Mathematical modeling supports substantial mouse neural progenitor cell death

Abstract

Background: Existing quantitative models of mouse cerebral cortical development are not fully constrained by experimental data.

Results: Here, we use simple difference equations to model neural progenitor cell fate decisions, incorporating intermediate progenitor cells and initially low rates of neural progenitor cell death. Also, we conduct a sensitivity analysis to investigate possible uncertainty in the fraction of cells that divide, differentiate, and die at each cell cycle.

Conclusion: We demonstrate that uniformly low-level neural progenitor cell death, as concluded in previous models, is incompatible with normal mouse cortical development. Levels of neural progenitor cell death up to and exceeding 50% are compatible with normal cortical development and may operate to prevent forebrain overgrowth as observed following cell death attenuation, as occurs in caspase 3-null mutant mice.

Figure 1

Appropriate ventricular zone (VZ) output requires greater than 5% neural progenitor cell (NPC) death. (A, B) Two simple cell decision paradigms are used to identify a plausibility window and calculate viable levels of NPC death. Beginning at 'P' and following the arrows clockwise, in model D_G1 (A) the NPC population P_i-1 is doubled at the i^th mitosis, then NPC death, 'D,' and differentiation, 'Q,' are imposed, so that the P population for the subsequent cell cycle (CC) is given by P_i = 2(1 - d_i - q_i)P_i-1. (B) In model D_G2 dying NPCs are removed from the NPC population prior to doubling the population, and P_i = (1 - q_i)2(1 - d_i)P_i-1. (C, D) Model output from P₀ = 550,000. VZ output (total Q cells, y-axis) is plotted after 11 CCs when the indicated fraction of NPCs die at each CC (Death, x-axis). The window of plausible VZ output, between 1 × 10⁷ and 2.72 × 10⁷, is indicated by dashed lines. (C) Using model D_G1, the center of the plausible VZ output range (total Q cells = 1.86 × 10⁷) corresponds to 19% NPC death, as indicated by the solid line that descends from plotted VZ output to the x-axis. (D) Similarly, the center of the plausibility window using model D_G2 corresponds to 17% NPC death.

Figure 2

Model D_G1 is insensitive to perturbations in q_i and d_i. (A) Most (>90%) model realizations give plausible ventricular zone (VZ) output (gray bars). The distribution of 10,000 simulations of VZ output (where = 0.19 for all i) is plotted as a histogram. Simulation outputs follow a log-normal distribution. The solid vertical line indicates the geometric population mean, and dashed vertical lines indicate 1 and 2 standard deviations from the mean. (B) Three examples of points in 22-dimensional parameter space where plausible VZ output is observed. The q fractions are plotted along a dashed line and d fractions are plotted along a solid line. The upper panel is from a bin below the mean, the center panel is from the bin at the mean, and the lower panel is from a bin above the mean. CC, cell cycle.

Figure 3

Intermediate progenitor cells (IPCs) modestly increase the requirement for neural progenitor cell (NPC) death. (A, B) IPCs (IP) are incorporated into (A) model D_G1 and (B) model D_G2 similarly. A fraction of Q cells (q^') become IPCs, undergo one CC (2 *IP), then re-enter the Q population. (C, D) Ventricular zone (VZ) output plotted as described in Figure 1. (C) In model D_G1 the plausibility window is centered at 22% NPC death. (D) In model D_G2 the plausibility window is centered at 24% NPC death.

Figure 4

Expansion cell cycles permit much more neural progenitor cell (NPC) death at later cell cycles (CCs). Ventricular zone (VZ) output plotted as described in Figure 1. During expansion CCs, d_i = 0.05; NPC death during subsequent CCs is indicated on the x-axis. (A, B) After three expansion CCs the plausibility window is centered at 29% using model D_G1 (A) and 24% using model D_G2 (B). (C, D) After four expansion CCs the plausibility window is centered at 37% using model D_G1 (C) and 28% using model D_G2 (D). (E, F) After five expansion CCs the plausibility window is centered at 53% using model D_G1 (E) and 36% using model D_G2 (F).

Figure 5

Constrained models require substantial levels of progenitor cell (NPC) death. Ventricular zone (VZ) output as plotted in Figure 1, including intermediate progenitor cells (IPCs) as in model D_G1 (Figure 3A) and model D_G2 (Figure 3B). (A, B) After three expansion cell cycles (CCs) the plausibility window is centered at 33% using model D_G1 (A) and 33% using model D_G2 (B). (C, D) After four expansion CCs the plausibility window is centered at 42% using model D_G1 (C) and 39% using model D_G2 (D). (E, F) After five expansion CCs the plausibility window is centered at 59% using model D_G1 (E) and 47% using model D_G2 (F).