Basic Research—Technology

Does the Geometric Location of Odontoblast Differentiationand Dentinal Tubules Depend on a Reaction-Diffusion Systembetween BMP2 and Noggin? A Mathematical ModelJavier L. Ni~no-Barrera, DDS,* and Diego A. Garz�on-Alvarado, MSc, PhD†

Abstract

Introduction: The mesenchymal differentiation toodontoblasts is a complex process that determines theformation of dentinal tubules. This process involvesa carefully regulated sequence of changes in thebehavior of mesenchymal cells coordinated by theexpression of different molecular factors that includesmainly the Noggin and bone morphogenetic proteintype 2 (BMP2). Methods: We investigated a bio-regulatory mathematic model based on a set of equa-tions of reaction-diffusion to predict the geometry ofthe formation of the dentinal tubules. Results: Wefound that odontoblast location and the dentinal tubulesformation are determined by the spatial distribution ofa set of molecular signals that compete among them-selves to maintain places of the greatest concentrationof BMP2, which determines the step from mesenchymalcells to odontoblasts and the formation of the dentinaltubules. Conclusions: This mathematic model suggestsa regulatory loop between BMP2 and Noggin, which ishighly stable and repeatable and determines the rightlocation patterns of the odontoblasts and the formationof dentinal tubules. This mathematic approach allows usto understand biological phenomena and biochemicalactivity during the period of pulp differentiation. (J En-dod 2012;38:1635–1638)

Key WordsBone morphogenetic protein type 2, Noggin, mathe-matic models, reaction-diffusion, tubules

From the *Faculty of Dentistry, Area of Endodontics, andthe †Research Group on Numerical Methods for Engineering,Universidad Nacional de Colombia, Cundinamarca, Colombia.

Address requests for reprints to Dr Diego A. Garz�on-Alvar-ado, Universidad Nacional de Colombia, Cundinamarca,Colombia. E-mail address: dagarzona@unal.edu.co0099-2399/$ - see front matter

Copyright ª 2012 American Association of Endodontists.http://dx.doi.org/10.1016/j.joen.2012.08.016

JOE — Volume 38, Number 12, December 2012

The odontoblastic process begins with the differentiation of odontoblasts from thedental papilla cells. The odontoblast moves inside the papilla and leaves behind

a cytoplasmic extension that is responsible for mineralizing their environment, formingthe dentinal tubules (1–7). The differentiation of odontoblasts from the papilla isa highly organized process that develops well-established patterns of its location andshape. The geometric arrangement of the differentiation of these cells suggests a patternthat establishes the average distances between the functional odontoblasts, which, in thefuture, form the dentinal tubules.

Among the growth factors involved in odontoblast differentiation is bone morpho-genetic protein type 2 (BMP 2), which sequentially is expressed by cells of epithelial andmesenchymal origin. The basement membrane allows its transport toward the mesen-chymal cells that finally differentiate into odontoblasts (4–10). Despite the extensivestudy of the biochemical factors that allow the differentiation of odontoblasts, thegeometric pattern of the process has not been fully elucidated. Ruch et al (5) postulatedthat only the preodontoblasts that have completed a number of cell cycles become post-mitotic and are competent to differentiate into functional odontoblasts. Mitsiadis andGraf (11) suggested that the differentiation of dental cells is induced by different chem-ical signals. This influence can induce cells to differentiate into a specialized cell ata specific area or simply choose their place of differentiation stochastically and thechemical signals only support their survival and proliferation. Therefore, the spatiallocation of the growth factors, specifically BMP 2 on the basement membrane, willdetermine the potential areas of differentiation of the competent preodontoblasts.

The central hypothesis of this article states that BMP2 forms a regulatory loop inconjunction with other proteins that regulate its expression in the embryonic tissues(eg, Noggin). The hypothesis of this article was based on the ability that BMP 2 andNoggin have to create geometric patterns that allow the correct location of odontoblasts.Therefore, the aim of this article was to propose a mathematic model of a biochemicalcharacter that simulates the pattern differentiation of the odontoblastos and the futureformation of the dentinal tubules.

Materials and MethodsWe assumed the existence of a reaction-diffusion system whose species are BMP2

and Noggin whose distribution in space can lead to a stable pattern over time andunstable in space, which is similar to the patterns of differentiation of the odontoblasts.A competitive process of type activator-inhibitor or activator-substrate developsbetween these 2 proteins. This regulation is supposed to be a loop that is highly coupledand is represented by equations of reaction-diffusion parameters, which are in thespace of Turing (12). This mathematic model allows us to simulate patterns of spatialdistribution that are repeatable with different types of initial conditions that will lead tothe same distribution, similar to the high repeatability of the process of odontoblastdifferentiation and the formation of dentinal tubules.

Model DescriptionThe regulatory process proposed in this article is outlined in Figure 1 and is based

on a reaction-diffusion system (partial differential equations) of type activator substrate(also called exhaustion model) (12, 13). The process indicates that there is a regulatoryloop between Noggin (activating factor) and BMP2 (substrate) in which Noggin is

Location of Differentiated Odontoblasts and Dentinal Tubules 1635

Figure 1. The control system of the molecular process of BMP2 and Noggin.The diagram shows the relationship of the molecular signals produced by theepithelial tissue and the basement membrane. The solid lines represent acti-vation and the dotted lines inhibition. The continuous curved lines representself-activation and the dotted curved lines self-inhibition.

Basic Research—Technology

self-regulating and competes with BMP2 production (10, 14–16). Thus,it is assumed that the BMP2 self-inhibits but activates the production ofNoggin, and, in turn, Noggin inhibits the production of BMP2. Thishypothesis is based on the results of Zhu et al (14), Walsh et al (16),and Plikus et al (15) who established the existence of a regulatoryloop between BMP2 and Noggin. Thanks to the action of BMP2 overthe undifferentiated cells, it begins the process of differentiation thatends with the location of the odontoblasts and the subsequent formationof the dentinal tubules. Thus, the odontoblasts differentiate and develophighly repeatable patterns similar to those found in Turing patterns.

The definition of the relationships indicated in Figure 1 can bequantified by means of equations that provide local changes in thefactors expressed by the undifferentiated cells. These equations areas follows:

vSN

vt|{z}Temporalvariation

¼ C

0BBB@ a1|{z}

Production by cells

� mSN|{z}autoregula

|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflReaction term

vSB

vt|{z}Temporalvariation

¼ C

0BBB@ a2|{z}

Production by cells

�

Nre|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflffl

Reaction term

vC0

vt|{z}Temporalvariation

¼ hSnB

SnB þ SnT|fflfflffl{zfflfflffl}BMP2 effect on differentifrom C to C0

1636 Ni~no-Barrera and Garz�on-Alvarado

C is the concentration of epithelial cells and mesenchymal tissue ex-pressing BMP and Noggin factors. SB and SN represent the concentra-tions of BMP and Noggin, respectively. The remaining model parametersare as follows:a1 (in equation 1) anda2 (in equation 2) are terms thatquantify the production of each molecular factor for epithelial andmesenchymal tissues; m is a constant that quantifies the inhibition inthe production of Noggin by its excess; g0 governs the nonlinear inter-action between the concentration of Noggin-BMP2 and quantifies theactivation or inhibition of each molecular factor; and DB and DN arethe diffusion coefficients of BMP2 and Noggin, respectively.

Equations 1 and 2 establish the time evolution of SB and SNbecause of the action of a reactive term and the diffuse transport. Inequation 1, we can observe the constant production term of Nogginby the expression of the epithelial and mesenchymal cells (given byCa1). Also in equation 1, the term�mCSN establishes the regulatingof Noggin by cells caused by the excess of this factor. In equation 2,BMP2 is expressed by mesenchymal and epithelial cells by Ca2. Inaddition, the term g0S

2NSB (in equation 1) represents the nonlinear

activation of SN (the production of Noggin in the presence of BMP2)and in equation 2 the nonlinear consumption of SB (by the presenceof Noggin).

Equation 3 represents the differentiation term of the mesenchymalcells to odontoblasts by the presence of BMP2, which is regulated astime t passes. In this equation, h is a constant that regulates cell differ-entiation, SnTl represents the value of the concentration of BMP2 withwhich begins the process of differentiation, Ta is the time required toperform the differentiation, and tr represents the time limit of actionof the BMP2.

To verify the potential of the proposed model in predicting thedistribution pattern of the dentinal tubules, numeric tests were per-formed in a 2-dimensional quadrilateral with a length of 11 mm. Theparameters of the reaction-diffusion model were selected to obtaindifferentiation patterns with a periodicity consistent with those presentin the formation of dentinal tubules. In the finite element mesh applied,we used 10,201 nodes and 2,500 quadrilateral elements. In all simula-tions, we used incremental time steps of Dt ¼ 0:1423s.

tion

þ g0S2NSB|fflfflffl{zfflfflffl}

Nonlinear coupledregulation

1CCCA

fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

þ DNV2SN|fflfflffl{zfflfflffl}

Difussion term

(1)

g0S2NSB|fflfflffl{zfflfflffl}

onlinear coupledgulation

1CCCA

fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

þ DBV2SB|fflfflffl{zfflfflffl}

Difussion term

(2)

ation

� Tra

Tra þ tr|fflfflffl{zfflfflffl}Required time for the process

(3)

JOE — Volume 38, Number 12, December 2012

Basic Research—Technology

ResultsBecause of the biochemical interaction between the 2 molecular

factors and the numeric results, we determined that spatial patternswere stable over time. The concentration of the molecular factors inthe ectomesenchyme and the action of the diffusion process allow theformation of a pattern that replicates in the entire domain so that struc-tures can be obtained with a wave number (4, 4) (17, 18). The numberof waves allows us to define the frequency and distribution of thenumber of areas of mesenchymal cell differentiation intoodontoblasts producing dentin and therefore the ability to form thedentinal tubule. The results of Figure 2 show the formation of 4 half-waves in each of the x and y directions. Figure 2 shows the results ofthe places where the differentiation process because of the action ofBMP2 takes place. Similarly, Figure 2E shows a micrograph showingthe geometry of the dentinal tubules.

From the reaction-diffusion mechanism, we can determine thechange of the concentration of Noggin and BMP2 for each instant oftime. The concentrations of Noggin (SN) and BMP2 (SB) within themesenchymal tissue evolve according to their diffusivity, interaction,and expression by mesenchymal cells. Both Noggin and BMP2 areconcentrated in high amounts in specific areas, allowing cell differen-tiation similar to what happens in the biological process.

DiscussionFrom tissue engineering, important advances have been developed

in the understanding of the interaction between growth factors, the

Figure 2. Results of numeric simulation. (A–D) Geometric areas in which the fodifferentiation. The areas of black color show the formation of dentinal tubules, andof exposed dentinal tubules. (Courtesy of Martha Calle, Universidad Nacional de C

JOE — Volume 38, Number 12, December 2012

cellular environment, and the extracellular (matrix) of animal organsand tissues. This new branch of knowledge helps to understand theprocesses involved in human and animal development at the in vitroand in vivo level (19). Additionally, and as a complement, the informa-tion obtained by tissue engineering can be systematized and analyzed bymathematics and new computational techniques. In particular, mathe-matics applied to biology allows us to understand and quantify thesephenomena of growth and development in the light of the universallaws of physics, biology, and biochemistry (20). Therefore, mathematicbiology constitutes a fundamental tool to explain complex phenomena,raise hypotheses, and isolate variables and biochemical and biologicaleffects to test in silico. In particular, this article develops the hypoth-eses, the formulation, and the computational solution of a mathematicmodel of the interaction between BMP2 and Noggin that produce well-established patterns of the location of odontoblasts and thereforedentinal tubules. Therefore, this article is of importance to dentists,biologists, clinicians, and researchers who want to understand froma mathematic perspective the process of odontoblast differentiation.

From the regulatory loop between BMP2 and Noggin (14), thisarticle proposes a reaction-diffusion equation system that generatesspatial patterns that indicate the location of odontoblasts. Thus, Nogginkeeps the cells in an undifferentiated state by inhibiting the expressionof BMP2. BMP2 induces the expression of Noggin and helps cells todifferentiate into odontoblastos (21). This regulatory loop (14)(Fig. 1) allows cell differentiation into odontoblasts, which are respon-sible for building the dentinal tubules. It is important to note that BMP 2has an interaction with Noggin during the differentiation process of the

rmation of the dentinal tubules will be carried out as a consequence of cellthe white areas show the intertubular dentin. (E) Scanning electron microscopyolombia.)

Location of Differentiated Odontoblasts and Dentinal Tubules 1637

Basic Research—Technology

odontoblast as reported by Qin et al (21). In addition, Aberg et al (8)have reported that these factors are expressed by the dental epitheliumand mesenchyme. Additionally, Qin et al (21) and Zhu et al (14) havereported that Noggin is a major inhibitor of the biochemical activity ofthe BMPs, particularly BMP2 and BMP4.

From Figure 2E, we can see that the location of the dentinal tubules(and therefore the odontoblasts) (22) have a regular pattern and arewell established in space. In places in which BMP2 has a higher concen-tration of Noggin, mesenchymal cells differentiate into odontoblasts.Conversely, where there is the greatest concentration of Noggin, the cellsremain in the undifferentiated state. It is important to note that the math-ematic model sets the correct place of the odontoblasts differentiationthat will later form the dentinal tubules. For the mathematic model toallow this distribution, we use parameters that are in the space of Turingand that form patterns with the same name. The application of thereaction-diffusion model with parameters in the Turing space is anarea of constant work and controversy in biology. Garz�on-Alvaradoet al (17, 18), Courtin et al (23), Murray (12), and Barrass et al(13) used reaction-diffusion models in their work to simulate differentbiological processes and found that the use of these systems may help toexplain various highly complex biological phenomena in which there ispattern formation.

From the results presented, we can conclude that chemical feed-back between the 2 reactant molecular factors (activator substrate)maybe the major cause of the production of a regular pattern as observed inodontoblast differentiation and dentin tubule formation. These patternsare highly stable and repeatable among different individuals andspecies, showing clearly defined areas of differentiation for the correctformation of the future tubule. In areas in which there is a greaterconcentration of BMP2 compared with the concentration of Noggin,the process of cell differentiation is presented. In this way, cells willdifferentiate following the BMP2 pattern. However, it is clear that theseresults have been obtained with a mathematic model based on assump-tions and simplifications.

First, it is evident that the relation of Noggin-BMP2 is not the onlyone governing factors that control the entire process of differentia-tion(1–6); the existence of an activator-substrate mechanism ensureshigh stability for the development of this biological process. Indeed,Kloen et al (24) have shown that BMP is important in the consolidationof bone fractures, and, on the contrary, their inhibitors prevent thisprocess. Therefore, the imbalance between these 2 types of factors(BMP and its inhibitors) may lead to the process of differentiation ornot. This article uses the concept of imbalance between BMP2 andNoggin for the location of differentiation of odontoblasts. Additionally,the BMP2 values for which cells differentiate have not been quantified,and, therefore, the numeric experiment presented heremust be corrob-orated by in vitro experimentation.

Second, during the process of cell differentiation and the construc-tion of the dentinal tubules may present a greater number ofphenomena that can alter their formation. For example, the mechanicphenomena are important during the developing processes and differ-entiation, which have been studied in other biological cases (20). Inaddition, genetics may have an important influence that has not beenquantified in this mathematic model. Therefore, this model is the basisfor the construction of a more complete one that can simulate morecomplex phenomena such as the process of mineralization around

1638 Ni~no-Barrera and Garz�on-Alvarado

the dentinal tubules and odontoblast elongation. Finally, despite allthe limitations and simplifications, the mathematic model proposedis able to reproduce in detail the right location areas of cell differenti-ation into odontoblasts.

AcknowledgmentsThe authors deny any conflicts of interest related to this study.

References1. Arana-Chavez VE, Massa LF. Odontoblasts: the cells forming and maintaining

dentine. Int J Biochem Cell Biol 2004;36:1367–73.2. Carda C, Peydro A. Ultrastructural patterns of human dentinal tubules, odontoblasts

processes and nerve fibres. Tissue Cell 2006;38:141–50.3. Linde A. Dentin mineralization and the role of odontoblasts in calcium transport.

Connect Tissue Res 1995;33:163–70.4. Lisi S, Peterkova R, Peterka M, et al. Tooth morphogenesis and pattern of odonto-

blast differentiation. Connect Tissue Res 2003;44(suppl 1):167–70.5. Ruch JV, Lesot H, Begue-Kirn C. Odontoblast differentiation. Int J Dev Biol 1995;39:

51–68.6. Thesleff I, Mikkola M. The role of growth factors in tooth development. Int Rev Cytol

2002;217:93–135.7. Thesleff I. Epithelial-mesenchymal signalling regulating tooth morphogenesis. J Cell

Sci 2003;116:1647–8.8. Aberg T, Wozney J, Thesleff I. Expression patterns of bone morphogenetic proteins

(Bmps) in the developing mouse tooth suggest roles in morphogenesis and celldifferentiation. Dev Dyn 1997;210:383–96.

9. Lesot H, Lisi S, Peterkova R, et al. Epigenetic signals during odontoblast differenti-ation. Adv Dent Res 2001;15:8–13.

10. Yang W, Harris MA, Cui Y, et al. Bmp2 is required for odontoblast differentiation andpulp vasculogenesis. J Dent Res 2012;91:58–64.

11. Mitsiadis TA, Graf D. Cell fate determination during tooth development and regen-eration. Birth Defects Res C Embryo Today 2009;87:199–211.

12. Murray JD. Parameter space for turing instability in reaction diffusion mechanisms:a comparison of models. J Theor Biol 1982;98:143–63.

13. Barrass I, aC, E. J. and Maini, P. K. Mode transitions in a model reaction-diffusionsystem driven by domain growth and noise. Bull Math Biol 2006;68:981–995.

14. Zhu W, Kim J, Cheng C, et al. Noggin regulation of bone morphogenetic protein(BMP) 2/7 heterodimer activity in vitro. Bone 2006;39:61–71.

15. Plikus MV, Zeichner-David M, Mayer JA, et al. Morphoregulation of teeth: modu-lating the number, size, shape and differentiation by tuning Bmp activity. EvolDev 2005;7:440–57.

16. Walsh DW, Godson C, Brazil DP, et al. Extracellular BMP-antagonist regulation indevelopment and disease: tied up in knots. Trends Cell Biol 20:244–56.

17. Garzon-Alvarado DA, Martinez AM, Segrera DL. A model of cerebral cortex forma-tion during fetal development using reaction-diffusion-convection equations withTuring space parameters. Comput Methods Programs Biomed 2011;104:489–97.

18. Garzon-Alvarado DA, Ramirez Martinez AM. A biochemical hypothesis on the forma-tion of fingerprints using a turing patterns approach. Theor Biol Med Model 2011;8:24.

19. Hara K, Yamada Y, Nakamura S, et al. Potential characteristics of stem cells fromhuman exfoliated deciduous teeth compared with bone marrow-derived mesen-chymal stem cells for mineralized tissue-forming cell biology. J Endod 2011;37:1647–52.

20. Peinado Cortes LM, Vanegas Acosta JC, Garz�on Alvarado DA, et al. A mechanobio-logical model of epiphysis structures formation. J Theor Biol 2011;287:13–25.

21. Qin W, Yang F, Deng R, et al. Smad 1/5 is involved in bone morphogenetic protein-2-induced odontoblastic differentiation in human dental pulp cells. J Endod 2012;38:66–71.

22. Nanci A. Ten Cate’s Oral Histology Development, Structure and Function, 7th ed.Philadelphia: Mosby; 2008:215.

23. Courtin B, Perault-Staub AM, Staub JF. Spatio-temporal self-organization of bonemineral metabolism and trabecular structure of primary bone. Acta Biotheor1995;43:373–86.

24. Kloen P, Lauzier D, Hamdy RC. Co-expression of BMPs and BMP-inhibitors inhuman fractures and non-unions. Bone 2012;51:59–68.

JOE — Volume 38, Number 12, December 2012