Revealing age-related changes of adult hippocampal neurogenesis using mathematical models

ABSTRACT New neurons are continuously generated in the dentate gyrus of the adult hippocampus. This continuous supply of newborn neurons is important to modulate cognitive functions. Yet the number of newborn neurons declines with age. Increasing Wnt activity upon loss of dickkopf 1 can counteract both the decline of newborn neurons and the age-related cognitive decline. However, the precise cellular changes underlying the age-related decline or its rescue are fundamentally not understood. The present study combines a mathematical model and experimental data to address features controlling neural stem cell (NSC) dynamics. We show that available experimental data fit a model in which quiescent NSCs may either become activated to divide or may undergo depletion events, such as astrocytic transformation and apoptosis. Additionally, we demonstrate that old NSCs remain quiescent longer and have a higher probability of becoming re-activated than depleted. Finally, our model explains that high NSC-Wnt activity leads to longer time in quiescence while enhancing the probability of activation. Altogether, our study shows that modulation of the quiescent state is crucial to regulate the pool of stem cells throughout the life of an animal.

(S1.1) where n ∈ N is the maximum number of asymmetric NSC divisions, c 0 denotes quiescent NSCs and c k 1 (0 ≤ k ≤ n) cycling NSCs with k divisions remaining. The parameter r describes the activation rate of quiescent NSCs, p is the division rate of proliferating NSCs and q denotes the rate of transformation to astrocytes of NSCs that had already divided k times ( Supplementary Information Fig. S1). Following the hypothesis of Encinas et al. (2011), we assume n = 3. Using the available proliferation rate measurements (4.1) allow to fit separately the model to different data sets (see Supplementary Information Figs. S2a & S2b).
To validate the models, we compare the dynamics of the clonal fit to the time-varying composition of the NSC pool. Model (S1.1) predicts that the fraction of cycling NSCs vanishes within three months ( Supplementary Information Fig. S3a), what contradicts the constant fraction of BrdU incorporating NSCs during aging as observed in the experiments (Encinas et al. (2011) and Fig. 2f). In contrast, model (2.1) is in line with the data showing a constant fraction of proliferating stem cells ( Supplementary  Information Fig. S3b). Figure S1: Graphical representation of model (S1.1).
Quiescent NSCs can become activated to enter the cell cycle and subsequently perform a series of three asymmetric divisions by producing a NSC and another cell (X) before entering the post-mitotic stage and transforming into an astrocyte (A).

S2 Astrocytic Transformation
We want to give an explanation for the claim that NSC apoptosis is almost nondetectable in the case of 1 − θ = 52.7% of the NSC decline resulting from apoptosis.
Consider the stem cell model (2.1) including aging effects (2.2). To model the dynamics of NSC apoptosis, we add a new compartment c A accounting for apoptotic, i.e. biologically dead but physically present stem cells. The dynamics of NSCs at a certain age τ is thus given by d dt where δ phag is the rate at which apoptotic cells are cleared via phagocytosis within, on average, 1/δ phag = 1.
apoptotic stem cells in the entire dentate gyrus. Considering the usual sampling fraction of one sixth of the DG, there only remain about two apoptotic stem cells to be detected.
To analyze the scenario that the accumulation of astrocytes could be explained with a higher transformation rate θ if in addition astrocytes are allowed to undergo apoptosis, we consider the modified dynamics where d 2 is the death rate of astrocytes. Estimating θ and d 2 simultaneously yields θ = 0.364 and d 2 = 1.4 × 10 −5 d −1 , showing that there is no justification for such scenario. In addition, assuming θ = 1 and only estimating d 2 results in an AICc score of 373.1, which compared to the AICc of 353.3 for the no-apoptosis model further indicates that there is no support for this scenario from a model selection viewpoint.

S3 Dynamics of Progenitor Cells
To model the dynamics of progenitor cells, we again make use of the study of Encinas et al. (2011). We consider the data set in which the authors label dividing cells with BrdU and track the number of BrdU labeled progenitors ( Supplementary Information  Fig. S4). Because of the rapid increase and subsequent decrease of labeled cells, they concluded that progenitors perform a series of symmetric self-renewing divisions, followed by subsequent transformation into neuroblasts. We model the dynamics of progenitors with the equations Here, P i is the number of progenitor cells with i remaining divisions and p > 0 is the proliferation rate. Moreover, we assume that at the start of the experiment, all progenitors have N remaining divisions, i.e. P N (0) = n for some n > 0 and P k (0) = 0 for k = N . For quantifying p, we consider the corresponding cell cycle length t c , which is linked to p via (4.1). The cell cycle length of progenitor cells has been measured in different studies, however with contrasting results ranging from 12−14 h to about 22 h (Hayes and Nowakowski 2002; Farioli-Vecchioli et al. 2014). We thus employ an unbiased approach for quantification by assuming different cell cycle lengths t c and compute the R 2 of the fit dependent on N , the maximum number of progenitor divisions ( Supplementary  Information Fig. S5).
The best fit can be obtained for N = 2, 3 or 4, but only N = 2 allows for a cell cycle length in the range of what is experimentally observed with the maximum R 2 at t c = 14.4 h. To achieve a better compromise between our model assumption and the measured cell cycle lengths, we relax the condition of an optimal R 2 . A visual assessment of the fit shows that R 2 = 0.95 provides a reasonable fit to the data. For N = 2, the maximal t c for which R 2 = 0.95 can be achieved is t c = 15.6 h ( Supplementary Information Fig. S6), which we assume for our subsequent analysis.

S4 Sensitivity Analysis
Model (2.1) contains three unknown parameters (a, q and r), which needed to be estimated from the data. It is interesting to analyze how variability in those parameters affects downstream findings that depend on estimated values. We investigate two questions. First, we check whether the clonal data of Fig. 3 can be fitted using the population-level estimates of model (2.1) by allowing only one parameter to vary. This would indicate that uncertainty in the estimate of one parameter could indeed explain population-level and clonal dynamics simultaneously. Second, how uncertainty in the population-level estimates of model (2.1) transfers to the analysis of different scenarios to explain the saturating decline of NSC numbers during aging.

S4.1 Clonal Fit with One Varying Population-Level Parameter
We fit the clonal data as outlined in the Materials and Methods section using the population-level estimates of parameters q and r shown in Table 1 and the previously stated assumption a = 0.525. Changing parameter q results in a rapid decline of the fraction of quiescent clones ( Supplementary Information Fig. S7a). Estimating r leads to a model fit with a fraction of activated and depleted clones, which does not match the one year time point data ( Supplementary Information Fig. S7b). Finally, if only parameter a varies, the one year time point cannot be matched for any of the three time series ( Supplementary Information Fig. S7c).

S4.2 Saturating Stem Cell Decline
Validating the models accounting for different scenarios to explain the saturation of the decline of NSC numbers, we re-estimate the parameters q and r (Table 4). Hence, model findings are independent of the original population-level estimates. However, findings depend on the assumption of 5% of stem cell divisions being symmetric, i.e. a = 0.525. To investigate the sensitivity of the results in respect to this assumption, we re-estimate parameters of all age-related scenarios, assuming either a lower selfrenewal rate (a = 0.5) or a higher one (a = 0.6). As can be seen from Supplementary Information Table S1, such variation of the self-renewal fraction does not change the results of model selection. The alternative scenarios remain to have at least 2.5 points higher AICc score.