Using cumulative labeling of newly synthesized DNA and explicit counting of each labeled cell migrating from the VZ and subventricular zone, the fraction of terminally differentiating cells has been measured experimentally 1418. We denote these fractions q_i, where i = 1, 2,..., 11 for the 11 cell cycles (CCs) estimated to occur during the neurogenic interval (Table 1). These data, q_i, and the estimated initial population of NPCs (P₀), allow us to determine, through mathematical modeling, what fractions of NPC death at each CC, d_i for I = 1, 2,..., 11, yield VZ output consistent with experimental counts of the total number of excitatory neurons in the mature mouse cerebral cortex. To this end, we require an estimate for the plausible range of total VZ neuronal output.

Measurements of CC duration together with the fraction of NPCs that differentiate at experimentally defined ages permits extrapolation of model parameter q_i for each i 14. Sensitivity analysis 36 provides a means of determining the relationship between inherent uncertainty in these estimates of q_i and uncertainty in simulated VZ output.

Using a Monte Carlo approach, we sampled each q_i and d_i as normally distributed random variables with means and taken from 1437 and variance given by and (Materials and methods). For example, for and ν₁ = 0.5 we have q_i ~N(0.42, 0.122). Sampling the 22-dimensional parameter space for q_i and d_i, and then computing VZ output for each sample point in parameter space, allowed us to determine that ν₁ = ν₂ = 0.5 gives plausible VZ output for >90% of the model realizations (Figure 2A).

For ν₁ = ν₂ = 0.5, we computed the first-order sensitivities of VZ output to variation in each q_i and d_i (Materials and methods). The first-order sensitivities and are measures of how uncertainty in each q_i and d_i drives uncertainty in VZ output. For example, when is close to zero for a given q_i, significant variation in VZ output can still occur despite fixing this q_i. We find that death rates in the first few CCs and differentiation rates in the middle CCs can account for the most significant portion of model variability; yet even the largest and are only approximately 10% (Table 2). As such, variation in VZ output is hardly attributable to variation in any one q_i or d_i. An obvious explanation is that, given exponential growth conditions, outlying values of either q_i or d_i in early CCs are easily compensated by outliers in later CCs. Interestingly, significant variation in q_i and d_i from the mean values and can still yield plausible VZ output (Figure 2B).

NPC mitoses occur on the ventricular surface, but 'non-surface' mitoses are also observed in the developing cortex. Initially, these cell divisions were erroneously considered non-neuronogenic 10, but proliferative intermediate progenitor cells (IPCs) – daughter cells of NPCs that have migrated to the subventricular zone and intermediate zones, and are immunoreactive for the transcription factor Tbr2 – are now known to contribute additional excitatory neurons to the mouse cerebral cortex 111213.

To determine how IPCs may affect the level of NPC death required for plausible VZ output, IPC subpopulation levels were extrapolated from the literature (Table 1 and references therein) and these constraints were incorporated into both models. We take a fraction of the Q cell population after the i^th division – that is, we denote q_i'Q_i as the size of an IPC subpopulation – and these cells undergo an additional CC, doubling their contribution to the total Q population (Figure 3A,B). In initial modeling, we assume that additional IPC CCs and IPC death either do not occur or at least offset each other, and we include those IPCs generated after the last NPC CC in the VZ output calculated for CC₁₁. With the inclusion of IPCs, VZ output without NPC death increases to 1.1 × 10⁹ in model D_G1 and more than doubles to 2.1 × 10⁹ in model D_G2 (Figure 3C,D). With 5% NPC death, model D_G1 calculates VZ output to be 7.3 × 10⁸ and model D_G2 calculates 1.3 × 10⁹ (Figure 3C,D). Since the q_i' fraction is largest in the last three CCs, IPCs have a relatively small effect on the plausibility window; it expands slightly to accommodate 17 to 29% NPC death in model D_G1 and 20 to 29% NPC death in model D_G2. Importantly, a larger IPC population (from a larger IPC fraction or additional IPC CCs without IPC death) could increase, but not decrease, the requirement for NPC death.

Cortical development with 5% NPC death at early CCs requires more than 50% NPC death at later CCs

A primary objection to the idea of substantial NPC death is the intuition that ≥ 50% NPC death throughout neurodevelopment would preclude expansion of the NPC pool. However, the level of NPC death need not be constant at each CC. For example, ISEL+ labels approximately 5% of cells at E10 and the percentage increases significantly thereafter 2035. Moreover, NPC lineage analyses using a genomically encoded marker found that clone size increased during early development but then diminished after E14 25, suggesting significant cell death with corpse elimination at later, but not earlier, CCs. These empirical observations are also consistent with increased model sensitivity to the levels of death at earlier CCs (Table 2). Together, these data suggest that early 'expansion' CCs occur; initially, low PCD levels – for example, setting d_i = 0.05 for 1 ≤ i ≤ 4 provides for four expansion CCs – further constrain VZ output. Here we calculate the corresponding amount of NPC death required during later CCs for plausible VZ output.

The fourth neurogenic CC takes place on E12, a time when significant ISEL+ labeling was observed 20. VZ output with three, four, or five expansion CCs amplifies differences between models D_G1 and D_G2. In model D_G1 with three expansion CCs, the plausibility window permits 21 to 42% NPC death; four expansion CCs permit 26 to 56% NPC death; and five expansion CCs permit 36 to >80% NPC death (Figure 4A,C,E). Using model D_G2, the plausibility window is less broad and accommodates less NPC death. After three expansion CCs the D_G2 plausibility window permits 17 to 34% NPC death; after four expansion CCs it permits 20 to 41% NPC death; and after five expansion CCs it permits 25 to 54% NPC death (Figure 4B,D,F).

We note here that inclusion of four or five expansion CCs makes model D_G1 and model D_G2 readily distinguishable from one another. This distinction is a consequence of the differential involvement of cell death when expansion CCs begin and end, and illustrates some additional sensitivity to the size of the founding NPC population. As described above, during the first CC in model D_G2 the founding population is reduced before mitosis; therefore, for uniformly constant cell death, model D_G2 with initial population P₀ is equivalent to model D_G1 with initial population d₁P₀. Although this distinction leads to only subtle differences in most simulations, when considering the relationship between expansion CCs and plausible levels of NPC death, model D_G1 with four or five expansion CCs illustrates additional potential for biological variation.

Early expansion CCs and IPCs together are most compatible with experimental data and further support high levels of NPC death

Both expansion CCs and IPC CCs occur during normal mouse brain development. When NPC population kinetics include expansion CCs (as shown in Figure 4) and IPCs (as shown in Figure 3), the plausibility window is broadest. For model D_G1 with IPC CCs, three expansion CCs permit 26 to 46% NPC death and four permit 32 to 60% NPC death (Figure 5A,C). After five expansion CCs, VZ output already exceeds the minimum calculated estimate of cortical neurons (data not shown), with = 43% NPC death required during the remaining CCs to restrain VZ output (Figure 5E). As above (Figure 3), model D_G2 with IPC CCs yields a lower and narrower range of NPC death levels: 28 to 41% with three expansion CCs, 32 to 49% with four expansion CCs, and 38 to 61% with five expansion CCs (Figure 5B,D,F).

Apart from initial modeling by Takahashi, Caviness, and colleagues (referenced throughout), Gohlke, Faustman, and colleagues 1744 have used a Kolmogorov forward equation to compute the probability distribution for mouse VZ output explicitly, as a continuous-time Markov chain. In contrast, we estimate this same probability distribution by repeatedly simulating a deterministic difference equation with random perturbations taken from assumed distributions. This allows us to illustrate how significant variation in q_i and d_i fractions, over the course of 11 CCs, can occur and still yield plausible VZ output (Figure 2B). In population measurements using stereological counting, variation at the level of subpopulations of NPCs would probably go unnoticed, yet this may be an important feature of cerebral cortical development. While such variation is implicit in the Kolmogorov forward equation approach, Monte Carlo allows us to view sample trajectories that illustrate this variation.

The Gohlke models also observe that the original Takahashi models lead to VZ output that exceeds experimental counts of cortical neurons by at least threefold 44. However, these authors use two additional parameters (in addition to low-level NPC death) to reduce cortical neuron production to plausible levels: a diminished growth fraction insofar as not all VZ cells are progenitor cells and a clearance time for dying NPCs. Many NPCs co-label for bromodeoxyuridine and ISEL+ 202442; therefore, Gohlke and colleagues have likely recast some additional NPC death as a 'diminished' growth fraction. These parameters have the effect of reducing the proliferative population from which any given q fraction is taken. Consequently, the overproduction observed using the Takahashi model is limited and plausible levels of VZ output are obtained from the Gohlke mouse model 1744.

The Gohlke mouse model is reportedly compatible with levels of NPC death up to 24% 17. This value is in good agreement with 13 to 26% NPC death calculated using models D_G1 and D_G2 (Figure 1). Despite this, Gohlke and colleagues 1745 report subsequent model analysis using NPC death rates at or near 0% NPC death. The contrasting higher levels of NPC death required in Gohlke models of primate cortical neurogenesis 45, relative to murine cortical neurogenesis, may simply reflect an underestimation of mouse NPC death.

A strong implication of all modeling experiments presented here is that TUNEL significantly underestimates NPC death during mouse cerebral cortical development. This reflects technical differences in sensitivity between the two procedures, with ISEL+ being approximately ten times more sensitive than the originally reported TUNEL technique 2122. ISEL+ detects more dying cells not only amongst NPCs, but also in other tissues like the thymus and small intestinal villus 43. Consistent with a tenfold reduced sensitivity relative to ISEL+, TUNEL detects as few as one-tenth of the dying NPCs (5% versus 50%).

In order to reconcile the approximately 5% NPC death proposed by prior models 18 with the forebrain overgrowth phenotype observed in caspase 3-deficient mice, it has been proposed that NPC death may occur normally in a small population of neuroepithelial stem cells or radial glia at an early age (<E12) 46. The claim is that additional survival of a few such cells could underlie forebrain overgrowth in caspase 3-deficient mice because each of these individual cells might ultimately give rise to many neurons (approximately 140 according to estimates from concurrent models). However, this notion is inconsistent with experimental data demonstrating marked overgrowth of NPCs in caspase 3-null embryos by E12, before large numbers of NPC progeny emigrate to the cortex 2642.

Our models are consistent with changes in NPC and neuronal populations that have been observed following experimental attenuation of PCD. For example, a 20% reduction in NPC death from the midpoint of any plausibility window corresponds to VZ output that exceeds the upper bound for that plausibility window. This is strikingly consistent with a 30% reduction in ISEL+ labeling of NPCs observed in caspase 3-deficient mice 42, where exceptional forebrain overgrowth is observed. We suggest that an in vivo correlate of 'exceeding the plausibility window' is forebrain overgrowth.

The breadth of the plausibility window provides a comparative measure of model robustness with respect to viable levels of NPC death. In some scenarios the slope of VZ output is steep (Figure 1D), and a 10% difference in NPC death leads to marked differences in VZ output, while a similar 10% change has little impact on VZ output in models where the slope is less steep (for example, Figure 4E). Notably, early expansion CCs (Figure 4), rather than the size of the founding NPC population or additional IPC progeny (compare Figures 2 and 3), significantly extend the range of NPC death that is compatible with plausible VZ output. Although modeled during the first three to five CCs here because of experimental evidence for pre-E12 expansion CCs 2025, transient low-level NPC death could theoretically operate during any CC. It is tempting to speculate that high levels of NPC death at later CCs follow from low levels of NPC death during early expansion CCs. A similar process has been reported in embryonic stem cells earlier in development 4748 and may be related to a permissive decatenation checkpoint observed in NPCs 48.

One might predict that other perturbations leading to transient high levels of NPC death at early CCs could lead to low level NPC death at later CCs. In this scenario, developmental accommodation of atypical NPC death might occur at the level of stem cell niches 4950, providing sufficient, but not necessarily 'normal,' VZ output. Perhaps local control of NPC death could insulate cerebral cortical development against genetic differences and chemical or environmental insults. Given genetic diversity among NPCs, produced in part by chromosomal aneuploidy 51 and retrotransposition 52, differences in NPC death amongst individuals suggests selection and/or survival mechanisms that influence the mosaic composition of an individual's cerebral cortex.