Mitotic Events in Cerebellar Granule Progenitor Cells that Expand Cerebellar Surface Area Are Critical for Normal Cerebellar Cortical Lamination in Mice

Abstract

Late embryonic and postnatal cerebellar folial surface area expansion promotes cerebellar cortical cytoarchitectural lamination. We developed a streamlined sampling scheme to generate unbiased estimates of murine cerebellar surface area and volume using stereological principles. We demonstrate that during the proliferative phase of the external granule layer (EGL) and folial surface area expansion, EGL thickness does not change and thus is a topological proxy for progenitor self-renewal. The topological constraints indicate that during proliferative phases, migration out of the EGL is balanced by self-renewal. Progenitor self-renewal must, therefore, include mitotic events yielding either 2 cells in the same layer to increase surface area (β-events) and mitotic events yielding 2 cells, with 1 cell in a superficial layer and 1 cell in a deeper layer (α-events). As the cerebellum grows, therefore, β-events lie upstream of α-events. Using a mathematical model constrained by the measurements of volume and surface area, we could quantify inter-mitotic times for β-events on a per-cell basis in post-natal mouse cerebellum. Furthermore, we found that loss of CCNA2, which decreases EGL proliferation and secondarily induces cerebellar cortical dyslamination, shows preserved α-type events. Thus, CCNA2-null cerebellar granule progenitor cells are capable of self-renewal of the EGL stem cell niche; this is concordant with prior findings of extensive apoptosis in CCNA2-null mice. Similar methodologies may provide another layer of depth to the interpretation of results from stereological studies.

INTRODUCTION

Understanding the growth dynamics of the central nervous system (CNS) is of significant importance to public health. For example, the post-natal human brain doubles in size, reaching 75% of adult brain weight by post-natal year 1 (1). Although not all mammals show such marked post-natal CNS growth as humans, all mammals share in common a significant increase in cerebellar growth during the late embryonic and postnatal epochs. Indeed, human premature birth disrupts normal cerebellar growth in the third trimester (2), underscoring the importance of late embryonic as well as postnatal developmental stages for human development. In mammals, the cerebellum plays important roles in motor learning and motor coordination, and has also been implicated in primate cognition (3, 4). Therefore, it is not surprising that perinatal human infants in particular show susceptibility to defects in cerebellar development. For example, postnatal administration of glucocorticoids for treatment of lung immaturity inhibits proliferation of cerebellar granule neuron progenitor cells (CGNP) (5). Such treatments have been linked to long-term neurocognitive deficits in school-age children (6). Studies of experimental models of cerebellar growth often implement microscopic analyses but an unbiased stereological workflow capable of extracting meaningful data from simple hematoxylin and eosin-based assays has not been implemented to date. A major motivation of this work was to generate a workflow scheme capable of deriving insights into cerebellar external granular layer (EGL) changes based on simple metrics. We hypothesized that by implementing mathematical modeling, we could understand growth dynamics in the CGNP niche.

The cerebellum is derived from 2 germinal zones, the ventricular/subventricular zone surrounding the fourth ventricle and the EGL, which is populated by CGNP. Sonic hedgehog (SHH) signaling is required for both cerebellar morphogenesis and CGNP proliferation (7, 8). CGNP cells emerge from the rhombic lip and then migrate to the EGL. Purkinje neurons originate in the hindbrain ventricular/subventricular zone (9, 10) and develop to become the source of SHH proteins (11). Seminal experiments by Altman and Anderson showed that preventing EGL proliferation by brain irradiation leads to cerebellar pathology, including small cerebellar size and Purkinje cell dyslamination (12). Although radiation has numerous sequelae, these early experiments underscored a relationship between EGL proliferation and total cerebellar growth. Decreasing EGL proliferation intrinsically in CGNPs decreases cerebellar foliation resulting in secondary dyslamination of Purkinje cells (13). Thus, proliferation of CGNPs in the EGL, Purkinje cell cytoarchitectural organization, and cerebellar foliation are intrinsically linked developmental processes.

CGNP proliferation drives EGL volume expansion. In rats, EGL volume peaks at P15, and decreases with complete EGL regression by P25; nevertheless, cerebellar surface area plateaus at P30 (14). In mice, adult foliation is present by P7 and characterized by 10 folia (termed I–X); by 1 month of post-natal age, the cerebellum is entirely formed (15). Using clonal analysis techniques in mice, Espinosa et al demonstrated that symmetrical proliferation of the CGNPs in response to Purkinje cell-derived SHH occurs until the third postnatal week (16). Indeed, the level of SHH expression is linked to the extent of cerebellar foliation (17). Antagonizing CGNP proliferation, either extrinsically by decreasing SHH signaling (7), or by intrinsically affecting CGNP proliferation (13), results in secondary Purkinje cell dyslamination. SHH-induced CGNP proliferation is physiologically blocked by cyclic AMP activation or by treatment with basic fibroblast growth factor and bone morphogenetic protein 2 (8, 18), presumably by antagonizing SHH signaling (19). Such mechanisms of blocking SHH-mediated proliferation ultimately decrease the number of symmetrical divisions of CGNPs, promoting the cells to exit cell cycle and migrate down Bergman glial filaments into the internal granule layer (IGL).

In summary, cerebellar cortical foliation and cytoarchitectural lamination are intrinsically linked and driven by expansion of CGNP in the EGL. The foliation of the murine cerebellar cortex, therefore, poses a model system for quantitatively describing brain growth to ascertain how the balance of symmetric and asymmetric cell division modulates cortical surface area expansion and, indirectly, neuronal lamination. Here, we utilized the mouse as a model organism because of its importance as an experimental paradigm in translational neuropathology. We developed a stereological method capable of obtaining unbiased estimates of volume, surface area, and thickness of the EGL that we interpreted quantitatively using a simple mathematical model that is parameterized partially through the unbiased measurements and partially through the enforcement of experimentally determined topological and geometrical constraints. Using the mathematical model, we were able to determine mitotic rate parameters that yielded insight on the process of cerebellar foliation, thereby illustrating the power of combining mathematical modeling with neuroanatomical analyses.

MATERIALS AND METHODS

Experimental Animals and Tissue Processing

Timed pregnant CD1 mice were purchased from Charles River under the authority of an IACUC-approved protocol. For post-natal time points, animals were killed via CO₂ inhalation in a controlled chamber (flow rate = 20% of the chamber volume/minute). The brains were dissected from the cranium and immersion fixed in 4% paraformaldehyde in phosphate buffered saline (PBS) overnight at 4°C. Following fixation, tissue was equilibrated in 30% sucrose in PBS for 24 hours at 4°C. Tissue was then embedded in OCT (Tissue Tek, Thermo Fisher Scientific, Waltham, MA) cryomolds and frozen in a dry ice-ethanol bath in either coronal or sagittal orientations. After freezing, the tissue was stored in a −70°C freezer. Data derived from conditional cyclin A2 deficient (CCNA2) animals were used from analyses of previously published dataset (13). The generation of CCNA2 targeted mutation in these mice are reported by Kalaszcynska et al (20). Routine genotyping was performed by genomic DNA extraction of tail snips followed by polymerase chain reaction using Clontech Terra Direct Red Dye Premix (Clontech Laboratories, Mountain View, CA) using the following settings on an Express Gene Gradient Cycler™ (Danville Scientific Inc., Charlotte, NC): 98°C for 2 minutes, 35 cycles of 98°C for 10 seconds, 58°C for 15 seconds, and 68°C for 30 seconds, followed by incubation at 4°C. Primers for genotyping were as follows: CCNA2 LoxP Genotype: 5’-GTCTTGTGGACCTTCACCAGACCT-3’, 5’-TGTACAGCATGGACTCCGAGCGAC-3’, 5’-CACTCACACACTTAGTGTCTCTGG-3’, which yields a wild-type and knock-in band at 445 bp and 745 bp, respectively in the absence of recombination, and deleted band of 580 bp after cre-mediated recombination. The Cre primers were: Cre1-Forward 5’-AAAATTTGCCTGCATTACCG-3’, Cre1-reverse 5’-AATCGCGAACATCTTCAGGT-3’.

Histology, Confocal/Brightfield Microscopy, Morphometry and Image Capture

OCT-embedded tissue blocks were mounted onto an HM550 cryostat for sectioning. For confocal microscopy applications, the tissues were cryosectioned at 12 to 15 µm and loaded onto Superfrost™ plus slides (Fisher) and immunostained using anti-Cyclin A2 (Santa Cruz Biotechnology, Santa Cruz, CA, SC-596) and anti-phospho-Histone H3 (Cell Signaling, Danvers, MA, 9706S); secondary antibodies were purchased from Molecular Probes (Billerica, MA), and nuclei were counterstained with DAPI (Sigma, St. Louis, MO). An LSM700 Zeiss Confocal Microscope was used and images were captured as .czi files using Zen™ software. Post-capturing, images were opened in FIJI, colors were separated, and files saved as TIFF images. Collages were generated using Adobe Photoshop CS6 and Adobe Illustrator CS6.

Unbiased Stereology

For unbiased stereology applications, tissue section thickness was 50 µm, and sections were floated onto a PBS bath and mounted on glass slides treated with Vectabond reagent (Vector Laboratories, Burlingame, CA). The entire cerebellum was sectioned for each sample. Because our measurements are isotropic, specific tissue orientation is not necessary for these procedures. Tissue sections were stained with hematoxylin and eosin, dehydrated in graded ethanol washes followed by xylene treatment; glass cover slips were mounted with Permount (Fisher). For stereological quantification of postnatal sections, every tenth section was evaluated. For stereological quantification of embryonic sections, every fifth section was evaluated. Cavalieri estimations were used to generate volume estimates of the external granule layer under the following parameters: counted in 10× magnification (objective was a Zeiss EC Plan-Neo Fluar 10×/0.3), section thickness = 50 µm, mounted thickness = 20 µm, grid size = 30 µm, sampling angle is randomized by StereoInvestigator™ (MBF Bioscience, Williston, VT), shape factor = 4.00. In the event that a section was damaged or lost during tissue processing, estimations of lost sections were performed automatically by StereoInvestigator™.

The isotropic fakir workflow for cerebellar surface area estimation was based on Kubinova and Janacek (21). Three orthogonal isotropic fakir probes are generated in an unbiased fashion by the computer program (StereoInvestigator™ v11), and are represented by lines that are solid and then dashed (Fig. 1). Areas of intersection are selected by the user if optically focused cerebellar folia intersect at the point where the solid line converts to a dashed line. StereoInvestigator™ v11 calculates the intersections of the structure (in our case cerebellar folia) for an estimated surface area. The parameters used for this estimation are as follows: counted in 10× magnification, section thickness = 50 µm, mounted thickness = 20 µm, line separation = 100 µm, volume per unit length of probe = 10000 µm³. The summation of surface area of sections for each age group was multiplied by a factor of 10 to account for the sampling interval for postnatal specimens. In the case of the data derived from CCNA2 mice, surface area estimates were derived from every fifth tissue section, and the summation of surface area of sections for each age group was multiplied by a factor of 5 to account for the sampling interval. In the event that sections were lost during tissue processing, we took the mean value of intersections from the sections prior and after the missing sections as an estimation of intersections for the missing section (e.g. if section 3 was lost in processing and section 1 = 5 intersections and section 3 = 11 intersections, section 2 was estimated at ~8 intersections). Such estimated intersections were not utilized to calculate the coefficient of error.

Figure 1.

Postnatal cerebellar growth measured by isotropic fakir methodology. (A–E) Photomicrographs of post-natal murine cerebellar sections stained with hematoxylin and eosin. Post-natal ages are delineated on the left. Despite sectioning orientation, unbiased estimates of folial surface area and external granular layer (EGL) volume can be generated. (F, G) Isotropic fakir workflow in StereoInvestigator™. Isotropic fakir, which was originally utilized for single cells, is used on thick sections to obtain the surface area estimates. The Fakir probes are lines with a solid and dashed component. A point in which the solid and dashed components of the probe intersect the focused edge of the cerebellar folia is shown at higher power in (G). This area is selected by clicking a mouse at this point of intersection by the user.

Neuronal Tracing Experiments

Neuronal tracing experiments were performed in adult nestin-cre, CCNA2^fl/fl mice. Control animals were CCNA2^fl/fl-preserved littermates of the nestin-cre, CCNA2^fl/fl mice. Animals were anesthetized with ketamine/xylazine, and perfused transcardially with 4% paraformaldehyde. Whole hindbrain/cerebella were then dissected under a stereomicroscope and prepared by Golgi’s method using the protocol of the manufacturer (FD Neurotechnologies, Columbia, MD); slides were prepared with a 100-µm cryosection thickness. Tracing was performed with Neurolucida™ v11 on Z-stack images obtained from StereoInvestigator™ v11. Specifically, contours for tracing were drawn at 10× magnification, and the Meander Scan function was utilized to generate a collage of Z-stack (1 picture per micron Z-step) photomicrographs (13600 × 13600 pixels) that were then saved as tiff images linked to a .dat file. The .dat file was opened in Neurolucida™ v11 for tracing. The tracings were then exported for illustration as a PDF file for graphical layout in Adobe Illustrator CS6. Neuronal tracings were performed with Neurolucida Explorer™. Low-power images of brainstems and cerebellum was also performed using the Meander scan function of StereoInvestigator™ v11 and the collaged image opened in Neurolucida™ v11. A Microsoft screen capture was taken and pasted to a PowerPoint (ppt) file, which was than selected and placed into an Adobe Illustrator CS6 file.

Statistical Analysis and Mathematical Simulations

For each post-natal time point, a total of 3 independent animals were utilized, a common sample size in cell biology (22, 23). Embryonic CCNA2 descriptive and inferential statistics were previously described (13). Statistical hypothesis testing was performed using ANOVA/Tukey HSD test or T-test, when appropriate, in Rv2.12. Mathematical simulations were performed using SciPy 0.14.0 under Python v2.7.7. Coefficient of error for stereology applications was determined as described by Gunderson and Jenson (24) using either StereoInvestigator™ v11 or Microsoft Excel™ for Mac v14.4.

Construction of the Mathematical Model

As a starting point, we considered the construction of a model for the development of the EGL in which we track the number of cells in each of K = 10 distinct layers (from pia to cerebellar white matter). The EGL is divided into a proliferative component (EGL-a) with Ki67-positive/Pax6-positive CGNP, and an EGL-b with “pre-cerebellar granule neurons” (pre-CGN). These cells are transit amplifying and Ki67-positive as well as cerebellar granule neurons (Ki67-negative, Zic1-positive) that migrate out of the EGL-b into the IGL. The average nuclear diameter of these cells in the EGL is ~4 µm. As a proxy for the number of cells, we model the surface area of particular types of cells in each of the layers. The dependent variables in this model are the surface areas of proliferative s_k, committed pre-cerebellar granule neuron (CGN) $s_k^{\prime }$ and post-mitotic $s_k^{″}$ cells that are occupied in each layer k Their dynamics are described through the ordinary differential equations:

$$\frac{d{s}_k}{dt}=({\beta }_k-{\mu }_k-{\delta }_k-{\epsilon }_k)s_k+({\alpha }_{k-1}+{\delta }_{k-1})s_{k-1}$$ (I)

$$\frac{d{s}_k^{\prime }}{dt}=({\beta }_k^{\prime }-{\mu }_k^{\prime }-{\delta }_k-{\epsilon }_k^{\prime })s_k^{\prime }+({\alpha }_{k-1}^{\prime }+{\delta }_{k-1})s_{k-1}^{\prime }+2{\epsilon }_k{s}_k+{\gamma }_{k-1}{s}_{k-1}$$ (II)

$$\frac{d{s}_k^{″}}{dt}=2{\epsilon }_k^{\prime }{s}_k^{\prime }+{\gamma }_{k-1}^{\prime }{s}_{k-1}^{\prime }-({\nu }_k+{\mu }_k^{″})s_k^{″},$$ (III)

where the rate parameters, as described in Table 1, are both time-dependent and space dependent, as denoted by the subscript k. As a corollary to these equations, one can compute the net output of the EGL using the differential equation $ẋ={\nu }_k{s}_k^{″}$, where the dot denotes a time derivative. In these equations we number the layers starting from the most superficial layer (starting at k = 1), and set s₀ = 0 for notational convenience. These formulae imply that the change in the total surface area $a_k=s_k+s_k^{\prime }+s_k^{″}$ of layer k can be expressed ${ȧ}_k={ṡ}_k+{ṡ}_k^{\prime }+{ṡ}_k^{″}$.

Table 1.

Variables and Rate Parameters in the Full Model Given by Equations I–III

Symbol	Description
sk(t)	Surface area of proliferative CGNP cells in layer k
$s_k^{\prime }(t)$	Surface area of committed cells (“pre-CGN”) in layer k
$s_k^{″}(t)$	Surface area of committed post-mitotic cells (CGN) in layer k
h(t)	The mean thickness of the EGL
ha(t)	The thickness of the EGL-a
${\alpha }_k(t),{\alpha }_k^{\prime }(t)$	Out-of-plane symmetric mitosis in layer k
${\beta }_k(t),{\beta }_k^{\prime }(t)$	In-plane symmetric mitosis in layer k
${\gamma }_k(t),{\gamma }_k^{\prime }(t)$	Out-of-plane asymmetric mitosis in k
${\epsilon }_k(t),{\epsilon }_k^{\prime }(t)$	In-plane symmetric mitosis to more committed state
${\mu }_k(t),{\mu }_k^{\prime }(t),{\mu }_k^{″}(t)$	Apoptosis rate in layer k
δk(t)	Movement from layer k to k + 1
νk(t)	Migration rate of post-mitotic cells out of EGL at layer k

The dependent variables are $s_k,s_k^{\prime }$. Rates are given per surface area (per cell) per unit time.

CGNP, cerebellar granule neuron progenitor cells; EGL-a, proliferative component of external granular layer; pre-CGN, pre-cerebellar granule neurons.

Informed by the data, we assume that at P0 there are K = 10 cell layers in the EGL, with the first 4 layers comprising the EGL-a, which consists of proliferative cells. We assume that the next 6 layers comprise the EGL-b, of which 2 layers contain transiently amplifying pre-CGN cells. Using this formulation, the volume of the EGL, $V(t)=\rho {\sum }_{k=1}^K(s_k(t)+s_k^{\prime }(t)+s_k^{″}(t))$, can be calculated where ρ represents the thickness of a cell layer.

Given the measurable physiological growth parameters, this model may be used to predict the time-course of cell population counts in the EGL. We note here that these equations can be interpreted as a mean-field approximation of a finer-grained stochastic model for proliferation. These differential equations are accurate in this approximation in the case where the number of cells is large, which is the case in the post-natal EGL, where the number of cells alive at P0 is exponentially large, as the tissue is macroscopic in scale. Unfortunately, because of the large number of parameters, using the exact model is often impractical. By making further simplifications, we sought to reduce the complexity of the model. We began by reducing the number of effective rate parameters.

We choose to constrain the relationship between the rate parameters (α, β, μ, γ, δ) by making certain physical and biological assumptions that are supported by both our observations and by observations in earlier literature. First, we require that cells are incompressible and that holes do not develop in the EGL. That is to say that all the layers maintain the same surface area, with the exception of the deepest layers, which are allowed to degrade through migration of cells into the IGL. We ensure this criteria by setting $s_k(0)+s_k^{\prime }(0)=s_1(0)+s_1^{\prime }(0)$ for all k < h(t), and ${ṡ}_k+{ṡ}_k^{\prime }={ṡ}_1+{ṡ}_1^{\prime }$ for all k < h(t). These conditions constrain the migration rates δ_k (t) and ν_k (t).

We also assume that members of each of the cell types behave largely the same except in certain positional contexts. In doing so, we greatly reduce the degrees of freedom of the problem. Prior studies have described the EGL-a as largely a homogeneous group of proliferating cells (25), although other studies have determined that the EGL-a consists of specific “anchoring centers” that proliferate in order to expand the surface area (28). Because we do not know the identity of these cells or the size of the pool from histological observations, we model these cells as the same in order to determine their average kinetics. That is to say: β_k (t) = β(t) and α_k (t) = α(t) Likewise, we assume that proliferating cells all undergo apoptosis at the same rate μ regardless of location in the EGL. We also assume that the pre-CGNs behave the same so that ${\beta }_k^{\prime }=\beta \prime$.

With all of these assumptions in place, we simplify the system of equations (I–III) as follows for the proliferative phase of P0 to P14, where the thickness of the EGL is held fixed:

$$\frac{da}{dt}=(\beta (t)-\mu )a$$ (IV)

$$\frac{ds}{dt}=4\frac{da}{dt}$$ (V)

$$\frac{ds\prime }{dt}=(\beta \prime (t)-\mu -\epsilon \prime (t))s\prime +\gamma (t)s$$ (VI)

$$\frac{ds″}{dt}=2\epsilon \prime s\prime -\mu s″-\nu s″$$ (VII)

$$\frac{dx}{dt}=\nu s″.$$ (VIII)

In these equations, s denotes the total surface area of CGNP cells in the EGL-a; s′ and s″ denote the total surface area of committed and post-mitotic cells in the EGL, respectively. As before, the variable x refers to the cumulative total number of cells that migrate out of the EGL. The total volume of the EGL can be computed as: V (t) = ρ (s_a + s_b + s′ + s″) = 2ρs_a. Assuming that the physical parameters β, μ,γ and ν are known, this reduced model now has only 2 remaining unresolved rate parameters, ε′, β′. We can resolve these parameters by imposing 2 additional topological constraints. As constraints, we fix the ratios between the 3 different cell types so that EGL-a consists of 4 layers of CGNP cells and EGL-b consists of 2 layers of pre-CGN cells and 4 layers of post-mitotic cells.

In the phase where the EGL decays (P14–P21), we assume that the CGNP and pre-CGN cells lose their proliferative ability and undergo differentiation. We model this phase through the following equations:

$$\frac{da}{dt}=0$$ (IX)

$$\frac{ds}{dt}=-\epsilon s$$ (X)

$$\frac{ds\prime }{dt}=2\epsilon s+\gamma s-\epsilon \prime s\prime$$ (XI)

$$\frac{ds″}{dt}=2\epsilon s\prime +\gamma \prime s\prime -(\nu -\mu )s″$$ (XII)

$$\frac{dx}{dt}=\nu s″.$$ (XIII)

The system of equations (V–XIII) now provides a model for computing the surface area and volume of the EGL driven by particular mitosis rates under the assumption that the ratio between the cell-types remains constant. Because inter-EGL migrations do not change the number of cells produced, they have been neglected.

RESULTS

Unbiased Estimation of Cerebellar Surface Area

In the mouse, by embryonic day 15 (E15), the bilateral thickenings of the rhombic lip fuse in the midline (26). Foliation is completed by the third post-natal week and is mediated by both differential proliferation of cerebellar granule neuron progenitor cells in the EGL (27), as well as by the generation of specific “anchoring centers (28).” Significant expansion of the cerebellar folial surface area occurs during post-natal mouse development (Fig. 1). To apply constraints to our mathematical model, we developed a methodology and workflow capable supplying mean EGL volume, mean EGL thickness, and folial surface area (i.e. the surface area of the entire cerebellum). Unbiased estimates of EGL volume were determined by Cavalieri estimation, and unbiased estimates of folial surface area were generated by our isotropic fakir workflow (Fig. 1). EGL thickness is derived from calculations dividing volume by surface area. Estimation of EGL volume by Cavalieri estimation demonstrates that EGL volume peaks at 7 days and then returns to baseline, as expected. The mean coefficients of error, Gunderson m = 1, are P0 = 0.109, P1 = 0.053, P7 = 0.021, P14 = 0.021, for P21 = 0.026. Isotropic fakir estimates showed the following mean coefficients of error, Gunderson m = 1: P0 = 0.269, P1 = 0.088, P7 = 0.070, P14 = 0.040, for P21 = 0.063. We conclude that the isotropic fakir workflow is capable of generating unbiased estimates of cerebellar surface area with acceptably low coefficients of error. The results of the measurements are plotted in Figure 2. We note that these results (p21 mean surface area = 483 mm², SEM = 24), although larger, are similar to those in prior studies using folial dissection techniques that would be expected to underestimate surface area (adult mean surface area = 146 mm², SEM = 4.3, reported by Sultan and Braitenberg [29] and re-illustrated by Sultan [30]). In contrast, surface area calculations based on in vivo magnetic resonance imaging (MRI) significantly underestimate cerebellar surface area by an order of magnitude (12.7 mm², SD = 0.8 [31]).

Figure 2.

Metrics for post-natal cerebellar folia. (A–C) Measurements of external granular cell (EGL) volume (A), folial surface area (B), and EGL thickness (C). Statistical hypothesis testing was performed using ANOVA/Tukey HSD tests. For volume estimations, all comparisons showed statistical significance with the exception of the following: P0–P1, p = 0.65; P0–P21, p = 0.998; P1–P21, p = 0.74; P7–P14, p = 0.1. For surface area estimations, all comparisons showed statistical significance with the exception of the following: P0-–P1, p = 0.13; P1–P7, p = 0.063; P14–P21, p = 0.999. For thickness estimations, the comparisons to show statistical significance included: P0–P21, p = 0.017; P14–P21, p = 0.041; P7–P21, p = 0.011.

EGL volume is a major driver of folial surface area. Furthermore, during the first 2 weeks of murine post-natal development, folial surface areas increase exponentially and then plateau. To obtain mean EGL thickness, we divided the EGL volume by the folial surface area. We conclude that during the exponential growth phase of the EGL and folial surface area, the mean thickness of the EGL is constant. For the EGL thickness to remain constant, CGN migration out of the EGL would have to be replaced by CGNP self-renewal. Therefore, thickness can be used as a topological proxy for CGNP niche self-renewal. To help give meaning to the volumetric, surface area, and EGL thickness data, we analyzed it within the context of a simple mean–field kinetic model for cellular proliferation in the EGL while utilizing constraints obtained from our unbiased estimations of folial surface area, EGL volume, and EGL thickness. Although different mitotic rates are noted in different regions of the cerebellum in humans (32) and mice (27), for purposes of modeling growth dynamics of the entire cerebella, we ignore the complex and intricate topology of the foliations of the EGL and the heterogeneities in proliferation between vermis, rostral and caudal folia to make the assumption that the EGL is composed of 3 types of cells (proliferative, committed, and post-mitotic) arranged in stacked sheets or layers. The EGL is divided into an EGL-a composed of proliferative CGNP; the EGL-b is composed of transit amplifying pre-CGNs and Zic1-positive post-mitotic cerebellar granule neurons undergoing migration to the IGL (13, 33). These variables are listed in Table 1 and illustrated in Figure 3. Post-proliferative cells would be Zic1-positive CGNPs X and Y of the EGL-b in the schematic (Fig. 3B), and their migration rate outs of the EGL-b are denoted as ν. The entire EGL are arranged in a certain number of distinct layers or sheets K, which may change, where each layer is a single cell thick.

Figure 3.

Definition of mathematical parameters. (A) Cerebellar granule neuron precursor (CGNP) cell proliferation in the external granular layer (EGL). The top row illustrates confocal photomicrographs of the EGL at different developmental stages. Animal post-natal ages are delineated on the top. The molecular markers utilized on the left are color coded to match the images. Cyclin A2 is an S-phase cyclin; pH3 labels cells undergoing mitosis. (B) Schematic drawing. The parameters utilized in the formulae of the text are described visually. CGNPs are illustrated as black circles. In this example, cells A, B, and W undergo mitoses. A represents a CGNP progenitor cell undergoing symmetric division leading to 2 progenitor cells (a β-event). Cell B represents a vertical mitosis resulting in no net loss of cells in layer 1 (an α-event). C directly migrates from its layer to a lower layer (a δ-event). Two apoptotic cells, D and Z, are illustrated as thinner circles with smaller font (a μ-event). Cells X-and Y represent CGNPs of the EGL-b that then migrate to the IGL (a ν-event). (C) Parameter definitions in the mathematical model and the terms utilized.

Based on our estimations of EGL thickness (Fig. 2), we note that during the proliferative phases P0–P14, the number of cell layers K does not change significantly, but is drastically reduced from P14–P21. Mitotic events occur in both the EGL-a and the EGL-b, with mitoses in the latter likely resulting in a transit amplifying pool of CGNP cells. We denote the exponential rate of mitoses of such committed CGNPs as β′, represented schematically as cell W in Fig. 3B. The folial surface area contributed by self-renewing CGNPs of each layer k we denote as s_k, and the folial surface area contributed by committed CGNPs as $s_k^{\prime }$. Within each layer, the surface area, or the number of cells, is controlled by the interplay between mitosis, migration, and apoptosis rates (μ) are represented as cells D and Z in Figure 3B, although we note that based on prior studies in rodent cerebellum, μ is very low (<1% of EGL cells [34]).

Each mitotic event may result in several distinct outcomes, which we define theoretically as follows: Events resulting in either 2 daughter cells in the same layer (a symmetric division of a progenitor cell resulting in 2 progenitor cells) are designated as β; the rate of transit amplification of pre-CGN cells of the EGL-b is termed β′. Events resulting in either 2 daughter cells in the 2 different layers (i.e. a symmetric division of a progenitor cell resulting in 2 progenitor cells but without contributing to folial surface area) are designated as α; the rate of transit amplification of committed CGN cells of the EGL-b is termed α′. Topologically, β-events increase surface area whereas α-events replenish the stem cell niche. Asymmetric divisions resulting in a daughter cell in the same layer as the parent cell and a daughter cell in the next layer (i.e. an asymmetric division of a progenitor cell resulting in a committed CGN in the next layer) is termed γ; alternatively, a transit amplification of committed CGNP cells in a deeper EGL layer, we denote as γ’. Cells that after division transition from 1 progenitor mother cell in 1 layer to 2 committed CGNs in the same layer are denoted as ε. All events can be said to occur at certain time-dependent rates, and changes in the number of cells present in each layer result from changes in the balance between each of the different events. Mean CGNP cell nuclear diameter does not change significantly over time; therefore, changes in volume directly represent changes in cell number.

In Figure 3A, β-event proliferations are noted in P1. At P7, the orientation a cell undergoing either an α or ε-event cell is identified by its cyclin A2-positive cytoplasm, which is maintained anchored to the subpial surface (in this context, cyclin A2 represents cells between S-phase and M-phase of the cell cycle). We note that based on cytoarchitecture alone we cannot distinguish between the theoretical α or ε events. Ultimately, due to geometrical constraints, β-type mitoses are needed, as without these surface area would never expand laterally. Because apoptosis (or μ) is known from prior studies to be low, folial surface area expansion approaches the value of β. Furthermore, since self-renewal is balanced by migration out of the EGL, transposition of cells from 1 layer to a deeper layer (δ) has to be balanced by proliferation in deeper layers. We take this geometrical constraint in consideration of our mathematical model by assuming that layers are continuous (i.e. no holes or gaps are present). This is experimentally described as β′-type mitoses, which represents symmetric divisions of a transit amplifying pool of pre-CGNs (a committed cell population).

Mathematical Model Predictions of Progenitor Cell Proliferation

Based on the formulation of the models, we address the question of how changes in surface area and volume arise in the EGL. We assume that the overall surface area of the EGL is the surface area of the layer with the largest surface, i.e. the most superficial EGL layer. Changes in volume occur whenever there are changes in either the surface area or the mean thickness of the EGL. The mean thickness of the EGL changes as cells in the EGL lose their proliferative ability and begin their migration into the IGL (Fig. 2). During the growth phase, the EGL maintains its thickness; thus, EGL thickness is a proxy measure of CGNP self-renewal.

Surface Area Dynamics

First, we analyzed the change in surface area of the EGL, which according to our model is equivalent to the change in surface area of the cells in the most superficial layer s₁. We assume that during the growth phase the cells in the EGL-a are entirely proliferative. So, we can say that $s_1^{\prime }(0)=0$. Surface area changes are due to the balance between in-plane mitosis and the processes by which cells leave the first layer. Assuming that cells in the first layer of the EGL remain proliferative between P0 and P14 we can write the relationships between the mean rate parameters as:

$${\int }_0^1[\beta (t)-\mu ]dt\approx 1.38\frac{1}{6}{\int }_1^7[\beta (t)-\mu ]dt\approx 0.106\frac{1}{7}{\int }_7^{14}[\beta (t)-\mu ]dt\approx 0.0814$$

It is immediately evident that the rate of cellular divisions that lead to surface area growth slows on a per-cell basis as time progresses. In the 24 hours between P0 and P1 the mean inter-mitotic wait time of β-events of at most 17.3 hours per our model, which approximates the 19 hour generation time reported by Fujita et al (35), and the 15 hour generation time found by Yoshioka et al (36). Between P1 and P7, due to the logarithmic relationship between growth rates and surface area changes, growth rate is drastically reduced and continues to decelerate through P14, with predicted inter-mitotic wait times of 227 hours from P1 to P7, and an inter-mitotic wait time of 295 hours from P7 to P14. Between P14 and P21, the surface area is seen to be steady. The observation is expressed quantitatively through the relationship:

$${\int }_{14}^{21}({\beta }_1(t)-{\mu }_1(t))=0.$$

Volume Dynamics

While the surface area dynamics of this problem can be investigated by considering the surface area of just the first cell layer, the volume dynamics depend on the evolution of the entire EGL. The thickness of the EGL does not change significantly between P1 and P14 (Fig. 2C), yet growth (i.e. EGL volume) is robust during this same time period (Fig. 2A). Therefore, migration of the EGL into the IGL is balanced by self-renewal of CGNPs in the EGL-a. Thus, geometrically speaking, EGL thickness is a measure of self-renewal capacity of the CGNP niche of the EGL-a. From P1 to P14, where the surface area of the EGL is growing, the thickness of the EGL is roughly constant at ~10 cell layers. This fact implies the relationship: V̇/ȧ = V/a = ρ (V̇ = change in volume over time, ȧ = change in area over time, ρ = thickness of a cell layer).

The volume dynamics between P14 and P21 are notable because the surface area is held constant while the overall volume of the EGL drops precipitously. This observation implies that the EGL is thinning, which can be attributed to either by migration out of the EGL (ν) or apoptosis (μ). Although interruption of apoptosis extends EGL volume past 21 postnatal days (37), the overall apoptotic rate in the EGL is low, with the majority of apoptotic events at P21 found in white matter and the molecular layers of the cerebellum (38). Therefore, the total rate of volume thinning from P14-P21 is driven primarily by ν, overwhelming the regenerative capability of the EGL. Further assumptions and constraints on the parameters are listed in Table 2. The volume of the EGL can then be found by solving the system of ordinary differential equations constituting our model. The results of this computation are shown in Figure 4, where the simulated surface area and volume as well as the cumulative migration into the IGL are plotted.

Table 2.

Assumptions on Variables and Parameter Values

Parameter	Assumptions
β	In P0, β is the overall mitotic rate of cells undergoing in-plane mitoses and is the principal driver of folial surface area expansion. For the purposes of this model we infer its rate from surface area data under the assumption that δ1 = 0. Assumed uniform in all layers except for in the EGL-b where it is zero.
γ	A mitotic event that does not contribute to folial surface area expansion. Found by conserving overall mitotic rate between P0–P14, same for all layers except for at the boundary between the EGL-a/b and in the EGL-b, where it is zero.
δ	Transposition of cells from one layer to another without a mitotic event. For the purposes of this model, it is computed in each layer to match surface areas with s1. Same for for all cell types in each layer. Also assume that δ1 = 0, δ10 = 0.
ε	Assume that ε = 0 except after P14. After P14, ε is chosen to match the decay kinetics of the data.
α	Assume that there are no symmetric out-of-plane divisions so α = 0.
μ	Is constant for all cell types in all layers, at 1% of the overall mitotic rate
β′	Computed to preserve the ratio between the number of layers constituted by CGNPs and pre-CGNs.
ε′	Computed to preserve the ratio between the number of layers constituted by CGNPs and pre-CGNs.
ν	Chosen to match literature value (36).

The assumptions are used in order to create the simplified model given by equations V–X. CGNP, cerebellar granule neuron progenitor cells; EGL-a, proliferative component of external granular layer; EGL-b, external granular cell layer with “pre-cerebellar granule neurons” (pre-CGN) that are transit amplifying and that migrate out of the EGL-b into the internal granular layer.

Figure 4.

(A, B) Simulated development of cerebellar external granular layer (EGL). (C) Predicted internal granular layer (IGL) volume due to migration from the EGL of cerebellar granule neuron precursor (CGNP) cells. The metric plotted is indicated on the left of the Y-axis for each graph. Simulation uses parameters given in the text (a), 40% reduction in proliferation (b) (5), and 300% increase in apoptosis rate (c) (5).

Mathematical Modeling of Extrinsic and Intrinsic Cerebellar Pathologies

We utilized our mathematical model to understand the implications of cerebellar growth dynamics in 2 experimental paradigms. As an extrinsic cerebellar pathology, we utilize data obtained from Heine and Rowitch (5). In their experiments, animals were treated with dexamethasone (DEX) and CGNP proliferation and apoptosis were evaluated. They found that DEX treatment resulted in 40% reduction in CGNP proliferation, an acute 300% increase in apoptosis; these detrimental effects on CGNPs by DEX have been proposed as a mechanism for cerebellar dysfunction in premature babies treated with DEX. We simulated the effect on cerebellar volume, surface area, and projected IGL volume resulting from CGN migration out of the EGL into the IGL using data points from this study (Fig. 4). In these simulations, all other parameters were held fixed; the migration parameters δ and ν were adjusted to satisfy the model constraints (i.e., that by P21 the volume expansion diminishes due to EGL thinning). By perturbing CGNP proliferation and apoptosis, cerebellar surface area would decrease, and, as expected, CGN contribution to IGL would decrease. Thus, our model predicts that decreasing CGNP proliferation has more pronounced effects than increasing apoptosis. Due to the intrinsic relationship between foliation and cerebellar cortical lamination, the significance of the data obtained from these simulations is that decreased CGNP proliferation would result in profound defects in Purkinje cell-dependent circuits.

We also utilized this model to evaluate intrinsic cerebellar pathologies, namely, the dependence on cerebellar growth on CCNA2, which our group has previously shown to regulated EGL proliferation, post-natal cerebellar growth, and, secondarily, cerebellar cortical lamination (13). For this analysis, we focused on comparing the late embryonic growth characteristics of the cerebellar folia between mice with a targeted deletion of CCNA2 to neural progenitors (Nestin-cre, CCNA2^fl/fl) and CCNA2 preserved littermates (Fig. 5A–C). Tracings of cerebellar cortex obtained from Golgi-impregnated post-weaning mice are illustrated in Figure 5D, and underscore the diminished foliation and cortical dyslamination in CCNA2-null cerebella. This marked cerebellar dysmorphism occurs due to events in the embryonic period. The morphometric values during embryonic cerebellar growth are plotted in Figure 5A–C. In all instances cerebellar surface area is decreased in the CCNA2–null embryos, yet EGL thickness does not significantly change. Therefore, despite stunted growth by the CCNA2-null cerebella, α-type events are balanced by ν-type events; therefore, the CCNA2-null CGNPs are capable of this type of self-renewal. β-type events (on a per cell basis) show a trend to be decreased (0.31 in the control and 0.25 in CCNA2-null cerebella), but this was not significant (p = 0.14). However, because the CCNA2-null cerebella show a total decrease in EGL volume, and therefore a decrease in total cells, the total β-type events in CCNA2-null cerebella are decreased.

Figure 5.

Decreased β-type mitoses in CCNA2-null cerebella. (A–C) Embryonic cerebellar morphometric data obtained from a prior dataset used to calculate cerebellar external granular layer (EGL) volume was re-analyzed with new stereological workflow to determine EGL volume, cerebellar folial surface area, and EGL thickness. Methodologies and descriptive statistics of this dataset are described in Otero et al and references therein (13). Error bars denote SEM; control, CCNA2-preserved littermates of Nestin-cre, CCAN2^fl/fl embryos. The mean coefficients of error, Gunderson (m = 1) (24) for the respective parameters and samples are as follows: E14 control volume = 0.024, E17.5 control volume = 0.015, E14 Nestin-cre, CCAN2^fl/fl EGL volume = 0.054, E17.5 Nestin-cre, CCAN2^fl/fl EGL volume = 0.021, E14 control folial surface area = 0.146, E17.5 control folial surface area = 0.105, E14 Nestin-cre, CCAN2^fl/fl folial surface area = 0.26, E17.5 Nestin-cre, CCAN2^fl/fl folial surface area = 0.141. Two-tailed, 2-sample unequal variance T-test shows significant differences between E14 EGL volume (p = 0.04), E14 folial surface area (p = 0.03), and E17.5 folial surface area (p = 0.04). E17.5 EGL volume did not reach threshold for significance (p = 0.057). No differences were seen in EGL thickness at E14 or E17.5 (p = 0.24 and 0.40, respectively). (D) Adult hindbrain cytoarchitectural analysis. Hindbrain were dissected in adult mice after perfusion with 4% paraformaldehyde with subsequent Golgi impregnation. Genotype is denoted on the top. The blue-dashed line denotes the cerebellar folia, which is complex in the control. In contrast, the Nestin-cre, CCAN2^fl/fl mouse shows only 1 folium. Neurolucida drawings of control and Nestin-cre, CCAN2^fl/fl mice are derived from the box labeled “tracing.” These drawings demonstrate disorganization in the mutant, with cells with stellate cell morphology deep to clusters of cerebellar granule neurons. ML, molecular layer; PCL, Purkinje cell layer; PIA, pia mater.

DISCUSSION

Although significant strides have been made in our understanding of the molecular underpinnings regulating cerebellar development, these discoveries are often challenging to implement practically in human clinical settings. The post-natal human infant is particularly susceptible to defects in CGNP proliferation, and at this time the only conceivable modality to assay brain development is through imaging using MRI technologies that give simple metrics such as volume and surface area. In view of the intrinsic relationship between cerebellar foliation and cerebellar cortical lamination, we set out to create a framework based upon a mathematical model in an experimental system where certain variables are known. Although this anatomical technique is applicable to translational neuropathologists interested in mouse models of brain development, we designed our study to utilize simple morphometric data that ultimately could be easily obtained from non-invasive medical procedures (such as an MRI). To our knowledge, this represents the first instance that a workflow capable of generating unbiased estimation of brain surface area from microscopic mouse sections has been developed. We utilized data from our own study and the scientific literature to constrain a simple mathematical model of cerebellar development that demonstrates a geometric constraint to the mitotic rate of CGNPs capable of undergoing mitoses that increase folial surface area. This allowed us to revisit previously held assumptions based on classical data in rodent cerebellar development and to test hypotheses based on simulations from our mathematical model. For example, classical studies have demonstrated α-type mitoses predominate over β-type mitoses (39). Taken in isolation, the data of Zagon et al would result in increased EGL thickness with minimal folial expansion during post-natal development. Quantifying mitotic spindle orientation over the entire cerebellum using standard microscopic imaging modalities would prove to be a challenging and time consuming endeavor, however. Instead, by using a mathematical approach, these types of questions can be more easily addressed. Per our model, mitoses resulting in increased folial surface area (β-events) decrease significantly with advanced postnatal ages, whereas mitoses perpendicular to the folia (α–events) increases. Thus, our models are aligned well with the literature. We conclude that mathematical modeling of brain development permits the identification of biologically relevant processes for embryonic development.

Implications of Mathematical Models for Pediatric Neurology

An estimated 63,000 infants are born yearly in the United States with a birth weight <1500 g (very low birth rate) (40). The incidence of children with very low birth rates has shown trends towards increased yearly incidence, and these children often show significant cerebellar pathologies (41). The cerebellar granule neuron located in the IGL is the most numerous neuron in the CNS, accounting for over 95% of the neurons in the adult cerebellum (42). Our mathematical model predicts that decreased proliferation has a significantly more adverse outcome for IGL cell number than extrinsic processes that increase CGNP cell death (Fig. 4C). A further strength of our model is that the parameters upon which it is based are simple morphometric values. As we transition this to future translational applications, we envision that these morphometric values would be extracted from non-invasive medical imaging in children. Further advantages offered by such mathematical modeling include the ability to make predictions of particular outcomes as a result of interventions. This ability allows us to simulate the effects that various pathologies can have on the development of the EGL (Figs. 4, 5). These data thus provide for us a framework upon which to build similar models of human post-natal cerebellar development. Ultimately, such models could be used to identify children at-risk of not meeting cerebellar developmental morphometric metrics and to institute in such children neuro-psychiatric interventions that may help them overcome those deficits.

Implications for Changing Rates of β-type and α-type Mitoses

Cerebellar granule neurons in the IGL send neuronal processes to the molecular layer to form parallel fibers. Parallel fiber formation occurs during the migration phase of the CGN as the soma moves out of the EGL towards the IGL. These connections carry important consequences for normal Purkinje cell development and Purkinje cell dendritic arborization. Parallel fibers are important for excitatory synapses with Purkinje cells. For an action potential to occur in a Purkinje cell, many parallel fibers must be activated by granule cells, which are activated by mossy fibers. Impairment of the formation of parallel fibers would thus be detrimental to signaling out of the cerebellar cortex. For example, glutamate-receptor-like molecule δ2 (GluR δ2) has been shown to play a key role in synaptic plasticity in the cerebellum. GluR δ2 is highly expressed in Purkinje cells, exclusively in the parallel fiber-Purkinje cell synapse. GluR δ2 regulates parallel fiber-Purkinje cell synapses, long terminal depression induction, and motor learning (43). Without proper parallel fiber integration into this system, the GluR δ2 pathway would inhibit synapse formation and ultimately detrimentally affect motor learning. Thus, without proper formation of parallel fibers, long terminal depression and ultimately, synaptic plasticity, would be impaired. In summary, exit from the EGL by CGN’s is coupled to parallel fiber-Purkinje cell synaptic events, and disequilibrations in the EGL niche would adversely affect such circuitry, as illustrated by our neuronal tracings of CCNA-null cerebella (Fig. 5).

Purkinje cell generation from the fourth ventricle is completed before birth (26) and they begin to form a noticeable layer during the early postnatal days that is several cell layers thick. The mature neuronal cytoarchitecture of the cerebellar cortex thus requires both lateral expansion (i.e. β-events) to “spread out” these Purkinje cells into the adult monolayer, as well as synaptogenesis from migrating CGNs during migration (ν-events) for proper dendritic arborization of Purkinje cells. As expected, our simulations indicate that the rate of ν-events increases during post-natal age (Fig. 4C). To keep the EGL niche stable during the growth phase, ν-events must therefore be balanced by α–events. As β-events diminish, either physiologically as the animal increases post-natal age or experimentally by intrinsic cerebellar pathologies such as CCNA2 ablation, α-events ultimately decrease and ν-events deplete this stem cell niche. Therefore, we argue that it is logical that a mechanism evolved in the EGL niche to “ramp up” the appropriate type of proliferation so as to balance CGN exit out of the EGL. Our mathematical model is aligned with this notion in that it predicts that α-event mitoses increase during the growth phase (Table 3). Thus, perturbing either of these processes would result in both Purkinje cell dyslamination and decreased arborization, which is commonly seen in processes that reduce global CGNP proliferation in experimental animals (13, 44). Indeed, our data in the CCNA2-null cerebella also suggest that equilibration of ν and α events without sufficient β-events ultimately lead to cerebellar cortical dyslamination (Fig. 5). From this we can infer that although α-events are more numerous, β-events are the principal drivers of cerebellar growth and lie upstream of α-events in the equilibration of the stem cell niche.

Table 3.

Mean Parameter Values Given by Postnatal Period

	α	β	μ
P0–P1	0.00 (ND)	1.400	0.014
P1–P7	1.29	0.107	0.014
P7–P14	1.31	0.082	0.014
P14–P21	0	0.027	0.014

All rates are given in units of day⁻¹. β is inferred from surface area changes of the external granular layer (EGL) illustrated in Figure 2. The parameter α is chosen to conserve the overall proliferation rate of cerebellar granule neuron precursor cells during the main proliferative time points from P1 to P14 to a rate corresponding to the initial value of β. This parameter is assumed to be the same for all cell types within each given layer. Given our morphometric data, we are unable to make a prediction of α at the first time point (ND = not determined), so it was chosen to be equal to 0. The δ parameters (not shown) are computed to obey the topological constraints mentioned in the text. The emigration rate was chosen ν = 1.78 was calculated to yield a nearly complete decay of the EGL in the P14 to P21 phase.

With these models we have developed a framework upon which to build future models that predict the rate of cerebellar growth during the post-natal period of children. These models would allow us to identify children at-risk of improper cerebellar growth and ultimately to introduce medical interventions that could decrease the morbidity of the sequelae experienced by children with cerebellar growth defects.