# Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions

^{1}

^{2}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Reaction–Diffusion as a Framework to Understand Size-Regulated Symmetry-Breaking

## 3. A Minimal Model for Size-Regulated Symmetry-Breaking

## 4. Critical Size for Polarisation Can Be Utilised to Enact Cell State Transitions

#### 4.1. Cell Size Dependent Transition from Asymmetric to Symmetric Division in the Early C. elegans Embryo

#### 4.2. Size-Dependent Polarity Establishment in Budding Yeast

## 5. Sequential Pattern Formation and Polarisation Can Be Coordinated by a Growing Domain

#### 5.1. Neuronal Sequential Bipolarisation Coordinated by Membrane Growth

#### 5.2. Size-Dependent Sequential Patterning in Mammalian Development—Insights from Gastruloids

#### 5.3. Sequential Patterning of Phalanges in Developing Digits Is Coordinated by Coupling Patterning to Growth

## 6. Regulating Pattern Size and Lifetime in Growing Systems

#### 6.1. Case 1: Transient Polarity Pattern Due to Growth-Induced Dilution

#### 6.2. Case 2: Pattern Scaling Due to Proportional Growth of System Size and the Subunit Pool

#### 6.3. Case 3: Pattern Splitting

## 7. Using Growth as a Timer: Transient Symmetry Breaking at Intermediate Size

## 8. Overcoming Size Constraints: Scaling Patterns in Growing Systems

#### 8.1. Autocatalysis as a Mechanism to Preserve Patterns in the Face of Growth

#### 8.2. Expander-Coupled Systems Can Scale Patterns to Domain Size Irrespective of History

## 9. Discussion

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Meinhardt, H. Models of biological pattern formation: From elementary steps to the organization of embryonic axes. Curr. Top. Dev. Biol.
**2008**, 81, 1–63. [Google Scholar] [PubMed] - Green, J.B.; Sharpe, J. Positional information and reaction-diffusion: Two big ideas in developmental biology combine. Development
**2015**, 142, 1203–1211. [Google Scholar] [CrossRef] [PubMed][Green Version] - Saiz-Lopez, P.; Chinnaiya, K.; Campa, V.M.; Delgado, I.; Ros, M.A.; Towers, M. An intrinsic timer specifies distal structures of the vertebrate limb. Nat. Commun.
**2015**, 6, 1–9. [Google Scholar] [CrossRef] [PubMed][Green Version] - Briscoe, J.; Small, S. Morphogen rules: Design principles of gradient-mediated embryo patterning. Development
**2015**, 142, 3996–4009. [Google Scholar] [CrossRef] [PubMed][Green Version] - Waters, C.M.; Bassler, B.L. Quorum sensing: Cell-to-cell communication in bacteria. Annu. Rev. Cell Dev. Biol.
**2005**, 21, 319–346. [Google Scholar] [CrossRef] [PubMed][Green Version] - Jörg, D.J.; Kitadate, Y.; Yoshida, S.; Simons, B.D. Competition for Stem Cell Fate Determinants as a Mechanism for Tissue Homeostasis. arXiv
**2019**, arXiv:1901.03903. [Google Scholar] - Altschuler, S.J.; Angenent, S.B.; Wang, Y.; Wu, L.F. On the spontaneous emergence of cell polarity. Nature
**2008**, 454, 886–889. [Google Scholar] [CrossRef][Green Version] - Chara, O.; Tanaka, E.M.; Brusch, L. Mathematical modeling of regenerative processes. In Current Topics in Developmental Biology; Elsevier: Amsterdam, The Netherlands, 2014; Volume 108, pp. 283–317. [Google Scholar]
- Ebisuya, M.; Briscoe, J. What does time mean in development? Development
**2018**, 145, dev164368. [Google Scholar] [CrossRef] [PubMed][Green Version] - Turing, A.M. The chemical basis of morphogenesis. Phil. Trans. R. Soc. Lond.
**1952**, 237, 37–72. [Google Scholar] - Kondo, S.; Miura, T. Reaction-Diffusion Model as a Framework for Understanding Biological Pattern Formation. Science
**1999**, 329, 1616–1620. [Google Scholar] [CrossRef][Green Version] - Butty, A.; Perrinjaquet, N.; Petit, A.; Jaquenoud, M.; Segall, J.; Hofmann, K.; Zwahlen, C.; Peter, M. A positive feedback loop stabilizes the guanine-nucleotide exchange factor Cdc24 at sites of polarization. EMBO J.
**2002**, 21, 1565–1576. [Google Scholar] [CrossRef] - Kauffman, S.A. Pattern formation in the Drosophila embryo. Phil. Trans. R. Soc. Lond. B
**1981**, 295, 567–594. [Google Scholar] - Meinhardt, A.; Gierer, J. Applications of a theory of biological pattern formation based on lateral inhibition. J. Cell Sci.
**1974**, 15, 321–346. [Google Scholar] [PubMed] - Sick, S.; Reinker, S.; Timmer, J.; Schlake, T. WNT and DKK determine hair follicle spacing through a reaction-diffusion mechanism. Science
**2006**, 314, 1447–1450. [Google Scholar] [CrossRef] [PubMed][Green Version] - Nakamura, T.; Mine, N.; Nakaguchi, E.; Mochizuki, A.; Yamamoto, M.; Yashiro, K.; Meno, C.; Hamada, H. Generation of robust left-right asymmetry in the mouse embryo requires a self-enhancement and lateral-inhibition system. Dev. Cell
**2006**, 11, 495–504. [Google Scholar] [CrossRef] [PubMed][Green Version] - Newman, S.A.; Frisch, H.L. Dynamics of skeletal pattern formation in developing chick limb. Science
**1979**, 205, 662–668. [Google Scholar] [CrossRef] [PubMed][Green Version] - Miura, T.; Shiota, K. TGFbeta2 acts as an “activator” molecule in reaction-diffusion model and is involved in cell sorting phenomenon in mouse limb micromass culture. Dev. Dyn.
**2000**, 217, 241–249. [Google Scholar] [CrossRef] - Raspopovic, J.; Marcon, L.; Russo, L.; Sharpe, J. Digit patterning is controlled by a Bmp-Sox9-Wnt Turing network modulated by morphogen gradients. Science
**2014**, 345, 566–570. [Google Scholar] [CrossRef] - Meinhardt, H.; de Boer, P.A.J. Pattern formation in Escherichia coli: A model for the pole-to-pole oscillations of Min proteins and the localization of the division site. Proc. Natl. Acad. Sci. USA
**2001**, 98, 14202–14207. [Google Scholar] [CrossRef] [PubMed][Green Version] - Bement, W.M.; Leda, M.; Moe, A.M.; Kita, A.M.; Larson, M.E.; Golding, A.E.; Pfeuti, C.; Su, K.C.; Miller, A.L.; Goryachev, A.B.; et al. Activator–inhibitor coupling between Rho signalling and actin assembly makes the cell cortex an excitable medium. Nat. Cell Biol.
**2015**, 17, 1471–1483. [Google Scholar] [CrossRef][Green Version] - Michaux, J.B.; Robin, F.B.; McFadden, W.M.; Munro, E.M. Excitable RhoA dynamics drive pulsed contractions in the early C. elegans embryo. J. Cell Biol.
**2018**, 217, 4230–4252. [Google Scholar] [CrossRef] [PubMed][Green Version] - Koch, A.; Meinhardt, H. Biological pattern formation: From basic mechanisms to complex structures. Rev. Mod. Phys.
**1994**, 66, 1481. [Google Scholar] [CrossRef] - Marcon, L.; Sharpe, J. Turing patterns in development: What about the horse part? Curr. Opin. Genet. Dev.
**2012**, 22, 578–584. [Google Scholar] [CrossRef] [PubMed] - Menshykau, D.; Kraemer, C.; Iber, D. Branch mode selection during early lung development. PLoS Comput. Biol.
**2012**, 8, e1002377. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mori, Y.; Jilkine, A.; Edelstein-Keshet, A. Wave-Pinning and Cell Polarity from a Bistable Reaction-Diffusion System. Biophys. J.
**2008**, 94, 3684–3697. [Google Scholar] [CrossRef] [PubMed][Green Version] - Halatek, J.; Brauns, F.; Frey, E. Self-organization principles of intracellular pattern formation. Phil. Trans. R. Soc. Lond B
**2018**, 373, 20170107. [Google Scholar] [CrossRef] - Munro, E.; Nance, J.; Priess, J.R. Cortical flows powered by asymmetrical contraction transport PAR proteins to establish and maintain anterior-posterior polarity in the early C. elegans embryo. Dev. Cell
**2004**, 7, 413–424. [Google Scholar] [CrossRef][Green Version] - Goehring, N.W.; Trong, P.K.; Bois, J.S.; Chowdhury, D.; Nicola, E.M.; Hyman, A.A.; Grill, S.W. Polarization of PAR proteins by advective triggering of a pattern-forming system. Science
**2011**, 334, 1137–1141. [Google Scholar] [CrossRef][Green Version] - Tostevin, F.; Howard, M. Modeling the establishment of PAR protein polarity in the one-cell C. elegans embryo. Biophys. J.
**2008**, 95, 4512–4522. [Google Scholar] [CrossRef][Green Version] - Goryachev, A.B.; Pokhilko, A.V. Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett.
**2008**, 582, 1437–1443. [Google Scholar] [CrossRef][Green Version] - Savage, N.S.; Layton, A.T.; Lew, D.J. Mechanistic mathematical model of polarity in yeast. Mol. Biol. Cell
**2012**, 23, 1998–2013. [Google Scholar] [CrossRef] [PubMed] - Goryachev, A.B.; Leda, M. Many roads to symmetry breaking: Molecular mechanisms and theoretical models of yeast cell polarity. Mol. Biol. Cell
**2017**, 28, 370–380. [Google Scholar] [CrossRef] [PubMed] - Howard, M.; Rutenberg, A.D.; de Vet, S. Dynamic compartmentalization of bacteria: Accurate division in E. coli. Phys. Rev. Lett.
**2001**, 87, 278102. [Google Scholar] [CrossRef] [PubMed][Green Version] - Kruse, K. A dynamic model for determining the middle of Escherichia coli. Biophys. J.
**2002**, 82, 618–627. [Google Scholar] [CrossRef][Green Version] - Huang, K.C.; Meir, Y.; Wingreen, N.S. Dynamic structures in Escherichia coli: Spontaneous formation of MinE rings and MinD polar zones. Proc. Natl. Acad. Sci. USA
**2003**, 100, 12724–12728. [Google Scholar] [CrossRef] [PubMed][Green Version] - Gierer, A.; Meinhardt, H. A theory of biological pattern formation. Kybernetik
**1972**, 12, 30–39. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hiscock, T.W.; Megason, S.G. Mathematically guided approaches to distinguish models of periodic patterning. Development
**2015**, 142, 409–419. [Google Scholar] [CrossRef][Green Version] - Frohnhöfer, H.G.; Krauss, J.; Maischein, H.M.; Nüsslein-Volhard, C. Iridophores and their interactions with other chromatophores are required for stripe formation in zebrafish. Development
**2013**, 140, 2997–3007. [Google Scholar] [CrossRef][Green Version] - Murray, J.; Oster, G. Cell traction models for generating pattern and form in morphogenesis. J. Math. Biol.
**1984**, 19, 265–279. [Google Scholar] [CrossRef] - Howard, J.; Grill, S.W.; Bois, J.S. Turing’s next steps: The mechanochemical basis of morphogenesis. Nat. Rev. Mol. Cell Biol.
**2011**, 12, 392–398. [Google Scholar] [CrossRef] - Granero, M.; Porati, A.; Zanacca, D. A bifurcation analysis of pattern formation in a diffusion governed morphogenetic field. J. Math. Biol.
**1977**, 4, 21–27. [Google Scholar] [CrossRef] [PubMed] - Zadorin, A.S.; Rondelez, Y.; Gines, G.; Dilhas, V.; Urtel, G.; Zambrano, A.; Galas, J.C.; Estevez-Torres, A. Synthesis and materialization of a reaction–diffusion French flag pattern. Nat. Chem.
**2017**, 9, 990–996. [Google Scholar] [CrossRef] [PubMed][Green Version] - Chiou, J.G.; Ramirez, S.A.; Elston, T.C.; Witelski, T.P.; Schaeffer, D.G.; Lew, D.J. Principles that govern competition or co-existence in Rho-GTPase driven polarization. PLoS Comput. Biol.
**2018**, 14, e1006095. [Google Scholar] [CrossRef] [PubMed][Green Version] - Otsuji, M.; Ishihara, S.; Co, C.; Kaibuchi, K.; Mochizuki, A.; Kuroda, S. A mass conserved reaction–diffusion system captures properties of cell polarity. PLoS Comput. Biol.
**2007**, 3, e108. [Google Scholar] [CrossRef][Green Version] - Banerjee, D.S.; Banerjee, S. Size regulation of multiple organelles competing for a shared subunit pool. bioRxiv
**2020**, bioRxiv:902783. [Google Scholar] [CrossRef][Green Version] - Brauns, F.; Halatek, J.; Frey, E. Phase-space geometry of reaction–diffusion dynamics. arXiv
**2018**, arXiv:1812.08684. [Google Scholar] - Goehring, N.W.; Hyman, A.A. Organelle growth control through limiting pools of cytoplasmic components. Curr. Biol.
**2012**, 22, R330–R339. [Google Scholar] [CrossRef][Green Version] - Hoege, C.; Hyman, A.A. Principles of PAR polarity in Caenorhabditis elegans embryos. Nat. Rev. Mol. Cell Biol.
**2013**, 14, 315–322. [Google Scholar] [CrossRef] - Kemphues, K.J.; Priess, J.R.; Morton, D.G.; Cheng, N. Identification of genes required for cytoplasmic localization in early C. elegans embryos. Cell
**1988**, 52, 311–320. [Google Scholar] [CrossRef] - Goldstein, B.; Hird, S.N. Specification of the anteroposterior axis in Caenorhabditis elegans. Development
**1996**, 122, 1467–1474. [Google Scholar] - Nguyen-Ngoc, T.; Afshar, K.; Gönczy, P. Coupling of cortical dynein and Gα proteins mediates spindle positioning in Caenorhabditis elegans. Nat. Cell Biol.
**2007**, 9, 1294–1302. [Google Scholar] [CrossRef] [PubMed] - Gönczy, P. Mechanisms of asymmetric cell division: Flies and worms pave the way. Nat. Rev. Mol. Cell Biol.
**2008**, 9, 355–366. [Google Scholar] [CrossRef] [PubMed] - Motegi, F.; Zonies, S.; Hao, Y.; Cuenca, A.A.; Griffin, E.; Seydoux, G. Microtubules induce self-organization of polarized PAR domains in Caenorhabditis elegans zygotes. Nat. Cell Biol.
**2011**, 13, 1361–1367. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sailer, A.; Anneken, A.; Li, Y.; Lee, S.; Munro, E. Dynamic opposition of clustered proteins stabilizes cortical polarity in the C. elegans zygote. Dev. Cell
**2015**, 35, 131–142. [Google Scholar] [CrossRef] [PubMed][Green Version] - Hubatsch, L.; Peglion, F.; Reich, J.D.; Rodrigues, N.T.; Hirani, N.; Illukkumbura, R.; Goehring, N.W. A cell-size threshold limits cell polarity and asymmetric division potential. Nat. Phys.
**2019**, 15, 1078–1085. [Google Scholar] [CrossRef] [PubMed] - Sulston, J.E.; Schierenberg, E.; White, J.G.; Thomson, J. The embryonic cell lineage of the nematode Caenorhabditis elegans. Dev. Biol.
**1983**, 100, 64–119. [Google Scholar] [CrossRef] - Chiou, J.g.; Balasubramanian, M.K.; Lew, D.J. Cell polarity in yeast. Annu. Rev. Cell Dev. Biol.
**2017**, 33, 77–101. [Google Scholar] [CrossRef] [PubMed] - Di Talia, S.; Skotheim, J.M.; Bean, J.M.; Siggia, E.D.; Cross, F.R. The effects of molecular noise and size control on variability in the budding yeast cell cycle. Nature
**2007**, 448, 947–951. [Google Scholar] [CrossRef] [PubMed] - Ozbudak, E.M.; Becskei, A.; Oudenaarden, A. A system of counteracting feedback loops regulates Cdc42p activity during spontaneous cell polarization. Dev. Cell
**2005**, 9, 565–571. [Google Scholar] [CrossRef] [PubMed][Green Version] - Okada, S.; Leda, M.; Hanna, J.; Savage, N.S.; Bi, E.; Goryachev, A.B. Daughter Cell Identity Emerges from the Interplay of Cdc42, Septins, and Exocytosis. Dev. Cell
**2013**, 26, 148–161. [Google Scholar] [CrossRef][Green Version] - Turner, J.J.; Ewald, J.C.; Skotheim, J.M. Cell size control in yeast. Curr. Biol.
**2012**, 22, R350–R359. [Google Scholar] [CrossRef] [PubMed][Green Version] - Howell, A.S.; Savage, N.S.; Johnson, S.A.; Bose, I.; Wagner, A.W.; Zyla, T.R.; Nijhout, H.F.; Reed, M.C.; Goryachev, A.B.; Lew, D.J. Singularity in Polarization: Rewiring Yeast Cells to Make Two Buds. Cell
**2009**, 139, 731–743. [Google Scholar] [CrossRef] [PubMed][Green Version] - Maroto, M.; Bone, R.A.; Dale, J.K. Somitogenesis. Development
**2012**, 139, 2453–2456. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sarrazin, A.F.; Peel, A.D.; Averof, M. A segmentation clock with two-segment periodicity in insects. Science
**2012**, 336, 338–341. [Google Scholar] [CrossRef] - LoTurco, J.J.; Bai, J. The multipolar stage and disruptions in neuronal migration. Trends Neurosci.
**2006**, 29, 407–413. [Google Scholar] [CrossRef] - Noctor, S.C.; Martínez-Cerdeño, V.; Ivic, L.; Kriegstein, A.R. Cortical neurons arise in symmetric and asymmetric division zones and migrate through specific phases. Nat. Neurosci.
**2004**, 7, 136–144. [Google Scholar] [CrossRef] - De Anda, F.C.; Gärtner, A.; Tsai, L.H.; Dotti, C.G. Pyramidal neuron polarity axis is defined at the bipolar stage. J. Cell Sci.
**2008**, 121, 178–185. [Google Scholar] [CrossRef][Green Version] - Menchón, S.A.; Gärtner, A.; Román, P.; Dotti, C.G. Neuronal (bi) polarity as a self-organized process enhanced by growing membrane. PLoS ONE
**2011**, 6, e24190. [Google Scholar] [CrossRef] [PubMed] - Van den Brink, S.C.; Baillie-Johnson, P.; Balayo, T.; Hadjantonakis, A.K.; Nowotschin, S.; Turner, D.A.; Arias, A.M. Symmetry breaking, germ layer specification and axial organisation in aggregates of mouse embryonic stem cells. Development
**2014**, 141, 4231–4242. [Google Scholar] [CrossRef] [PubMed][Green Version] - Arnold, S.J.; Robertson, E.J. Making a commitment: Cell lineage allocation and axis patterning in the early mouse embryo. Nat. Rev. Mol. Cell Biol.
**2009**, 10, 91–103. [Google Scholar] [CrossRef] [PubMed] - Harrison, S.E.; Sozen, B.; Christodoulou, N.; Kyprianou, C.; Zernicka-Goetz, M. Assembly of embryonic and extraembryonic stem cells to mimic embryogenesis in vitro. Science
**2017**, 356, eaal1810. [Google Scholar] [CrossRef] [PubMed][Green Version] - Warmflash, A.; Sorre, B.; Etoc, F.; Siggia, E.D.; Brivanlou, A.H. A method to recapitulate early embryonic spatial patterning in human embryonic stem cells. Nat. Methods
**2014**, 11, 847. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ten Berge, D.; Koole, W.; Fuerer, C.; Fish, M.; Eroglu, E.; Nusse, R. Wnt signaling mediates self-organization and axis formation in embryoid bodies. Cell Stem Cell
**2008**, 3, 508–518. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sagy, N.; Slovin, S.; Allalouf, M.; Pour, M.; Savyon, G.; Boxman, J.; Nachman, I. Prediction and control of symmetry breaking in embryoid bodies by environment and signal integration. Development
**2019**, 146, dev181917. [Google Scholar] [CrossRef][Green Version] - Turner, D.A.; Girgin, M.; Alonso-Crisostomo, L.; Trivedi, V.; Baillie-Johnson, P.; Glodowski, C.R.; Hayward, P.C.; Collignon, J.; Gustavsen, C.; Serup, P.; et al. Anteroposterior polarity and elongation in the absence of extra-embryonic tissues and of spatially localised signalling in gastruloids: Mammalian embryonic organoids. Development
**2017**, 144, 3894–3906. [Google Scholar] [CrossRef] [PubMed][Green Version] - Morgani, S.M.; Hadjantonakis, A.K. Signaling regulation during gastrulation: Insights from mouse embryos and in vitro systems. eLife
**2019**, 137, 391–431. [Google Scholar] - Juan, H.; Hamada, H. Roles of nodal-lefty regulatory loops in embryonic patterning of vertebrates. Genes Cells
**2001**, 6, 923–930. [Google Scholar] [CrossRef] - Shahbazi, M.N.; Siggia, E.D.; Zernicka-Goetz, M. Self-organization of stem cells into embryos: A window on early mammalian development. Science
**2019**, 364, 948–951. [Google Scholar] [CrossRef] - Cornwall Scoones, J.; Hiscock, T.W. A dot-stripe Turing model of joint patterning in the tetrapod limb. Development
**2020**, 147, dev183699. [Google Scholar] [CrossRef][Green Version] - Crampin, E.J.; Gaffney, E.A.; Maini, P.K. Reaction and diffusion on growing domains: Scenarios for robust pattern formation. Bull. Math. Biol.
**1999**, 61, 1093–1120. [Google Scholar] [CrossRef] - Postma, M.; Roelofs, J.; Goedhart, J.; Gadella, T.W.; Visser, A.J.; Van Haastert, P.J. Uniform cAMP stimulation of Dictyostelium cells induces localized patches of signal transduction and pseudopodia. Mol. Biol. Cell
**2003**, 14, 5019–5027. [Google Scholar] [CrossRef] [PubMed][Green Version] - Erzurumlu, R.S.; Jhaveri, S.; Benowitz, L.I. Transient patterns of GAP-43 expression during the formation of barrels in the rat somatosensory cortex. J. Comp. Neurol.
**1990**, 292, 443–456. [Google Scholar] [CrossRef] [PubMed] - Hecht, I.; Kessler, D.A.; Levine, H. Transient localized patterns in noise-driven reaction-diffusion systems. Phys. Rev. Lett.
**2010**, 104, 158301. [Google Scholar] [CrossRef] [PubMed][Green Version] - Ben-Zvi, D.; Shilo, B.Z.; Barkai, N. Scaling of morphogen gradients. Curr. Opin. Genet. Dev.
**2011**, 21, 704–710. [Google Scholar] [CrossRef] [PubMed] - Wolpert, L. Positional information and the spatial pattern of cellular differentiation. J. Theor. Biol.
**1969**, 25, 1–47. [Google Scholar] [CrossRef] - Kinney, J.B.; Atwal, G.S. Equitability, mutual information, and the maximal information coefficient. Proc. Natl. Acad. Sci. USA
**2014**, 111, 3354–3359. [Google Scholar] [CrossRef][Green Version] - Werner, S.; Stückemann, T.; Amigo, M.B.; Rink, J.C.; Jülicher, F.; Friedrich, B.M. Scaling and regeneration of self-organized patterns. Phys. Rev. Lett.
**2015**, 114, 138101. [Google Scholar] [CrossRef] [PubMed][Green Version] - Mark, S.; Shlomovitz, R.; Gov, N.S.; Poujade, M.; Grasland-Mongrain, E.; Silberzan, P. Physical model of the dynamic instability in an expanding cell culture. Biophys. J.
**2010**, 98, 361–370. [Google Scholar] [CrossRef] [PubMed][Green Version] - Banerjee, S.; Utuje, K.J.; Marchetti, M.C. Propagating stress waves during epithelial expansion. Phys. Rev. Lett.
**2015**, 114, 228101. [Google Scholar] [CrossRef] [PubMed] - Ravasio, A.; Le, A.P.; Saw, T.B.; Tarle, V.; Ong, H.T.; Bertocchi, C.; Mège, R.M.; Lim, C.T.; Gov, N.S.; Ladoux, B. Regulation of epithelial cell organization by tuning cell–substrate adhesion. Integr. Biol.
**2015**, 7, 1228–1241. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Size-regulated symmetry breaking in activator–substrate model. (

**a**) Pattern formation in a model of positive feedback coupled to a finite constituent pool. (

**b**) Patterns form above a critical system size (${L}^{*}$), corresponding to the largest mode where the homogeneous state becomes unstable and the system breaks symmetry. The order parameter $\mathcal{O}$ for symmetry breaking is defined as $\mathcal{O}\left(L\right)={\int}_{0}^{L}|dS/dx|/\left[\underset{L}{max}{\int}_{0}^{L}|dS/dx|\right]$, where $\mathcal{O}$ is zero for a homogeneous state and $\mathcal{O}=1$ for a symmetry broken patterned state. All parameters other than system size was kept constant in this analysis and initial perturbations were so chosen that $N/L$ is constant, where $N=\int P(x,t)+S(x,t)\phantom{\rule{0.166667em}{0ex}}dx$ is the total pool size. Parameters: ${D}_{P}=1$, ${D}_{S}=0.05$, ${k}_{\mathrm{on}}/{k}_{\mathrm{off}}=20$, ${S}_{0}^{2}=10$ and $N/L=1.5$. (

**c**) Phase diagram in the plane of autocatalytic activity $\kappa $ and the Hill saturation parameter ${S}_{0}^{2}$, showing three different phases: homogeneous state (black), symmetry broken saturated state (green), and symmetry-broken unsaturated state (red). Colormap (green to red) denotes the average value of the reaction rate ${F}_{av}=\int F\left(x\right)dx$, computed in the high density region, with ${F}_{av}=0$ in the saturated state and ${F}_{av}\ne 0$ in the unsaturated state. ${F}_{av}=0.01$ (blue points) defines the crossover value from the saturated state to the unsaturated state. Parameter values are the same as (

**b**) except for ${D}_{S}=0.01$ and $L=3$. Inset: Profiles of $S\left(x\right)$. (

**d**–

**e**) Order parameter $\mathcal{O}$ for the unsaturated (

**d**) and the saturated (

**e**) regimes, showing that the symmetry of the homogeneous state is broken beyond a critical system size. We used periodic boundary conditions for all numerical simulations, unless otherwise specified. For initial conditions, we assumed a homogeneous $P\left(x\right)$ and a sinusoidal $S\left(x\right)$ profile of large wavelength.

**Figure 2.**Sequential pattern formation in growing domains. (

**a**) Kymographs of $S(x,t)$ for increasing system size. Figures show spontaneous pattern formation via symmetry breaking of the homogeneous state above a critical system size ${L}^{*}$. As we simulate larger systems (Equations (1) and (2)), multiple patterns emerge in a size-dependent manner. Simulations were done as described in Figure 1b. (

**b**) Number of patterns as a function of system size, for both the saturated and unsaturated regimes of the S-P model. Parameter values for (

**a**,

**b**) are the same as Figure 1b,c, respectively. Small amplitude random (uniform distribution) initial conditions for S and P were used for (

**b**).

**Figure 3.**Size-regulated symmetry breaking in single cells. (

**a**) A phase-diagram for the PAR system, considering polarisation state as a function of the circumferential length of the embryo (“Length”), and the ratio of aPAR to pPAR pool size (“AP ratio”). The diagram demonstrates a bipolar state (grey region) becomes unstable below a critical circumferential embryo length (pink and blue regions). Schematics for each of the three states are overlayed, with aPARs denoted in pink and pPARs denoted in blue. This bifurcation point quantitatively matches the critical size for which dividing P-cells in the early C. elegans embryo transition from asymmetric to symmetric division. Figure adapted from the work in [56]. Adjacent is the feedback motif that drives pattern formation. (

**b**) In the budding yeast (S. cerevisiae), cell cycle commitment to Start is linked to the localisation of Cdc42 effectors at the presumptive bud site [58]. Cdc42 polarity establishment is related to the duration of the ${G}_{1}$ phase of the cell cycle, which ends at a critical cell size [59]. Models have shown that a growth process with positive feedback leads to Cdc42 polarisation at a single site [31].

**Figure 4.**Sequential pattern formation in developmental systems. (

**a**) In in vitro cultured neurones, polarity arises sequentially. A second polarity axis is formed after cell growth, and its orientation is “mirrored” off the first. A putative symmetry breaking circuit is presented adjacently [69], considering a membrane-protein (MP) activator coupled to modulators of endocytosis (ME), representing the effects of small GTPases. (

**b**) Gastruloids polarise and elongate only when initialised with a critical number of cells [70]. For seed numbers beyond this initial bifurcation value, gastruloids can self-organise more axes. T/Brachyury expression is localised to the protrusion in monopolarised gastruloids, and is speculated to also be localised to further protrusions in multipolar variants. A potential feedback circuit is drawn adjacently, which remains to be investigated. (

**c**) An activator–substrate model for lung branching [25], based on autocatalytic production of the signalling molecule Shh (activator) at the lung bud tip, via consumption of the substrate molecule Fgf10. (

**d**) A dot-stripe mechanism is proposed to pattern the joints of developing digits: a Turing-like dot-forming system specifies the positions of bones and orients through repression a Turing-like stripe-forming system to specify joints. Modelled on a growing domain, sequential joint specification emerges, with joints forming near the developing tip. A coupled Turing scheme is described adjacent, considering a dot-forming substrate-depletion module (A,S) coupled to a stripe-forming activator–inhibitor module (B,I).

**Figure 5.**Pattern scaling, splitting and transient pattern formation in growing domains. (

**a**) Feedback motif for an activator–substrate system coupled to a growing domain. (

**b**) (Left) When subunits are produced at a rate slower than the rate of domain growth, growth-induced dilution leads to transient pattern establishment. (Middle) When the production of subunit pool occurs at a rate comparable to system size growth, the pattern formed grows in proportion to system size, exhibiting a dynamic scaling behaviour. This is different from sequential pattern formation as the polarity is preserved during growth. (Right) In the case of strong autocatalysis of S, the pattern spontaneously splits. (

**c**) Phase diagram for pattern formation as functions of pool growth rate relative to the system, $\tilde{G}/\alpha $, and ${k}_{\mathrm{on}}/{k}_{\mathrm{off}}$. Colormap denotes the inverse of the pattern lifetime, $1/{T}^{c}$. (

**d**) Time evolution of structure size ${S}_{\mathrm{tot}}={\int}_{0}^{L}Sdx$ (blue) and system size L (red) for the case of transient pattern formation. The lifetime of the pattern ${T}^{c}$ is coupled to the system size at transition to the homogeneous state, ${L}^{c}$. They can be tuned independently of each other, for example, by changing growth rate $\alpha $ (case B) where only the transition time changes, or by changing autocatalysis rate (case C) where ${T}^{c}$ remains the same but transition happens at a different system size. (

**e**) Tunability of pattern lifetime can be utilised as a control mechanism for symmetric and asymmetric cell division (in terms of polarity protein content). When the pattern is transient (left) the dissolution of the structure will make the daughter cells symmetric in fate, containing the same amount of polarity proteins. If the pattern persists (right) then the division will lead to asymmetric fate inheritance. Parameter values: ${D}_{P}=1$, ${D}_{S}=0.005$, ${\kappa}_{0}=0.85$, ${S}_{0}^{2}=10$, $\alpha =0.01$, $L\left(0\right)=1$, and total pool density $\int (P+S)dx/L=2$, with G and ${k}_{\mathrm{on}}/{k}_{\mathrm{off}}$ variable.

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Cornwall Scoones, J.; Banerjee, D.S.; Banerjee, S. Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions. *Cells* **2020**, *9*, 1646.
https://doi.org/10.3390/cells9071646

**AMA Style**

Cornwall Scoones J, Banerjee DS, Banerjee S. Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions. *Cells*. 2020; 9(7):1646.
https://doi.org/10.3390/cells9071646

**Chicago/Turabian Style**

Cornwall Scoones, Jake, Deb Sankar Banerjee, and Shiladitya Banerjee. 2020. "Size-Regulated Symmetry Breaking in Reaction-Diffusion Models of Developmental Transitions" *Cells* 9, no. 7: 1646.
https://doi.org/10.3390/cells9071646