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

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## Abstract

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## 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

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**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/).

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**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