A Mathematical Model of Granule Cell Generation During ... · 2 Developmental Genetics Graduate Program, NYU School of Medicine, New York, NY, USA 3 Developmental Biology Program,

Bull Math Biol (2016) 78:859–878DOI 10.1007/s11538-016-0163-3

ORIGINAL ARTICLE

A Mathematical Model of Granule Cell GenerationDuring Mouse Cerebellum Development

Shoshana R. Leffler1,2 · Emilie Legué3 ·Orlando Aristizábal1 · Alexandra L. Joyner3 ·Charles S. Peskin4 · Daniel H. Turnbull1,2,5

Received: 20 November 2015 / Accepted: 29 March 2016 / Published online: 28 April 2016© Society for Mathematical Biology 2016

Abstract Determining the cellular basis of brain growth is an important problem indevelopmental neurobiology. In the mammalian brain, the cerebellum is particularlyamenable to studies of growth because it contains only a few cell types, includingthe granule cells, which are the most numerous neuronal subtype. Furthermore, in themouse cerebellumgranule cells are generated fromgranule cell precursors (gcps) in theexternal granule layer (EGL), from 1day before birth until about 2weeks of age. Thecomplexity of the underlying cellular processes (multiple cell behaviors, three spatialdimensions, time-dependent changes) requires a quantitative framework to be fullyunderstood. In this paper, a differential equation-based model is presented, which canbe used to estimate temporal changes in granule cell numbers in the EGL. The modelincludes the proliferation of gcps and their differentiation into granule cells, as well asthe process by which granule cells leave the EGL. Parameters describing these biolog-ical processes were derived from fitting the model to histological data. This mathemat-ical model should be useful for understanding altered gcp and granule cell behaviorsin mouse mutants with abnormal cerebellar development and cerebellar cancers.

B Charles S. [email protected]

B Daniel H. [email protected]

1 Skirball Institute of Biomolecular Medicine, New York University School of Medicine,540 First Avenue, New York, NY 10016, USA

2 Developmental Genetics Graduate Program, NYU School of Medicine, New York, NY, USA

3 Developmental Biology Program, Sloan Kettering Institute, New York, NY, USA

4 Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York,NY 10012, USA

5 Departments of Radiology and Pathology, NYU School of Medicine, New York, NY, USA

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860 S. R. Leffler et al.

Keywords Granule cell precursor cells · External granule layer · EGL · Differentialequations

List of symbols

αE Rate constant for granule cell exit from EGL (measured in h−1)αP Rate constant for gcp proliferation (measured in h−1)δ(t) Time-dependent probability that a gcp differentiates into a granule cellvc Volume of a gcp or differentiated granule cell (measured in µm3)a Initial slope of the probability function δ(t) (measured in h−1)Ai(t) Area of the iEGL, a time-dependent function (measured in µm2)Ao(t) Area of the oEGL, a time-dependent function (measured in µm2)EGL External granule layer of the cerebellumgcp Granule cell precursoriEGL Inner (differentiated) layer of the EGLIGL Inner granule layer of the cerebellumL Medial-lateral length of the cerebellar vermis, a constant (measured in

µm)ML Molecular layer of the cerebellumMRI Magnetic resonance imagingNi(t) Number of cells in the iEGL, a time-dependent functionNo(t) Number of cells in the oEGL, a time-dependent functionODE Ordinary differential equationoEGL Outer (proliferative) layer of the EGLp27 Molecular marker used to detect differentiated cells in the iEGLP Postnatal day (e.g., P6=postnatal day 6)SHH Sonic Hedgehog, a mitogen for gcp proliferationT = 1/a (measured in h), the time that the linear δ(t) Function becomes 1Td Doubling time of gcps (measured in h)TE Exit time of differentiated granule cells from the EGL (measured in h)

1 Introduction

Brain growth during fetal and postnatal development depends on maintaining a finebalance between neural cell proliferation and differentiation. Themammalian cerebel-lum provides a striking example of postnatal brain growth. In mouse, the cerebellumvolume increases approximately fivefold over the first 2weeks after birth (Szulc et al.2015), a period of marked molecular, cellular and large scale patterning changes (Sil-litoe and Joyner 2007; Sudarov and Joyner 2007; Martinez et al. 2013). At the cellularlevel, the granule cell precursor (gcp) cells proliferate in the outer layer of the EGL inresponse to the mitogen Sonic Hedgehog (SHH), secreted by nearby Purkinje neurons(Wechsler-Reya and Scott 1999; Dahmane and Ruiz i Altaba A, 1999; Wallace 1999).Following proliferation, the gcps differentiate to form granule cells, by first moving tothe inner layer of the EGL where they extend their T-shaped axons, and then migrat-

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A Mathematical Model of Granule Cell Generation during. . . 861

ing toward and past the Purkinje cell layer to take up their final positions in the innergranule layer (IGL). The granule cell axons, known as parallel fibers, contribute to themolecular layer (ML) between the EGL and Purkinje cell layer (Altman and Bayer1997). As the most numerous neuronal cell type in the mammalian brain, the granulecells contribute significantly to the growth of the cerebellum, adding volume to boththe IGL and ML during the first 2weeks of postnatal mouse development.

As a developmental model system, the mouse cerebellum provides a number ofadvantages. The number of cell types is relatively small, with a stereotyped layeredcytoarchitecture throughout the cerebellum (Altman and Bayer 1997; Sillitoe andJoyner 2007). Furthermore, the folded morphology of the postnatal cerebellum devel-ops similarly in all mammalian species, with 10 major lobules in the midline of micethat are further foliated in higher mammals, including humans. At the cellular level,the gcps and granule cells in mice have been studied extensively for close to 50years(Fujita 1967; Haddara and Nooreddin 1966; Mares et al. 1970; Seil and Herndon1970; Hatten et al. 1982; Goldowitz et al. 1997), including recent reports using themost advanced genetic cell labeling and tracking tools, providing important new datafrom clonal analysis (Espinosa and Luo 2008; Legué et al. 2015). Finally, a number ofgenetic factors have been identified, including SHH, that are involved in cerebellumgrowth and patterning, and many genetically engineered mouse models of abnormalcerebellum development have been produced and are available for in vivo studies(Joyner et al. 1991; Goldowitz et al. 1997; Goodrich et al. 1997; Corrales et al. 2006;Cheng et al. 2010).

Despite this wealth of biological data, it is still not clear how to predict cerebellumgrowth from the underlying neural cell behaviors. Mathematical models have thepotential tomake quantitative predictions of how changes in cell behaviors affect tissuegrowth. In the current study,wedeveloped and tested amodel basedon cellular kinetics,applying differential equations with rate constants derived from observed granulecell behaviors of proliferation and differentiation in the EGL. The model accuratelypredicted how granule cells are generated during mouse cerebellum development,including temporal changes of tissue layers within the EGL, and the average expectedclone size of individual gcp cells. Simulated changes in cell parameters were shownto affect the numbers and distributions of gcps and granule cells within the EGL.This model should be useful for studying how changes in these cell properties giverise to abnormal tissue development in mouse models of human neuro-developmentaldiseases including cancer.

2 Experimental Methods

2.1 Animals, Histological Sections and Staining

All mice used in this study were maintained under protocols approved by the Institu-tional Animal Care and Use Committees at New York University School of Medicineand Memorial Sloan-Kettering Cancer Center. Swiss-Webster mice were cardio-perfusedwith 4 ◦Cphosphate-buffered saline (PBS) and 4 ◦Cparaformaldehyde (PFA)at P2, P6, P10 and P14 (day of birth is denoted P0) and their brains dissected, post-fixed

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overnight at 4 ◦C in 4% PFA, cryoprotected in 30% sucrose in PBS and embed-ded/frozen in OCT (TissueTek) and cryosectioned (sagittal) at 12-µm thickness.

For antibody staining against p27, an antigen retrieval step was performed: slideswere equilibrated first in PBS for 5min, then in 10mM Na+ citrate buffer, 0.05%Tween20, pH6 for 10min at room temperature (RT). Slides were then transferredinto pre-heated 10mM Na+ citrate buffer, 0.05% Tween20, pH6 and maintained at95–100 ◦C for 30min. Slides were transferred back into RT Na+ citrate buffer, 0.05%Tween20, pH6 and let cool down for 10min, then rinsed in PBS three times before pro-ceedingwith the vendor-specified antibody staining protocol (primarymouse anti-p27;1:20,000, BD Biosciences 610241/secondary biotinylated goat anti-mouse; 1:500,ABC kit, Vector laboratories) using nickel-enhanced DAB revelation (DAB kit, Vec-tor Laboratories). Slides were counterstained with Strong Nuclear Fast Red beforebeing coverslipped and imaged.

2.2 Image Acquisition and Analysis

Images of the stained sagittal sections were captured on a digital whole slide scan-ner (Leica SCN400F). These scans were viewed and images of selected midsagittalsectionswere output to ImageJ (Fiji) usingSlidePathDigital ImageHub (LeicaBiosys-tems). Contours were drawn manually in ImageJ to measure the areas of the outer(oEGL) and inner (iEGL) EGL layers in lobule III. These area data were then storedin Excel files (Microsoft) for statistical analyses and plotting. Selected data were alsotransferred to Matlab (MathWorks) for fitting model parameters and to make compar-isons between the experimental data and model results.

3 Mathematical Model

A balance between gcp proliferation and differentiation is required to generate therequired number of granule cells in the mature cerebellum. To describe this processmathematically, we took into consideration the cellular behaviors observed in the EGL(Fig. 1). Gcps undergo symmetric division in the EGL during cerebellum development(Espinosa and Luo 2008; Legué et al. 2015). Proliferation occurs in the outer EGL

Fig. 1 (Color Figure Online) Granule cells are generated in the EGL during cerebellum development.Granule cell precursors (gcps; pink, denoted “o”) proliferate by symmetric division in the oEGL, afterwhich they divide terminally to generate two differentiated granule cells (dark purple, denoted “i”) inthe iEGL. The rate constant for proliferation is denoted αP , while δ is the probability that a dividinggcp differentiates. αE is the rate constant for the exit of differentiated granule cells from the iEGL to themolecular layer (ML)

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(oEGL), where a gcp generates two daughter gcp cells with each division. Eventually,each gcp divides terminally to generate two differentiating granule cells that move intothe inner EGL (iEGL). Finally, after an axon is extended, each differentiated granulecell exits the iEGL, migrating through the ML to take up its final position in the IGLof the mature cerebellum. To summarize, proliferation leads to the growth of the EGL(in the proliferative oEGL layer), while differentiation initially increases the size ofthe iEGL, but ultimately leads to depletion of the entire EGL as cells leave the cellcycle and exit this transient tissue layer.

3.1 Model for Granule Cell Generation in the EGL

To model the cellular behaviors described above, we need to describe mathematicallythe changes in the number of proliferating gcps, No, located in the oEGL, and thenumber of differentiated granule cells, Ni, located in the iEGL (Fig. 1). We introducethe parameter αP, the rate constant for the division of gcps, and δ, the probability that agcp divides terminally to generate two differentiated granule cells. Since cell divisionsin the EGL are assumed to be symmetric, (1 − δ) is the probability that a gcp dividesto generate two gcps. An additional rate constant, αE, is introduced to account for theexit of granule cells from the EGL. With these three parameters, we can write a pairof coupled ordinary differential equations (ODEs) for No and Ni:

dNo

dt=

Proliferation︷︸︸︷

αP (1 − δ)No

Differentiation︷︸︸︷

−αPδNo (1a)

dNi

dt=

Differentiation︷︸︸︷

2αPδNo

Exit︷︸︸︷

−αENi (1b)

Note that the different components of Eqs. (1a) and (1b) have natural and intuitiveinterpretations in terms of the addition or subtraction of cells in the oEGL and iEGLlayers due to proliferation, differentiation and exit from the EGL (see annotationsincluded with the equations).

3.2 Need for a Time-Dependent Probability Function, δ(t)

In the case that the probability δ is a constant, Eqs. (1a) and (1b) have analyticalsolutions. Specifically, the solution to Eq. (1a) is particularly simple:

No (t) = No(0)e(1−2δ)αPt (2)

where No(0) is the initial value of No at time t = 0. Since δ must have a value between0 and 1, it is easy to determine the behavior of No for different values of δ:

(i) For 0 ≤ δ < 1/2, No(t) is an increasing exponential function of time;(ii) For δ = 1/2, No(t) is a constant (= No(0));(iii) For 1/2 < δ ≤ 1, No(t) is a decreasing exponential function of time.

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None of these behaviors are consistent with observations (see Sect. 4), which indicatethat the number of oEGL cells initially increases and then decreases to zero duringdevelopment. We therefore conclude that δ must be a time-dependent function. Forthis case, the solution of Eq. (1a) is more complicated than Eq. (2), but can still bewritten analytically:

No (t) = No(0)eαP

∫ t0 (1−2δ(τ ))dτ (3)

To be consistent with observed oEGL behavior, the δ(t) probability function mustbe less than 1/2 initially, and then greater than 1/2 after some time. Since t = 0 is thetime just before gcps begin to differentiate into granule cells, we also assumed thatδ(0) = 0. We investigated three single-parameter (δ) functions, chosen to meet theseconditions:

(a) Linear Function

δ(t) ={

at, 0 ≤ t < T = 1/a1, t ≥ T

(4a)

where a is the slope (in units of h−1) and T = 1/a is the time (in h) when δ

becomes 1.(b) Rational Function

δ(t) = at

1 + at(4b)

(c) Exponential Function

δ(t) = 1 − e−at (4c)

3.3 Calculation of Granule Cell Clone Size

If we ignore the spatial location of the cells, Eq. (1) can be rewritten to describe thenumbers of proliferating gcps, denoted No (as before), and differentiated granule cells(whether they are in the EGL, IGL or in between), denoted Ng:

dNo

dt= αP (1 − 2δ) No (5a)

dNg

dt= 2αPδNo (5b)

These equations are equivalent to Eq. (1), the only difference being that the termdescribing exit from the EGL in Eq. (1b) is no longer included. Furthermore, Eqs. (5a)and (5b) can be added together to yield:

d

dt

(

No + Ng) = αPNo (6)

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which is independent of δ. This makes sense, since the right side of Eq. (6) is therate at which gcp cell divisions occur (regardless of outcome) and every cell divisionincreases the total number of cells by 1. Integrating Eq. (6) from 0 to∞, and using thefact that Ng(0) = 0 and No(∞) = 0, then the total number of granule cells generatedcan be written as:

Ng(∞) = No(0) + αP

∫ ∞

0No(t)dt (7)

The expected clone size of eachoriginal gcp is the ratio Ng(∞)/No(0), since Ng(∞)

is the total number of granule cells generated by the No(0) gcp cells present at t = 0.This ratio can be calculated by solving equation (7) numerically, after substitutingEq. (3) for No(t). Note that for the linear function δ(t), given by Eq. (4a), there is ananalytical solution for the clone size (see “Appendix”):

Ng(∞)

No(0)= 2 + √

(αPT ) πe

(

αPT4

)

erf

(√αPT

2

)

(8)

where erf(x) is the error function defined as (Abramowitz and Stegun 1964):

erf(x) = 2√π

∫ x

0e−v2dv (9)

3.4 Note on Time Scales and Conversions from Cell Numbers to Tissue LayerAreas

The time scale in the equations described above is defined in terms of an initial time(t = 0) just before differentiation begins. At t = 0 there are no differentiated granulecells (Ni(0) = Ng(0) = 0), but there are gcp cells (No(0) > 0). The time t = 0was assumed to be 1day before birth (Legué et al. 2015). The experimental data weremeasured in postnatal days (P2, P6, P10 and P14), so t = 0 in the model is actually“P-1”.

The model equations deal with numbers of cells (No, Ni, Ng), while the experi-mental data were measured from histological sections to determine areas of the oEGL(Ao) and iEGL (Ai) (measured in µm2). Conversion between cell numbers and tissueareas was made using the following simple formula:

No,i = Ao,i × L

vc(10)

where vc is the volume of a granule cell, assumed to be 300-µm3 (Mares et al. 1970;Seil and Herndon 1970; Altman and Bayer 1997), and L is the medial-lateral width(measured inµm) of the vermis (central cerebellum), containing lobule III fromwhichwemeasured thedata.Bothhistological andMRI studies have shown that L is relativelyconstant over the early postnatal developmental period, with most of the growth ofthe cerebellum occurring in the anteroposterior direction, along the length of each of

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the lobules (Legué et al. 2015; Szulc et al. 2015). From MRI data (Szulc et al. 2015),we estimated L (measured along the medial-lateral outer contour of the vermis) to be1775-µm(±20%) between P2 and P14.

3.5 Matlab Implementation of the Model

To solve the ODEs in Eq. (1), we used Euler’s method, implemented in Matlab (Math-Works). Parameter optimization was performed using fminsearch, which is Matlab’simplementation of the Nelder-Mead optimization algorithm (Nelder and Mead 1965).For this, we first defined an objective function, which was a root-mean-square errorestimate to be minimized:

Objective function =√

√

√

√1

8

{

4∑

n=1

[

Amo (n) − Ad

o(n)]2 +

4∑

n=1

[

Ami (n) − Ad

i (n)]2

}

(11)

where Amo,i denotes the model predictions and Ad

o,i the measured data values of theoEGLand iEGLareas (indicated by subscripts o and i, respectively). The index counter,n, is used to denote the postnatal stage of each measurement: n = 1, P2; 2, P6; 3,P10; 4, P14. The fminsearch function was called with 4 parameters: αP, αE, a (theslope of the function δ(t) in Eq. (4)), and Ao(0), the area equivalent to No(0) (Eq.(10)). fminsearch was used to find the optimal values of these four parameters, whichminimized the value of the objective function.

4 Results

4.1 The Mouse Cerebellum Grows Significantly Over the First 2Weeks AfterBirth

The mouse cerebellum undergoes tremendous growth over the first 2weeks of postna-tal life (Fig. 2), when the majority of granule cells are generated (Sudarov and Joyner2007; Legué et al. 2015; Szulc et al. 2015). Histological analysis clearly revealed theincrease in cerebellum size and complexity of foliation (Fig. 2a), and closer examina-tion of individual lobules showed that much of this increase was localized within theIGL andML, which include contributions from the gcps (generated by the EGL) in theform of the granule cell bodies and dendrites (IGL) and their parallel fibers (ML). Wefocused our analysis on lobule III (Fig. 2b), which has a relatively simple shape thatcould be identified and measured from the earliest developmental stages analyzed.

4.2 The Thickness of the oEGL and iEGL Changes Between P2 and P14

At the tissue level, histological analyses demonstrated changes in the thickness of thecell layers in the EGL during postnatal development (Fig. 3). Immunostaining for the

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Fig. 2 (Color FigureOnline)Themouse cerebellumgrows and forms lobules during postnatal development.a Midsagittal sections of the mouse cerebellum from postnatal day P2 to P14 demonstrated the significantgrowth and foliation (lobules denoted by Roman numerals I to X) that occurs over these developmentalstages. Scale bar 200µm for each panel in (a). Sections were immunostained for p27 (dark purple, earlydifferentiation marker) and counterstained with nuclear fast red (pink). b Magnified views of lobule III(arrows in a) show the developing layers (outer to inner): EGL external granule layer,ML molecular layer,IGL inner granule layer. Note that the EGL is largely depleted in lobule III by P14. Scale bar 100µm foreach panel in (b)

early differentiationmarker, p27, enabled segmentation of the oEGL (p27−) and iEGL(p27+) layers. Between P2 and P6, we observed relatively constant thicknesses of the2 layers. Both the oEGL and iEGL layers showed significantly reduced thickness atP10 and were depleted of granule cells by P14, with only a few scattered p27+ cellspresent at the outer edge of the ML at this time point.

4.3 Areas of the oEGL and iEGLWere Measured to Quantify Cellular Changes

The recent demonstration that gcps do not cross the borders between lobules duringpostnatal development (Legué et al. 2015) justified a single lobule analysis, since eachlobule represents a separate developmental unit where granule cells are generatedand maintained (Fig. 4). To quantify the growth characteristics of the oEGL and iEGLlayers, contourswere drawn inmidsagittal sections, covering the entire extent of lobuleIII at each stage (P2, P6, P10 and P14) (Fig. 4a; N = 6 sections from 3 animals ateach stage). The resulting data indicated relatively small variability between sectionsand mice and that the numbers of cells and resulting tissue area measurements inthe oEGL (Fig. 4b) and iEGL (Fig. 4c) initially increased (from P2 to P6) and thendecreased to near zero (from P10 to P14). We also observed a phase/time delay, which

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Fig. 3 (Color Figure Online)The thickness of the oEGL andiEGL undergo changes duringpostnatal cerebellumdevelopment. Magnified viewsof the sections shown in Fig. 2revealed dynamic changesbetween P2–P14 in the thicknessof the proliferating oEGL (pink,nuclear fast red) anddifferentiated iEGL (darkpurple, p27 staining) layers(dotted lines show approximatelayer borders), with themolecular layer (ML) below theiEGL at each stage. Scale bar50µm for each panel. Insertschematic (top) from Fig. 1

was expected, between the oEGL (larger at early time points) where gcps and granulecells are generated, and iEGL (larger at later time points), where the differentiatedgranule cells reside before exiting the EGL. These data were used to test the proposedmathematical model of granule cell generation.

4.4 Determining the Best Model Type and Parameter Values

The mathematical model contained several parameters, some of which have beenestimated previously but others that are unknown or not known with precision. Forexample, the rate constant for cell division, αP, is related to the gcp doubling time Tdthrough the formula:

αP = ln2

Td(12)

Previous studies have indicated that Td is ∼ 19h at P10–11 (Fujita 1967), whichprovides an initial estimate of αP ≈ 0.036 h−1. Also, the time required for differ-entiating granule cells to transit the EGL was estimated previously to be TE ∼28h

(Fujita 1967), corresponding to an exit rate, αE

(

= 1TE

)

≈ 0.036h−1. However, the

value of Ao(0) (the area equivalent to the initial number of gcps No(0)) has not beenreported in the literature, and the value of a (the initial slope of δ(t)), or even the most

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Fig. 4 (Color Figure Online) Areas of the oEGL and iEGL reflect developmental changes in the numbersof proliferating and differentiated granule cells in the EGL. a Areas of the oEGL and iEGL layers weremeasured in midsagittal sections of lobule III at P2, P6 and P10 (N = 6 sections from three animals at eachstage). Example contours measured at P6 are shown. Scale bar 100µm for each panel in (a). bQuantitativeoEGL areas (mean± standard deviation shown at each stage). cQuantitative iEGL areas. Note that the areasof both the oEGL and iEGL were observed to be zero at P14, so no contours/quantitation were generatedat that stage

appropriate functional form of δ(t), is not known at all. We therefore used an opti-mization algorithm to fit the 4 model parameters (αP, αE, a, Ao(0)) to the measureddata, minimizing an objective function defined by Eq. (11).

Three versions of the δ(t) function were implemented for optimization: linear,Eq. (4a); rational, Eq. (4b); and exponential, Eq. (4c). In each case, the fminsearchfunction converged within 60–70 iterations, resulting in a significant improvement inthe fit compared to the initial guess values (Fig. 5). The final values of the objec-tive function were slightly lower (∼10%) for the optimal rational and exponentialfunctions compared to the optimal linear function. The optimized parameter values(aopt, αopt

P , αoptE , Ao(0)opt, as well as the derived times Td and TE) determined by fmin-

search for each δ(t) function are listed in Table 1. The optimal linear function reached

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Fig. 5 A four-parameter optimization routine was used to fit the model to experimental data. An objectivefunctionwas defined in Eq. (11) to assess the (root-mean-square) error in the fit betweenmodel and data, andthe Matlab fminsearch function was used determine the optimal fit, varying the parameters αP, αE, a, andAo(0) to find the minimum value of the objective function for each of the three δ(t) probability functions.The final optimal model parameters are given in Table 1

Table 1 Optimal model parameters determined by fitting to measured data

δ(t) Function aopt (h−1) Ao(0)opt (µm2) αoptP (h−1) Td (h) α

optE (h−1) TE (h)

Linear 0.0029 1994 0.0348 19.9 0.0387 25.8

Rational 0.0059 1005 0.0558 12.4 0.0588 17.0

Exponential 0.0041 1411 0.0443 15.6 0.0474 21.1

Note that the parameters determined by fminsearch optimization were aopt, Ao(0)opt, αoptP and α

optE . The

gcp doubling time Td and the granule cell exit time TE were calculated from αoptP and α

optE , respectively

the value 1 (100% likelihood that a gcp will differentiate into 2 granule cells) closeto P13.5, approximately half a day before P14, when the EGL was observed to bedepleted. In comparison, the optimal rational and exponential functions only achievedvalues between 0.6 and 0.75 by P14 (Fig. 6). Interestingly, the three different δ(t) func-tions each reached the value of 0.5 (when No and Ao are maximized) at approximatelythe same time (∼ P6).

Using the optimal parameters (Table 1), model predictions were compared to mea-sured data (Fig. 7) for each of the δ(t) functions. Each model provided a good fitto the data acquired from oEGL (Fig. 7a) and iEGL (Fig. 7b), showing a delay ofapproximately 2days between the peak number of proliferating oEGL cells (∼P6)and the peak number of differentiating iEGL cells (∼P8). There were some commondiscrepancies between the experimental data and model predictions, including smallunderestimation at early stages and overestimation at later stages in the numbers ofiEGL cells (Fig. 6c), which also led to an overestimation of the time when the EGLwas depleted of granule cells (experimental ∼P14 vs. model ∼P16.5). This effect wasslightly less for the linear model compared to the two alternatives.

Given the similarity in fits between models using different δ(t) functions, we usedcomparison of model predictions to previous estimates of the rate constants αP and

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Fig. 6 Optimal forms of the δ(t) functions. The three time-dependent probability functions δ(t), (linear,Eq. (4a); rational, Eq. (4b); exponential, Eq. (4c)) were plotted between t = 0 (1day before birth, or “P-1”)and P20

αE to choose the best model. Using the linear δ(t) function, αoptP was determined

to be 0.0348 h−1 (Table 1), corresponding to a doubling time Td ≈ 20h using Eq.(12), which is very close to Fujita’s estimate of 19h (Fujita 1967). Similarly, theexit rate constant α

optE was 0.0387 h−1 (Table 1), indicating that TE(= 1/αopt

E ) wasapproximately 26h. This value is also close to Fujita’s estimate of 28h (Fujita 1967).In contrast, the optimal αP and αE rate constants and their associated times (Td, TE)

were significantly different than Fujita’s estimates when the rational and exponentialδ(t) functions were used (Table 1). For example, the doubling times were Td ≈ 12and 15h, while the exit times were TE ≈ 17 and 21h for the rational and exponentialfunctions, respectively, which were considered to be unrealistically small. Based onthese considerations, we decided that the model with the linear δ(t) function providedthe best parameter estimates and fit to measured data.

4.5 The Model Provides Predictions of Initial Number of gcps and gcp CloneSize

Using the model with the linear δ(t) function, Ao(0)opt was found to be 1994-µm2

(Table 1). Using Eq. (10), this implies that the initial number of gcp cells (No(0)) inlobule III is close to 12,000. Using Eq. (8) with the optimal (linear model) values of theparametersαP and T = 1/a (Table 1),we can predict that the average clone size of eachinitial gcp in lobule III is∼ 124. Therefore, the model presented in this paper providesquantitative predictions about the granule cell properties and behaviors that can becompared to past and future data acquired from the developing mouse cerebellum.

4.6 Model Simulations Provide Insights into Developmental Disorders

We used the model to simulate several cases of abnormal granule cell behaviors thatmight be expected to arise in the initial stages ofmedulloblastoma, a common pediatric

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Fig. 7 Model predictions with optimal parameters showed good agreement with measured data. Usingthe optimal parameters (Table 1), the model predictions were compared to the measured data. The modelpredictions for the oEGL (a) and iEGL (b) areas were in good agreement with the measured data (mean ±standard deviation at P2, P6 and P10; note that the areas of the oEGL and iEGL were both taken to be 0 atP14) for each of the optimized models

brain tumor of the cerebellum (Fig. 8) (Roussel and Hatten 2011). Previous studies ofmouse medulloblastoma models have demonstrated EGL thickening and persistenceof gcp proliferation at stages after the EGL would normally be depleted (Goodrichet al. 1997; Kim et al. 2003; Matsuo et al. 2013; Oliver et al. 2005; Suero-Abreu et al.2014). The optimalmodel parameters (Fig. 8a) weremodified to examine the effects ofan increase in the proliferation rate (Fig. 8b), a decrease in the differentiation rate (Fig.8c), and a delay in the exit of granule cells from the EGL (Fig. 8d–f). Each of thesesimulations resulted in an increase in the numbers of cells in the oEGL and/or iEGL,and a persistence of granule cells in the EGL beyond the normal stage of depletion,similar to what is observed in early stages of medulloblastoma.

These altered parameters can also effect the predicted clone sizes: Compared tothe normal cerebellum (clone size= 124), our simulation of increased proliferation

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Fig. 8 Simulations of mouse mutants with developmental disorders. The model described in this papercan be used to simulate aberrant granule cell proliferation and differentiation relevant to a variety of mousemutant models of developmental diseases, including medulloblastoma, a common pediatric brain tumorin which too many gcp-like cells are generated. To this end, model predictions of the growth dynamics ofthe oEGL (solid line) and iEGL (dashed line) were compared between normal EGL (a) and several casesof abnormal EGL which are often assumed to occur in the initial phases of medulloblastoma progression:

increased proliferation (b, αP = 1.5 × αoptP ), decreased differentiation (c, a = 0.75 × aopt) and delayed

exit (d;αE = 0.5 × αoptE ; e, αE = 0.05 × α

optE ; f, αE = 0). In every case except when there is no exit (f),

the oEGL and iEGL areas increase initially and then decrease to zero (similar to the normal EGL), but themagnitude and temporal dynamics of the growth curves are significantly different between simulations. Inthe case of no exit (f), all the granule cells generated are maintained in the EGL

(Fig. 8b, αP = 1.5× αoptP ) resulted in an increased clone size of 677, while decreased

differentiation (Fig. 8c,a = 0.75×aopt) resulted in a clone size of 387.Our simulationsof delayed exit from the EGL (Fig. 8d, αE = 0.5 × α

optE ; Fig. 8e, αE = 0.05 × α

optE ;

Fig. 8f, αE = 0) had no effect on the clone size, which makes sense since Eq. (8) doesnot depend on αE. Even a complete failure of all granule cells to exit (Fig. 8f) does notchange the final number of differentiated granule cells, but it does maintain them inthe EGL. While these simulations are able to recapitulate some of the early aspects ofmedulloblastoma formation, modeling the progression to overt tumors would require

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altering the underlying assumption in our model that δ(t) increases and ultimatelybecomes larger than 0.5. Indeed, the probability function δ(t) would need to remainless than 0.5 (as discussed in the Mathematical Model section) in order to modelsustained gcp proliferation and indefinite tumor growth.

5 Discussion

In this study, we started from first principles, using observed behaviors of gcps inthe EGL—symmetric cell division; proliferation in the oEGL; differentiation in theiEGL; ultimate exit from the EGL—to write a pair of coupled ODEs that model thetime-dependent generation of granule cells during early postnatal mouse cerebellumdevelopment. This model incorporates intuitive parameters related to the observed cellbehaviors—differentiation probability function, δ(t); proliferation rate constant, αP;EGL exit rate constant, αE—that were subsequently fit to experimental data derivedfrom histological sections. After fitting, the model showed excellent agreement to theexperimental data from the oEGL and iEGL. We further showed how the model couldbe extended to compute the number of progeny (clone size) generated by each gcpcell. Finally, we showed that the model provides a quantitative framework to analyzehow differences in cell behaviors can affect the generation of cells within the EGL andthe resulting granule cell clone size and/or tissue growth in developmental disordersof the cerebellum. Overall, experimental data have suggested that proliferation, dif-ferentiation and exit are important for EGL growth, but our model makes quantitativea plausible description of cerebellum growth control: The rate of gcp proliferationcontrols the early increase in the volume of the EGL, the transition from proliferationto differentiation is responsible for slowing the growth, and the exit of differentiatedgranule cells is responsible for the decline in volume and eventual disappearance ofthe EGL.

Quantitatively, the model was in close agreement to data on granule cell behaviorsavailable in the literature (e.g., Fujita 1967), providing added confidence in the validityof our optimization procedure. Recent data from clonal analyses have shown consid-erable variability between clones, but reported mean clone sizes, when measured fromembryonic day E17.5 (2days before birth) to adulthood, were 250 (Espinosa and Luo2008) and 450 (Legué et al. 2015) and were close to 120 when measured from P1to adulthood (Legué et al. 2015). The model estimate of 124, measured from E18.5(=“P-1”) to adulthood, is close to these measurements, and certainly within the vari-ability of the experimental data, especially considering that neither of the previousreports were focused only on lobule III. Our simulations with altered parameters (Fig.7) might point to the cellular processes responsible for the variability in the measuredclone size; i.e., individual clones may have higher or lower rates of proliferation,differentiation and/or EGL exit.

As described above, the current model captures many key features that underliethe generation of granule cells that are responsible for much of the growth of thecerebellum. Furthermore, our approach of fitting the model parameters to histologicaldata derived from a single lobule (III) was justified based on recent published resultsshowing that each lobule represents a separate developmental unit, with little or no

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mixing of granule cells between different lobules (Legué et al. 2015). One limitationof our approach is that we did not attempt to take into account the stage-dependentdifferences in gcp and granule cell density in the EGL (Fig. 3). Estimation of the celldensities at each stage would provide a more accurate conversion between area mea-surements and cell numbers (Eq. (10)), but would require more accurate stereologicalmethods of counting cells in histological sections. In addition, the current model doesnot incorporate any spatial information within or between lobules. Our choice of aconstant medial-lateral vermis width, L , is a reasonable first approximation, sinceboth histological and MRI studies have shown that the increases in L between P2 andP14 are small (Legué et al. 2015; Szulc et al. 2015). In future, it would be straightfor-ward to incorporate a time-dependent function for L in the model, based on additionalmeasured data. Another concern is the fact that known medial-lateral movements ofgcp cells within the EGL are not incorporated into the model (Sgaier et al. 2005;Legué et al. 2015). Again, additional experimental data on the extent and rates of suchmedial-lateral migrations could be used to refine the model in future.

We argued intuitively that the probability of a gcp differentiating into a granulecell must be represented by a time-dependent function, δ(t). We further investigatedseveral forms of this function, concluding that a linear δ(t) gave the best fit betweenknown parameter values and experimental data. The linear function assumed in thecurrent model enabled us to correctly predict the increase and decrease in the numberof EGL cells observed experimentally during postnatal development. The fact that ouroptimal parameters resulted in an overestimation of the time to depletion of the EGLsuggests that the model might be improved further in future. It will be important infuture to incorporate more of the unique properties of gcp proliferation and differenti-ation into the model. Previous studies have suggested that feedback regulation of celllineages may be used to model the regenerative properties of some tissues (Landeret al. 2009), but these models will likely need revision to be applied to the EGL, wherecell generation is achieved through symmetric cell divisions rather than the more usualasymmetric divisions of self-renewing stem cells. Another important area for futureimprovement should take into account the results of recent clonal analyses showingthat the timing of cell differentiation within each gcp clone is not independent, as cur-rently assumed in the model, but instead all the cells in a clone differentiate togetherwithin a short time period (Espinosa and Luo 2008; Legué et al. 2015). When moredata are available on the timing of this synchronized differentiation, this feature ofgranule cell behavior could be incorporated into the model.

In this paper, we showed that our mathematical model provided a good fit to theexperimental data from a single cerebellar lobule, lobule III. In the future, it wouldbe interesting to perform a similar analysis of other lobules in the developing mousecerebellum. We know from qualitative observations of histology (e.g., Fig. 2) andquantitative measurements from MRI (Szulc et al. 2015) that different lobules havedifferent growth characteristics. Consistent with this observation, the average clonesize is also larger in long versus short lobules (Legué et al. 2015). By fitting the modelto other lobules, we could predict the proliferation and differentiation parameters thatmight be responsible for these differences, providing new insights into the factorscontrolling cerebellar growth. Overall, the model illustrates the relative balance ofproliferation and differentiation processes to growth control in the normal EGL and

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provides new insights into how changes in the granule cell exit rate, in addition toproliferation and differentiation, may contribute to the early thickening of the EGLobserved in mouse medulloblastoma models. Beyond the cerebellum, similar mathe-matical models could be developed and applied to other tissues to study the balancebetween proliferation and differentiation during organogenesis. Taken together, theresults described in this paper provide an illustration of how mathematical modelingcan be used to provide insights into the cellular processes underlying tissue growthduring brain and organ development.

Acknowledgments This research was supported by Grants from the National Institutes of Health (NIH):R01NS038461 (to DHT) and R37MH085726 (to ALJ). Partial support was also provided by the NIHCancerCenter Support Grants at New York University Langone Medical Center (P30CA016087) and MemorialSloan-Kettering Cancer Center (P30CA008748). SRL was partially supported by the NYU DevelopmentalGenetics Graduate Program NIH Training Grant (T32HD007520), and CSP was partially supported bythe Systems Biology Center New York under NIH grant P50GM071558. We thank Dr. Kamila Szulc foranalyzing MRI data related to the vermis width and Jae Han (Andy) Lee for assistance with some of theanalysis of histological data. We also thank the Histopathology Core at NYU School of Medicine for helpwith scanning the histological sections at high resolution.

6 Appendix: Derivation of the Formula for Average Clone Size for theLinear δ(t) Probability Function

The total number of granule cells generated is given by:

Ng(∞) = No(0) + αP

∫ ∞

0No(t)dt (13)

where

No(t) = No(0)eαP

∫ t0 (1−2δ(τ ))dτ (14)

and

δ (τ ) ={

τ/T, 0 ≤ τ < T1, τ ≥ T

(15)

Substituting Eqs. (14) and (15) into (13), we have:

Ng(∞)

No(0)= 1 + αP

∫ T

0eαP

∫ t0

(

1− 2τT

)

dτdt + αP

∫ ∞

Te−αP

[∫ T0

(

1− 2τT

)

dτ+∫ tT dτ

]

dt (16)

Note that∫ T0

(

1 − 2τT

)

dτ = T − T 2

T = 0, so Eq. (16) becomes:

Ng(∞)

No(0)= 1 + αP

∫ T

0eαP

(

t− t2T

)

dt + αP

∫ ∞

Te−αP(t−T )dt

= 2 + αP

∫ T

0eαP

(

t− t2T

)

dt (17)

123


By expressing

(

t − t2

T

)

= T

4− 1

T

(

t2 − tT + T 2

4

)

= T

4− 1

T

(

t − T

2

)2

,

Equation (17) can be rewritten as:

Ng(∞)

No(0)= 2 + αPe

(

αPT4

) ∫ T

0e− αP

T

(

t− T2

)2

dt (18)

Now, we will define:

v =√

αP

T

(

t − T

2

)

, so that dv =√

αP

Tdt (19)

Then:

t = 0 ⇒ v = −√

αPT

2; and t = T ⇒ v =

√αPT

2(20)

Substituting Eqs. (19) and (20) into (18), we have:

Ng(∞)

No(0)= 2 + √

αPT e

(

αPT4

) ∫

√αPT2

−√

αPT2

e−v2dv

= 2 + 2√

αPT e

(

αPT4

) ∫

√αPT2

0e−v2dv

Therefore the average clone size is:

Ng(∞)

No(0)= 2 + √

(αPT ) πe

(

αPT4

)

erf

(√αPT

2

)

(21)

where erf(x) is the error function defined as (Abramowitz and Stegun 1964):

erf(x) = 2√π

∫ x

0e−v2dv (22)

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