . (a) The higher σ α , the lower the robustness R of a population. (b) The level of heterogeneity in the differentiation probabilities does not affect the differentiation rates D significantly.

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probability for overgrowth among the unstable cases increases from around 50% for σα = 0 to 100% for σα = 0.12. With increasing heterogeneity in differentiation probabilities, some stem cells are more likely to maintain stemness and have a high cell cycle activity. This causes a stem cell population to overgrow. Moreover, the long-term simulations revealed that the differentiation rates D of populations with a robustness ≥ 10% do not significantly change for various differentiation heterogeneity levels σ α (Fig. 3b; p-values > 0.005, Table A.2 in Appendix A.4). 3.2. Coupled cell cycle progression and differentiation Having analyzed uncoupled stem cell populations, we investigated stem cell populations with coupled cell cycle and differentiation probabilities. Again, we generated probabilities as described in Section 2.2 with c = 0 and σ α ∈ [0.0, 0.12]. Two representative distributions are shown in Fig. 4a and further representative distributions are shown in Fig. A.2. We studied the dynamics of stem cell populations with coupled cell cycle and differentiation activities (c = {0.9, −0.9}) and σα = 0.08. We found that for all simulation times, a negative coupling resulted in significantly higher average differentiation rates D than a positive coupling (Fig. 4b; pvalues < 0.005, Table A.3 in Appendix A.4). Evaluating the robustness R, we observed that for all cases, R decreased over time and that there is no life-long homeostasis for a population with cou-

pled cell cycle and differentiation activities (Fig. 4b). This trend is similar to the uncoupled case (Fig. 2b). However, the probability to maintain homeostasis decreased significantly more slowly for populations with a positive coupling than for populations with a negative coupling (p-values < 0.005, Table A.3 in Appendix A.4). Stem cell populations with a coupling between cell cycle activity and differentiation mainly consist of two subpopulations: slowly dividing stem cells and fast dividing stem cells, which are either more likely to maintain stemness or more likely to differentiate. For populations with a negative coupling, the slowly dividing stem cells are more likely to differentiate and the fast dividing stem cells are more likely to maintain stemness, which causes populations to overgrow. In addition, because these stem cell populations are larger, they enable the generation of more pre-differentiated progeny compared to populations with a positive coupling. For populations with a positive coupling, the fast dividing stem cells are more likely to differentiate, while the cells that are more likely to maintain stemness divide slowly, enabling higher robustness of a stem cell population in the long term. Focusing on long-term simulations (tend = 800h one month), we observed that the type of coupling between cell cycle activity and differentiation indeed had a strong impact on homeostasis. Compared to the uncoupled case, a positive coupling significantly increased the probability for a tissue to be maintained in the long term (Figure 5 a; p-values < 0.005 for σ α > 0.04, Table A.4 in

Fig. 4. (a) Two representative coupled stem cell populations with positive and negative correlation: c = −0.9 (orange) and c = 0.9 (red). The same colors are used in the histograms at the top and on the right hand side showing cell cycle activity and differentiation probability distributions. Where the two distributions overlap, the bars are dark orange. (b) Average differentiation rates D (top) and average robustness R (bottom) of two coupled stem cell population types (c = {0.9, −0.9}) for different simulation times tend . A negative coupling increases differentiation rates D, while a positive coupling increases the robustness R. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 5. Long-term (tend = 800h) robustness R and differentiation rates D of stem cell populations with coupled cell cycle and differentiation for various variabilities of differentiation probabilities σ α . (a) The robustness R decreases with increasing σ α , but a positive coupling increases the probability to maintain homeostasis. (b) The differentiation rates D do not change for increasing σ α , however, a negative coupling increases differentiation rates D, while a positive coupling lowers them.

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Appendix A.4). In contrast, a negative coupling decreased the probability for a tissue to be maintained in the long term. However, in most cases, the decrease was not significantly different from the uncoupled case (Table A.4 in Appendix A.4). Comparing the probability of overgrowth among the unstable cases for different couplings, we observed no significant difference (p-values > 0.005, Table A.5 in Appendix A.4). This result can be explained as follows: stem cells that are more likely to maintain stemness lead a population to overgrow. As these cells occur with higher differentiation heterogeneity, the overgrowth rate among the unstable cases increases with increasing differentiation heterogeneity σ α . However, the coupling between cell cycle activity and differentiation determines whether cells that are more likely to maintain stemness have a high or a low cell cycle activity. Therefore, overgrowth can be accelerated or slowed down by a negative or positive coupling, respectively. Taken together, the probability for overgrowth among the unstable cases depends on the heterogeneity level of differentiation probabilities, while the coupling type has an impact on the dynamics of overgrowth and by this on the robustness of a population. Evaluating the differentiation rates D, we found that for a positive coupling the number of differentiated progeny decreased significantly (Fig. 5b; p-values < 0.005 for σ α ∈ [0.04, 0.1), Table A.4 in Appendix A.4), while a negative coupling resulted in not significantly higher average differentiation rates D compared to the uncoupled case (p-values > 0.005, Table A.4 in Appendix A.4). 3.3. Single-cell RNA-Seq data of adult murine NSCs In our simulations, we observed that a positive correlation between cell cycle activity and differentiation produced the most robust stem cell populations (Fig. 5a). Analyzing published singlecell RNA-Seq data (Llorens-Bobadilla et al., 2015), we found a positive correlation between cell cycle and differentiation marker expression levels in the adult murine subventricular zone (SVZ) (Fig. 6a). These findings suggests that an adult NSC population might ensure its robust homeostasis by coupling cell cycle and differentiation rates. This positive correlation was confirmed when analyzing the subpopulation of stem cells displaying non-zero expression levels in both, cell cycle and differentiation markers (Fig. A.3). An underlying mechanism for a coupling of cell cycle and differentiation at the molecular level could be driven by proneural genes regulating both processes in a coupled manner (Fig. 6b).

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The proneural gene Ascl1 expressed in neuronal progenitor cells (Castro et al., 2011; Urbán and Guillemot, 2014), could be part of a molecular network coupling the regulation of cell cycle progression and differentiation of NSCs. Additional regulators could be other proneural bHLH transcription factors or CcnD genes (Urbán and Guillemot, 2014). In addition, signaling networks such as Notch and Hedgehog have been associated to both cell cycle progression and differentiation (Urbán and Guillemot, 2014; Ohnuma et al., 2001; Philpott, 2010). Determining which molecular players might be involved in the adult brain will require further experimental investigation. For instance, live imaging of NSCs tagged for cell cycle and differentiation markers could show whether their expressions are correlated. Furthermore, it was proposed that Ascl1 switches its targets from proliferation to differentiation (Castro et al., 2011) and a perturbation of this switch could possibly give further insights into a potential coupling.

4. Discussion Homeostasis is an important mechanism for the sustainment of adult tissues during life. Various organs regenerate via tissue homeostasis, and there is evidence that complex organs such as the brain use it. Here, we presented a stochastic model to study stem cell dynamics in the subventricular zone (SVZ) of the adult mouse brain, where neural stem cells (NSCs) divide mostly symmetrically (Obernier et al., 2018). We focused on the interplay between cell cycle activity and differentiation on the cellular level and its impact on stem cell population dynamics. We also analyzed published single-cell RNA-Seq data of adult mouse NSCs (LlorensBobadilla et al., 2015) which supported the results of our simulations. In the following, we present and discuss the main findings of this study. Our simulations showed that the probability to maintain homeostasis decreased with increasing differentiation heterogeneity. This effect is most likely due to stronger fluctuations at the cellular level in a heterogeneous population which have the potential to cause unstable dynamics on the population level. Furthermore, we found that the coupling of cell cycle and differentiation had an effect on stem cell population dynamics, and, depending on whether the coupling is positive or negative, increases or decreases the robustness of homeostasis in an adult tissue, respectively. A coupling between cell cycle activity and differentiation leads to the generation of two subpopulations within a stem cell population. There

Fig. 6. (a) Analysis of published single-cell RNA-Seq data of adult mouse NSCs (Llorens-Bobadilla et al., 2015) reveals a positive correlation between cell cycle and differentiation marker expression level (p-value < 0.001). (b) Schematic representation of a possible underlying molecular mechanism of a coupled regulation of cell cycle progression and differentiation.

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are slowly dividing and fast dividing stem cells, which are either more likely to maintain stemness or more likely to differentiate. In a population with a positive coupling, the fast dividing stem cells are more likely to differentiate whereas the slowly dividing stem cells are more likely to maintain stemness. This combination leads to an increase in the typical lifetime of the stem cell pool. However, how subpopulations within a stem cell population could be generated remains unclear. For instance, limited access to a stem cell niche in combination with a short range signal can possibly generate two or more subpopulations of stem cells (Simons and Clevers, 2011). As shown by our simulations, there are two ways for a stem cell population to achieve a high long-term robustness: either with homogeneous differentiation probabilities or with a positive coupling between cell cycle and differentiation rates for heterogeneous differentiation probabilities (Fig. 5a). A central question is, which of these two scenarios is more likely to occur in a NSC population in the adult brain. Based on observations in other systems, we suggest the existence of differentiation heterogeneity within a NSC population and therefore the second scenario to be more likely. For instance, there is evidence of heterogeneous gene expression amongst hematopoietic stem cell populations (Hu et al., 1997; Enver et al., 1998; Haas et al., 2018). Furthermore, proliferation dynamics and regional identity are heterogeneous among the NSCs in the mammalian brain (Chaker et al., 2016). However, it remains unclear, whether such heterogeneity is a reflection of intrinsic differences of NSCs or is caused by an external signal (Silva-Vargas et al., 2018). Single-cell RNA-Seq is a powerful method to compare gene expression levels of cells within a population. Our qualitative analysis of (Llorens-Bobadilla et al., 2015) showed a positive correlation between the expression of cell cycle markers and differentiation markers, indicating that cell cycle activity and differentiation could indeed be coupled. As RNA-Seq data has several limitations, such as providing a snapshot of gene expression in time and allowing for a relative quantification of gene expression levels only, we can neither conclude that the differentiation probabilities are heterogeneous and positively coupled nor that they are rather homogeneous. Nonetheless, the idea that both, cell cycle progression and differentiation, are controlled by the same molecular players could explain a positive correlation between these two key components of homeostasis. There are several candidates of proneural genes that could drive cell cycle progression and differentiation in a coupled manner (Urbán and Guillemot, 2014). Amongst those, the proneural gene Ascl1 has been associated with cell cycle progression and differentiation of neuronal progenitor cells (Castro et al., 2011). However, the existence and the details of such a regulatory network in the adult brain will require further experimental investigation, as well as the identification of additional molecular players. For instance, live imaging experiments marking adult NSCs separately for cell cycle and differentiation markers could show whether their expression levels occur in a correlated manner. In addition, the specific perturbation of possible regulating proneural genes could give further insights into a potential coupling of cell cycle activity and differentiation. In this study, our focus was on exploring the effect of a coupled internal regulation of cell cycle activity and differentiation on stem cell population dynamics. Explicit regulating factors are not part of our model, as they were intensively explored in former studies (Sun and Komarova, 2012; Yang et al., 2015). As there is no long-term homeostasis without external control, we propose that feedback signaling among the stem cells or signaling from the stem cell niche are the building

blocks to maintain a stem cell population. For instance, robustness can be achieved by a negative regulatory feedback that controls the differentiation probability by the number of stem cells (Sun and Komarova, 2015). Furthermore, stem cell quiescence imposed by the stem cell niche was recently found to be key for the maintenance and repair of the aging adult mouse brain (Kalamakis et al., 2019). In case of a brain injury, lost neurons can be replenished by the division and differentiation of NSCs (Ludwig et al., 2018). A significant number of differentiated cells has to be generated and integrated into the existing neuronal circuity (Sun, 2014). Thus, regeneration requires disabling the balance between stem cell self-renewal and differentiation for a certain time under precise control. Moreover, in case of an imbalance between stem cell self-renewal and differentiation in the long term, a stem cell population suffers overgrowth or depletion, resulting in fatal consequences for the organ such as tumorigenesis (Morrison and Kimble, 20 06; Moore, 20 06; Buszczak et al., 2014). Our simulation results have shown that a stem cell population whose cell cycle activity and differentiation are negatively correlated is rather unstable: it tends to overgrow, while producing many differentiated cells at the same time. This result suggests that a regulating molecular mechanism might possibly be altered in case of injury or disease and adopt a state that is similar to the negatively coupled scenario. With our stochastic model, we explored how fluctuations that emerge at the cellular level affect the dynamics of stem cell populations dividing exclusively symmetrically. We showed that a coupling of cell cycle activity and differentiation increases the robustness of a heterogeneous stem cell population. Our qualitative analysis of published single-cell RNA-Seq data of adult mouse NSCs (Llorens-Bobadilla et al., 2015) revealed a positive correlation between neural cell cycle and differentiation markers. Therefore, we propose that a positive coupling between cell cycle and differentiation is likely to be part of a homeostatic strategy for NSC populations. In summary, in terms of neurogenesis, injury and neural diseases related to tissue overgrowth, the molecular link between cell cycle progression and differentiation in NSCs is worthwhile to study further.

Materials Code availability For simulations and data analysis we used Python programming language (Python Software Foundation, https://www.python.org/). Our code is available on https://github.com/astopka/ StochasticModelNSC.git.

Data availability For the analysis of single-cell RNA-Seq data we used the public dataset from (Llorens-Bobadilla et al., 2015) and the processed data was downloaded from https://www.ncbi.nlm.nih.gov/ geo/query/acc.cgi?acc=GSE67833.

Acknowledgments This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.

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Appendix A A.1. Estimation of μα Here we show that we can approximate μα ≈ 0.5 for the uncoupled case in the parameter space that is relevant in our study. For that purpose, it is sufficient to consider the continuum version of the discrete equation k (1 − 2αk )k ; that is, we consider

 ( 1 − 2 α ) = 0

(A.1)

with the mean values  = μ , α = μα and the corresponding standard deviations σ  , σ α determining the normal distribution of cell cycle activities  and differentiation probabilities α , respectively. It is convenient to describe the cell cycle activity and the differentiation probability in terms of deviations from the mean values, i.e.,  = μ + δ , α = μα + δα . It follows δ  = 0, δα  = 0 as can be seen by averaging the equations for  and α . Inserting this into Eq. (A.1), multiplying and averaging, results in μ (1 − 2μα ) − 2δ δα  = 0. The variances σ2 = δ2  and σα2 = δα2  are given by the covariance matrix



  δ δα  σ2 = δα2  c σ σα

δ2  = δ δα 

c σ σα



σα2

(A.2)

with the correlation coefficient c ∈ [−1, 1] defining the type of correlation between cell cycle activity and differentiation. It is

μα =

c σ σα 1 − , 2 μ

(A.3)

and it follows

μα =

c σα 1 − , 2 3

(A.4)

where we used σ = 0.02 and μ = 0.06 from our simulations. As the maximum value of the correlation is |c| = 1 and σ α ∈ [0, 0.12] the correction term in Eq. (A.4) is at most 0.04. Thus, | cσ3α |  0.5 and we can approximate μα = 0.5 in our simulations for the uncoupled case. A.2. Average stem cell population dynamics We studied the average stem cell population dynamics, assuming constant average stem cell properties  = k k , α = αk k for any time. Eq. (4) then becomes

dB = [ · (1 − 2α )]B(t ) dt

(A.5)

with the solution

B(t ) = B0 · e[ ·(1−2α )]t .

(A.6)

We characterized the evolution of a stem cell population by bringing the initial population size B0 in relation to the final population size Btend :

ρB =

Btend . B0

(A.7)

There are three possible scenarios for the evolution of a stem cell population over time: it can be constant in size (ρB = 1), grow (ρ B > 1) or shrink (ρ B < 1), where the two latter cases occur under non-homeostatic conditions (Fig. A.1). We analyzed the non-homeostatic cases

Fig. A.1. Average stem cell population dynamics for the homeostatic cases (black lines) and the non-homeostatic cases for tend = 40h.

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and found that even a minimal deviation from the homeostatic cell cycle and differentiation activity leads to a vast change in the stem cell population size within the time of two average cell cycles (tend = 40h). Both, overgrowth and depletion, become stronger with an increasing average cell cycle activity  k k . A.3. Representative distributions of stem cell probabilities

Fig. A.2. (a) Uncoupled probabilities with σα = 0.04, c = 0. (b) Uncoupled probabilities with σα = 0.12, c = 0. (c) Coupled probabilities with σα = 0.08, c = 0.5. (d) Coupled probabilities with σα = 0.08, c = 1.0. (e) Coupled probabilities with σα = 0.08, c = −0.5. (f) Coupled probabilities with σα = 0.08, c = −1.0.

A.4. Statistical tests To determine if the difference between the results of two simulation samples are statistically significant, we used statistical tests to compare their robustness R and differentiation rates D. As each simulation results in either a robust or an unstable stem cell population, the robustness R of a sample is described by a Binomial distribution with n = 300 independent simulations and the probability of success, which in our case equals the robustness R. To compare the robustness R of two samples, we define the test score

t=



R1 − R2 R(1 − R )(n1 + n2 )/(n1 n2 )

(A.8)

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Table A.1 Statistical tests for simulations with uncoupled cell cycle and differentiation (Fig. 2b). T-values and p-values for the robustness R and differentiation rate D are listed.

σα = 0.02 σα = 0.08 t t t t

= 200h = 400h = 600h = 800h

Compared samples

T-value R

p-value R

T-value D

p-value D

= 200h, t = 400h = 400h, t = 600h = 600h, t = 800h = 200h, t = 400h = 400h, t = 600h = 600h, t = 800h σα = 0.02, σα = 0.08 σα = 0.02, σα = 0.08 σα = 0.02, σα = 0.08 σα = 0.02, σα = 0.08

3.37450 0.74502 3.99320 7.30366 3.51014 3.68506 2.20487 6.22612 8.83399 8.63924

< 0.001 0.45478 < 0.001 < 0.001 < 0.001 < 0.001 0.02634 < 0.001 < 0.001 < 0.001

1.31732 2.62696 −1.97871 2.61079 0.85605 −1.18532 −3.66140 −1.28774 −2.20358 −1.65811

0.18833 0.00887 0.04843 0.00939 0.39264 0.23704 < 0.001 0.19852 0.02824 0.09872

t t t t t t

Table A.2 Statistical tests for long-term simulations with uncoupled cell cycle and differentiation (Fig. 3). T-values and p-values for the robustness R and differentiation rate D are listed.

c = 0.0

Compared samples

T-value R

p-value R

T-value D

p-value D

σα σα σα σα σα

2.00261 0.09682 4.82442 3.59371 6.51275

0.04462 0.88765 < 0.001 < 0.001 < 0.001

−0.11575 −0.08199 −1.43244 1.02319 1.11628

0.90789 0.93469 0.15283 0.30717 0.26622

= 0.0, σα = 0.02 = 0.02, σα = 0.04 = 0.04, σα = 0.06 = 0.06, σα = 0.08 = 0.08, σα = 0.1

Table A.3 Statistical tests for simulations with coupled cell cycle and differentiation (Fig. 4b). T-values and p-values for the robustness R and differentiation rate D are listed.

c = 0.9

c = −0.9

t t t t

= 200h = 400h = 600h = 800h

Compared samples

T-value R

p-value R

T-value D

p-value D

t = 200h, t = 400h t = 400h, t = 600h t = 600h, t = 800h t = 200h, t = 400h t = 400h, t = 600h t = 600h, t = 800h c = 0.9, c = −0.9 c = 0.9, c = −0.9 c = 0.9, c = −0.9 c = 0.9, c = −0.9

6.17451 2.88136 3.98386 6.86357 3.27947 4.92537 6.86780 7.62006 8.04033 8.91711

< 0.001 0.00379 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001

−1.01156 0.88918 −0.08757 −0.14523 0.51351 −0.17290 −10.47145 −6.05999 −4.89871 −3.44692

0.31231 0.37438 0.93026 0.88463 0.60808 0.86304 < 0.001 < 0.001 < 0.001 < 0.001

Table A.4 Statistical tests for long-term simulations with uncoupled and coupled cell cycle and differentiation (Fig. 5). T-values and p-values for the robustness R and differentiation rate D for R ≥ 10% are listed.

σα = 0.02 σα = 0.04 σα = 0.06 σα = 0.08 σα = 0 . 1 σα = 0.12

Compared samples

T-value R

p-value R

T-value D

p-value D

c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0

0.57916 1.29340 1.87026 0.77313 1.99220 2.75868 3.38788 2.20661 5.55715 4.98448 3.77534 8.61941 7.60338 4.42470 11.30895 5.58649 1.42834 6.69961

0.55254 0.19303 0.06118 0.43245 0.04602 0.00559 < 0.001 0.02545 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 < 0.001 0.15621 < 0.001

−0.95657 −1.91454 −2.87738 −2.91461 −1.43119 −4.41099 −5.22418 −1.40713 −6.03493 −2.89577 −2.66439 −5.54615 −1.02492 – – – – –

0.33928 0.05621 0.00420 0.00374 0.15311 < 0.001 < 0.001 0.16047 < 0.001 0.00412 0.00842 < 0.001 0.30772 – – – – –

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A. Stopka and M. Boareto / Journal of Theoretical Biology 475 (2019) 60–72 Table A.5 Statistical tests for long-term simulations with uncoupled and coupled cell cycle and differentiation (Fig. 5). Overgrowth rates O among the unstable cases as well as corresponding T-values and p-values are listed.

σα = 0.02 σα = 0.04 σα = 0.06 σα = 0.08 σα = 0 . 1 σα = 0.12

Coupling

O [%]

Compared samples

T-value O

p-value O

c = 1.0 c = 0.0 c = −1.0 c = 1.0 c = 0.0 c = −1.0 c = 1.0 c = 0.0 c = −1.0 c = 1.0 c = 0.0 c = −1.0 c = 1.0 c = 0.0 c = −1.0 c = 1.0 c = 0.0 c = −1.0

61.17 75.84 66.65 73.28 84.03 82.38 91.73 89.87 91.38 96.98 95.75 96.81 100.0 98.79 96.76 100.0 98.91 99.66

c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0 c = 1.0, c = 0.0 c = 0.0, c = −1.0 c = 1.0, c = −1.0

1.87169 1.27391 0.70350 1.55789 0.28462 1.38934 0.46548 0.43834 0.09495 0.53481 0.55526 0.08420 1.39649 1.55649 2.30067 1.61541 1.07058 0.90037

0.05610 0.19885 0.47963 0.12586 0.78205 0.16669 0.63363 0.65383 0.92245 0.61160 0.58004 0.90998 0.06758 0.12488 0.02312 0.09848 0.26847 0.60579

n R +n R

with R = 1 n1 +n2 2 . The null hypothesis is that the two samples follow the same Binomial distribution: H0 : R1 = R2 . Under the null hy1 2 pothesis the mean value of t is zero and its variance equal to one. Furthermore, for large n1 and n2 , t is normally distributed. We define the significance as the probability of finding |t| > |tsim | in the null hypothesis, where tsim is the t score of the two simulated samples. In other words, we test how likely it is that, if the two samples follow the same Binomial distribution, we draw a t that is at least as large as tsim . The p-value is thus the probability of obtaining a result of |t| > |tsim |, given that the null hypothesis were true. To estimate the p-value, we perform a Monte Carlo simulation by generating 1,0 0 0,0 0 0 times two Binomial variables drawn from the same Binomial law with the success rate R, while we compute the corresponding t value for each repetition. As each unstable simulation either overgrowths or shrinks, the overgrowth rate O among the unstable cases of a sample is also described by a Binomial distribution. For comparing the overgrowth rates O, we adapted n to the number of unstable cases. Furthermore, we O1 −O2 used the test score t = √ to prevent numerical instabilities for extreme values of O (O = 0 or O = 1) and chose an ( +O(1−O ))(n1 +n2 )/(n1 n2 )

infinitesimal  = 10−100 . Then the mean value of t is equal to zero and its variance is approximately one for normal values of O. To test the statistical difference between average differentiation rates D, we performed a Welch’s T-test, which is an adaptation of the unpaired two-sample Student’s T-test for independent samples with unequal variances. We used the function scipy.stats.ttest_ind to perform the T-test for the means of two samples (https://www.scipy.org/). For both statistical tests we chose the same level of significance α . We chose a critical α level of 0.005 instead of the traditional level of 0.05 to reduce the probability of false positives, i.e. falsely rejecting the null hypothesis. Testing statistical significance with a significance level alpha = 0.005 provides a strong evidence for the alternative hypothesis being true. A.5. Subpopulation in single-cell RNA-Seq data of adult murine NSCs Here, we analyzed the subpopulation of cells that do have non-zero expression levels in both gene marker sets. For this analysis, we excluded cycling cells that do not express differentiation markers and cells that do express differentiation markers without cell cycle activity. The results show, that also for this subpopulation there is a positive correlation between cell cycle and differentiation in adult mouse NSCs (p-value < 0.001, Fig. A.3).

Fig. A.3. Cell cycle marker expression levels against cell differentiation marker expression levels of those adult mouse NSCs (Llorens-Bobadilla et al., 2015), that express both, cell cycle and differentiation markers.

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