Size-regulated symmetry breaking in reaction-diffusion models of developmental transitions
Jake Cornwall Scoones, Deb Sankar Banerjee, Shiladitya Banerjee
SSize-regulated symmetry breaking in reaction-diffusion models of developmental transitions
Jake Cornwall Scoones, Deb Sankar Banerjee, and Shiladitya Banerjee ∗ Division of Biology and Biological Engineering, California Institute of Technology, Pasadena, CA 91125, USA † Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213, USA † The development of multicellular organisms proceeds through a series of morphogenetic and cell-state transi-tions, transforming homogeneous zygotes into complex adults by a process of self-organisation. Many of thesetransitions are achieved by spontaneous symmetry breaking mechanisms, allowing cells and tissues to acquirepattern and polarity by virtue of local interactions without an upstream supply of information. The combinedwork of theory and experiment has elucidated how these systems break symmetry during developmental transi-tions. Given such transitions are multiple and their temporal ordering is crucial, an equally important questionis how these developmental transitions are coordinated in time. Using a minimal mass-conserved substrate-depletion model for symmetry breaking as our case study, we elucidate mechanisms by which cells and tissuescan couple reaction-diffusion driven symmetry breaking to the timing of developmental transitions, arguing thatthe dependence of patterning mode on system size may be a generic principle by which developing organismsmeasure time. By analyzing different regimes of our model, simulated on growing domains, we elaborate threedistinct behaviours, allowing for clock-, timer-, or switch-like dynamics. By relating these behaviours to ex-perimentally documented case studies of developmental timing, we provide a minimal conceptual framework tointerrogate how developing organisms coordinate developmental transitions.
I. INTRODUCTION
In developmental systems, it is important for the mechanis-tic constituents to ‘know about the size of the living system asa whole [1, 2]. This is most apparent in developmental transi-tions, which in many cases only proceed when cells or tissueshave reached a critical size. Such control strategies allow liv-ing systems to couple developmental time to their size andgeometry. What is the physical basis of these phenomena?One can envisage two broad classes of size-control mech-anisms. The first proposes size-dependent transitions are un-der external regulation: in developing tissues, this could bemanifested as a cell-intrinsic clock, whereby a transition isachieved after a pre-defined time interval [3]; or a gradient-based mechanism, whereby a distance critically far from thesource of signalling molecules triggers a transition [4]. Alter-natively, size-regulation could be an emergent property of col-lective decision-making, akin to quorum sensing [5–7]: com-munication between mechanistic constituents allows the sys-tem to sense its size.
A priori , these control mechanismshave several conceptual benefits: synchrony in a transitionis more robust, given decisions are made collectively; andsuch decision making does not rely on a subset of constituents(e.g. source cells in gradient generation), instead being de-centralised [1, 8]. Can we find examples of emergent size-regulation from collective decision-making in living systems?Many developmental transitions couple size- and temporal-control to a change in polarity regime: at a critical size, thesystem may spontaneously break symmetry to polarise, de-polarise, bipolarise, or even radically change its pattern. Wehypothesise that size regulation of such developmental transi-tions are an emergent property of the many mechanisms of po-larisation. This view provides a framework for understanding ∗ Correspondence: [email protected] † These authors contributed equally to this work developmental time [9], placing a critical emphasis on systemsize.Here we outline a minimal reaction-diffusion model forsize-dependent polarisation in developing systems, arguingthat the underlying regulatory motifs can be understood via asubstrate-depletion feedback motif coupled with the growth ofthe system. We then overview cases of size-dependent sym-metry breaking across scales, focusing on biochemical sys-tems. We consider size-dependent decision-making within in-dividual cells through to analogous processes in developingmulticellular systems, proposing that our minimal model canhelp unify these divergent processes within a common theo-retical framework. We conclude by speculating on the roleof these mechanisms in coupling size-dependent transitions todevelopmental time.
II. REACTION-DIFFUSION AS A FRAMEWORK TOUNDERSTAND SIZE-REGULATEDSYMMETRY-BREAKING
Pattern forming reaction-diffusion (RD) systems [10] arewidely used to characterise the complex networks of molec-ular and cellular interactions that underlie biological symme-try breaking. In these systems, pattern formation can arisespontaneously driven by feedback motifs between diffusiblemolecules (intracellular polarity proteins or extracellular mor-phogens). Since Turing’s insight in 1952, many different RDmotifs have been proposed as the physical basis for the emer-gence of developmental patterns across scales [11] – from po-larity establishment at the scale of a single cell [12] to patternformation on the scale of a whole organism [11, 13].Perhaps the most famous phrasing of an RD system is theactivator-inhibitor circuit, originally developed by Gierer andMeinhardt [14]. In activator-inhibitor systems, an activatormolecule promotes its own production as well as the produc-tion of its fast-diffusing inhibitor that suppresses autocatalyticproduction of the activator. This motif has remarkable ex- a r X i v : . [ q - b i o . T O ] J u l planatory power across contexts, being used to describe thespontaneous establishment of hair follicle spacing [15], left-right asymmetry establishment in vertebrates [16], skeletalpatterns in growing limbs [17–19], as well as pole-to-pole os-cillation of Min proteins during bacterial cell division [20],and self-organisation of Rho GTPases in the animal cell cor-tex [21, 22]. Substrate depletion models can also yield thespontaneous emergence of periodic patterns [23, 24]. In thesemodels, the activator consumes its own substrate to promoteits autocatalytic production, leading to out-of-phase pattern-ing of the activator and the substrate molecules (Fig. 1a). Forexample, a substrate-depletion model was been used to ex-plain lung branching, explaining out of phase patterns of geneexpression between Shh (the activator) and FGF (the sub-strate) [25].Substrate-depletion models are particularly relevant in thestudy of intracellular pattern formation and polarity establish-ment as feedback can be phrased in a mass-conserved man-ner. In such models, patterns emerge by the redistributionof polarity proteins [26, 27]: proteins that form a polaritypatch engage in self-recruitment, acting as activators, but thispositive-feedback is limited by a finite pool of (typically cyto-plasmic) subunits. Variants on mass-conserved substrate de-pletion models have been used to understand PAR polarityestablishment in the
C. elegans embryo [28–30] and Cdc42polarisation in
S. cerevisiae [31–33]. Further, such modelsexhibit dynamic regimes, for example helping to explain os-cillations in the
E. coli
Min-protein system [34–36].While the mechanistic constituents and precise feedback ar-chitectures of RD mechanisms differ, many rely on a centralmotif of local activation and long-range inhibition [37]. Thisconcept has acted as an important heuristic in framing mod-els of biological pattern formation, but importantly unifies di-verse RD systems within a common mathematical framework.Specifically, recent theoretical models have demonstrated thatmost RD models of pattern formation can be approximatedby the same mathematical formulation: the SwiftHohenbergequation [38]. Strikingly, other mechanisms of periodic pat-tern formation that rely on cell movement [39] or mechanicalinstabilities [40] also rely on local-activation and long-rangeinhibition [38, 41]. Hence many dynamical features of thesemodels are applicable across systems and length scales.In this perspective, we restrict our focus to RD models forbiochemical pattern formation, elucidating the biological sig-nificance of a common dynamical feature shared across manymotifs: the role of a critical system size for symmetry break-ing [23, 42, 43]. To demonstrate this, we develop a mathe-matical framework for a mass-conserving RD system with afeedback motif similar to activator-substrate models for cellpolarisation (Fig. 1a).
III. A MINIMAL MODEL FOR SIZE-REGULATEDSYMMETRY-BREAKING
To analyse the role of system size on the timing of sym-metry breaking in a biochemical system, we consider a min-imal model for a mass-conserved substrate-depletion system. Specifically we model the spatiotemporal dynamics of a regu-latory structure S in a living system of size L , and coupled to afinite pool of building blocks. Let P ( x , t ) denote the concen-tration of building blocks in the subunit pool at location x attime t , and S ( x , t ) is the concentration of building blocks in-corporated in the regulatory structure. S increases in amountby depleting the subunit pool P , and S can undergo dissoci-ation into P (Fig. 1a). The coupled dynamics of S and P aregiven by: ∂ t S = D s ∇ S + k on P f ( S ) − k off S , (1) ∂ t P = D p ∇ P − k on P f ( S ) + k off S , (2)where D p and D s are the diffusion constants of the subunitsand the structure S ( D s (cid:28) D p ), k on parameterises the as-sociation rate of subunits to S , and k off is the constant rate ofdisassembly of S . The function f defines the size-dependenceof the autocatalytic production rate of S . We assume the func-tional form f ( S ) = S n / ( S n + S n ) , where the constant n > controls the strength of cooperative assembly of S . The to-tal amount of P and S remains conserved at all times, i.e., N = (cid:82) P ( x, t ) + S ( x, t ) dx = constant, and is assumed toscale linearly with the system size L .As the structure S grows by locally depleting the pool P ,localized patterns of S will exist in low density regions of P as long as D P (cid:29) D S (Fig. 1a). The symmetry of the ho-mogeneous state breaks and patterns appear above a criticaldomain size, making this transition size-dependent (Fig. 1b).The critical size can be obtained from linear stability analysisas L ∗ = (cid:18) D S D P D S ∂ P F + D P ∂ S F (cid:19) (3)where F = k on P f ( S ) − k off S , and the derivatives are eval-uated at the homogeneous steady state. This property of sizedependent symmetry-breaking can serve as a decision-makingrule to enact developmental state transitions when the systemsize reaches a critical value L ∗ . As the system size gets largermore discrete structures will emerge (Fig. 3a). Such sequen-tial pattern formation may regulate size-dependent develop-mental transitions, as we discuss later.The S - P model introduced above (for n = 2 ) is simi-lar to previously studied mass-conserved RD models such aswave-pinning [26] and Turing-like autocatalytic model [31].These activator-substrate models exhibit two distinct dynamicregimes [44]: the wave-pinning regime, characterised by widemesa-like patterns and saturated subunit association kinet-ics; and the Turing regime that yields narrow concentrationpeaks by virtue of competition between structures. The Turingregime operates below saturation ( S (cid:28) S ), where a winner-take-all competition between structures asymptotically resultsin a single concentration peak [31, 45, 46]. By contrast, in theregime above saturation ( S (cid:29) S ), the structures can co-existfor very long timescales, with the timescale of coarsening de-termined by parameters such as the diffusion coefficients orthe reaction fluxes [47]. The S - P model exhibits both the sat-urated and the unsaturated regimes that can be obtained by Pool S a S Pool x system size L o r de r pa r a m e t e r O S ( x , t ) t x S ( x , t ) t x b L* S k on / k off S a t u r a t ed Unsaturated
S(x) S(x) c Homogeneous H o m ogeneou s κ = O L unsaturated d O L saturated e FIG. 1. Size regulated symmetry breaking in activator-substratemodel. (a) Pattern formation in a model of positive feedback cou-pled to a finite constituent pool. (b) Patterns form above a criticalsystem size ( L ∗ ), corresponding to the largest mode where the ho-mogeneous state becomes unstable and the system breaks symme-try. The order parameter O for symmetry breaking is defined as O ( L ) = (cid:82) L | dS/dx | / (cid:104) max L (cid:82) L | dS/dx | (cid:105) , where O is zero for ahomogeneous state and O = 1 for a symmetry broken patternedstate. All parameters other than system size was kept constant inthis analysis and initial perturbations were so chosen that N/L isconstant, where N = (cid:82) P ( x, t ) + S ( x, t ) dx is the total pool size.Parameters: D P = 1 , D S = 0 . , k on /k off = 20 , S = 10 and N/L = 1 . . (c) Phase diagram in the plane of autocatalyticactivity κ and the Hill saturation parameter S , showing three dif-ferent phases: homogeneous state (black), symmetry broken sat-urated state (green), and symmetry-broken unsaturated state (red).Colormap (green to red) denote the average value of the reactionrate F av = (cid:82) F ( x ) dx , computed in the high density region, with F av = 0 in the saturated state and F av (cid:54) = 0 in the unsaturated state. F av = 0 . (blue points) defines the cross-over value from the satu-rated state to the unsaturated state. The inset figures show the spatialprofiles of S in the different regimes. Parameter values are the sameas (b) except for D S = 0 . and L = 3 . (d-e) Order parameter O for the unsaturated (d) and the saturated (e) regimes, showing thatthe symmetry of the homogeneous state is broken beyond a criticalsystem size. We used periodic boundary conditions for all numericalsimulations, unless otherwise specified. For initial conditions, we as-sumed a homogeneous P ( x ) and a sinusoidal S ( x ) profile of largewavelength. tuning the strength of autocatalytic activity ( κ = k on /k off )and the Hill saturation parameter S (Fig. 1c). Both thesedynamical regimes exhibit size-dependent symmetry break-ing (Fig. 1d-e), as well as sequential pattern formation for increasing system sizes (Fig. 3). In the latter case, we as-sume that the subunit density is constant for increasing L , (cid:82) L ( S + P ) dx ∝ L , as macromolecular contents often scalewith cell size [48]. For appropriate choice of model parame-ters in the unsaturated regime, an increase in total subunit poolsize coupled with local depletion of the subunit pool gives riseto coexisting peaks of the same height over biologically rele-vant timescales, much shorter than the long timescale of struc-ture coarsening. IV. CRITICAL SIZE ON POLARISATION CAN BEUTILISED TO ENACT CELL STATE TRANSITIONS
Our minimal model demonstrates how a positive-feedbackmotif coupled to features of system size, such as a limitingcytoplasmic pool, can yield size-regulated symmetry break-ing of regulatory structures. In this section we explore bio-logical realisations of our model around the bifurcation fromunpolarised to polarised, arguing that cells may utilise thesesize-dependent properties to coordinate state transitions.
A. Cell size dependent transition from asymmetric tosymmetric division in the early C. elegans embryo
The polarisation of the early
C. elegans embryo has becomea paradigm in biological symmetry breaking [49]. Anterior-posterior (AP) polarity in
C. elegans is established before thefirst cell division [50]. Polarity establishment is achieved bythe segregation of two groups of partitioning-defective (PAR)proteins to the anterior versus posterior [28–30]. Initially an-terior PARs (aPARs) cover the entire membrane of the egg,but upon fertilisation at the posterior, serving as the symme-try breaking cue [51], aPARs segregate anteriorly, and poste-rior PARs (pPARs) posteriorly. Segregated PARs coordinatepolarised division, whereby the division plane is set by theboundary of the two PAR domains [52, 53].PARs bind the membrane from a common, finite cyto-plasmic pool and diffuse freely. Symmetry breaking isachieved by phosphorylation-dependent mutual inhibition be-tween aPARs and pPARs [57, 58], patterning the cell mem-brane into two polarisation domains. This double-negativefeedback structure reinforces biases in the localisation ofPARs and hence plays a similar role as the positive feed-back motif in our minimal model. Indeed explicit substrate-depletion models yield phenomenologically identical results[54].The boundary between PAR domains is regulated by therelative diffusivities of aPARs and pPARs, as well as the rel-ative off rates. Boundary length is set by L D = (cid:112) D/k off ,where D is the aPAR diffusion constant and k off is the aPARdissociation rate from the membrane. Hence, provided diffu-sion and dissociation rates are independent of system size, L D will also be independent of cell size. Modelling confirms thatthis holds true regardless of structural differences in the model[54]. This length-scale thus sets a minimum cell size that cansustain polarised PAR domains (grey region in Fig. 2a): below Leng t h AP ratio
P1P2P3P4 A sy mm e t r i c d i v i s i on S y mm e t r i c d i v i s i on Cell size threshold a Cdc42-GTP Pool b aPAR pPAR Pool
Pool size regulated
StartG time FIG. 2. Size-regulated symmetry breaking in single cells. (a) Aphase-diagram for the PAR system, considering polarisation state asa function of the circumferential length of the embryo (‘Length’),and the ratio of aPAR to pPAR pool size (‘AP ratio’). The dia-gram demonstrates a bipolar state (grey region) becomes unstable be-low a critical circumferential embryo length (pink and blue regions).Schematics for each of the three states are overlayed, with aPARsdenoted in pink and pPARs denoted in blue. This bifurcation pointquantitatively matches the critical size for which dividing P-cells inthe early
C. elegans embryo transition from asymmetric to symmet-ric division. Figure adapted from Ref. [54]. Adjacent is the feedbackmotif that drives pattern formation. (b) In the budding yeast (
S. cere-visiae ), cell cycle commitment to Start is linked to the localisationof Cdc42 effectors at the presumptive bud site [55]. Cdc42 polarityestablishment is related to the duration of the G phase of the cellcycle, which ends at a critical cell size [56]. Models have shown thata growth process with positive feedback leads to Cdc42 polarisationat a single site [31]. this critical size, diffusion overwhelms the capacity for PARsegregation, resulting in homogeneous localisation of eitheraPARs or pPARs (pink and blue regions of Fig. 2a)The physical critical size limit may be used by developing C. elegans embryos to coordinate a developmental transition.Sequential divisions of generating the first three posterior cells(P1-3) are asymmetric, each generating two daughters of dif-ferent fates, segregating germline determinants to only one.This pattern shifts to a symmetric mode by the third divi-sion of P4 (Fig. 2a), generating the two founding cells of thegermline lineage (Z2/Z3) [59]. Divisions are fast, meaningcell volume declines progressively, falling beneath the theo-retical critical cell size for polarisation by P4 [54]. By quan-tifying division symmetry using 3D reconstructions of PARdistributions, Habatsch et al. [54] found that the polarisationregime shifts from asymmetric to symmetric by P4. The tim-ing of this regime shift can be changed by reducing embryosize through genetic (ima3 RNAi) or physical (laser mediatedextrusion) perturbations. Thus a reduction in cell size coordi-nates a developmental transition in
C. elegans embryos fromasymmetric to symmetric cell division.
B. Size-dependent polarity establishment in budding yeast
Cell cycle commitment to budding in
S. cerevisiae followsfrom Cdc42 polarity establishment at the presumptive bud-ding site (Fig. 2b). The small Rho GTPase Cdc42-GTP formsa polarity patch to mark the bud location [60, 61]. This po-larity pattern emerges from an autocatalytic positive feed-back via Bem1 on the clustering of slowly diffusing mem-brane bound Cdc42-GTP [12], while the cytosolic Cdc42-GDP diffuses fast (Fig. 2b). Polarity establishment in the sys-tem can be captured by mass-conserved substrate-depletionmodel, with a slow-diffusing activator and a fast-diffusingsubstrate [26, 31, 45]. With appropriate choice of parame-ters, activator-substrate models would predict the formation ofsingle polarity cluster beyond a critical cell size [44] (Fig. 1).Thus the establishment of Cdc42 polarisation can be linkedto a critical cell size, consistent with models of critical cellsize threshold at the termination of G phase of the cell cy-cle [56, 62]. Molecular rewiring experiments have shown thatwhen the Bem1 is tweaked to diffuse very slowly, multipleCdc42 polarity patches are formed [63]. Since the onset ofpattern formation depends on L/L D , with L D the diffusionlength, slowing down diffusion is equivalent to increasing thesystem size, so multiple patterns emerge in accordance withpredictions from increasing domain length in our minimal RDmodel (Fig. 3a-b). V. SEQUENTIAL PATTERN FORMATION ANDPOLARISATION CAN BE COORDINATED BY A GROWINGDOMAIN
Across scales of biological organisation, structures of abipolar or iterative nature are abundant, and their developmentis often sequential. While clock-and-wavefront models havesuccessfully explained sequential patterning of axial segmen-tation in vertebrates [64] and recently also some invertebrates[65], sequential pattern formation can also arise from collec-tive decision-making. Our minimal mass-conserved substrate-depletion model illustrates how an RD mechanism can giverise to sequential periodic patterning: smaller domain sizescan sustain only a single pattern via an instability of the ho-mogeneous state, whereas domain growth provides sufficientspace to accommodate multiple patterns (Fig. 3a-b). Herewe implicitly assume that the RD system relaxes much fasterthan the timescale of domain growth, and that subunit con-centration remains unchanged. When domain growth rate iscomparable to the reaction rate, different dynamic patternsemerge as discussed in Section 6. In spite of differencesin their mechanistic bases, sequential patterning through do-main growth is common among many RD models (activator-inhibitor and substrate-depletion) [23]. In the following, weexpand our focus beyond just mass-conserved models, to il-lustrate how local-activation and long-range inhibition can ex-plain how growth can couple developmental tempo to statetransitions. a b
L/5 L 2L 3L
Increasing system size H i g h Lo w Pool S saturatedunsaturated L/L D nu m be r o f pa tt e r n s FIG. 3. Sequential pattern formation in growing domains. (a) Kymographs of S ( x, t ) for increasing system size. Figures show spontaneouspattern formation via symmetry breaking of the homogeneous state above a critical system size L ∗ . As we simulate larger systems (Eq. (1)-(2)), multiple patterns emerge in a size dependent manner. Simulation was done as described in Fig. 1b. (b) Number of patterns as a functionof system size, for both the saturated and unsaturated regimes of the S - P model. Parameter values for (a) and (b) are the same as Fig. 1b andFig. 1c, respectively. Small amplitude random (uniform distribution) initial conditions for S and P were used for (b). A. Neuronal sequential bipolarisation coordinated bymembrane growth
Neuronal polarisation is critical for brain development. Po-larisation commences as soon as neurones complete their finaldivision, by a process of neurite formation and selection.
Invitro studies have suggested neurones acquire a bipolar phe-notype, generating a leading neurite, key in guiding migration,and a trailing neurite which later acquires axonal fate [66, 67].Bipolarisation in vitro is achieved stochastically, whereby theposition of the first neurite is seemingly random, with the sec-ond being positioned opposite to the first [68]. How are thesepatterns coordinated?Menchn et al. [69] proposed an activator-inhibitor Turingmodel for cell polarisation. They argued that the necessaryfeedback architecture for a Turing instability is manifest indeveloping neurones: integral membrane proteins (the polar-isation cue) undergo cooperative self-recruitment i.e. local-activation; and also recruit more diffusive endocytosis modu-lators which facilitate their removal i.e. long-range inhibition(Fig. 4a). Indeed, in the right parameter regime in a finite do-main, simulations suggest neurones can spontaneously breaksymmetry. The polarity regime is critically dependent onmembrane size: a subcritical size prohibits symmetry break-ing (like in
C. elegans ); an intermediate size allows for a sin-gle polarity axis; and larger sizes allow for a bipolar pheno-type (Fig. 4a). Sequential and ”mirrored” polarisation can beachieved by membrane growth. A growing domain leads to atime-dependent bifurcation, whereby the cell transitions froma unipolar to bipolar stability regime. The ”mirroring” of thesecond neurite on the first can be rationalised in terms of thefeedback circuit: the region of membrane furthest from thefirst neurite will display the lowest concentration of inhibitor.Neurones thus coordinate the developmental timing of bipo-larity through a size-dependent process.
B. Size dependent sequential patterning in mammaliandevelopment – Insights from gastruloids
Unlike in
C. elegans , establishment of anterior-posteriorpolarity in the epiblast of mammalian embryos occurs well af-ter the first cell division, an axis that lays the ground plan forthe commitment of germ-layers during gastrulation. In mice,AP symmetry breaking has long thought to be coordinated bythe positioning of extra-embryonic cues to the posterior andhence specifying the future primitive streak [70]. This viewof sequential polarity hand-off has been thrown into questionin recent years by several in vitro systems, suggesting thatepiblast has the capacity to break symmetry spontaneouslyin the absence of extra-embryonic cues [71–75]. While theprecise genetic constituents of this symmetry breaking areunder contention [76, 77], several that argue some form ofreaction-diffusion system is at play, citing for example theco-expression of morphogens with their extra-cellular antago-nists, e.g. Wnt and its antagonist Dkk [78, 79].Consistent with this hypothesis, in vitro systems displaysize-dependence in symmetry breaking capacity and pattern-ing modality [72, 75]. This is seen in gastruloids, small aggre-gates of embryonic stem cells (ESCs) that can spontaneouslybreak symmetry [72], axially elongating and displaying po-larised expression of primitive streak marker T/Brachyury.In refining their protocol for generating gastruloids, van denBrink et al. [72] found that seeding microwells with dif-ferent numbers of ESCs yielded qualitatively different phe-nomenology (Fig. 4b): critically small ( ≤ cells) aggre-gates could not break symmetry; aggregates of intermediatesize ( ∼ − cells) could subsume a unipolar state;aggregates of double that size ( ∼ cells) displayed twooppositely positioned poles; and critically large aggregates( > cells) generated many poles. These results are con-sistent with a Turing-like system controlling polarity, wherebycritically small domains cannot sustain an instability, whereasincreasingly large domains can maintain progressively more’peaks. This is furthered by the observation that bipolar gas-truloids polarise sequentially, with the second pole seemingly Increasing seed numberSubcriticalsize Monopolar Bipolar Multipolar
Wnt Dkk
Monopolar Bipolar
MP MEShh Fgf10
Growth G r o w t h A SB I abcd FIG. 4. Sequential pattern formation in developmental systems. (a)In in vitro cultured neurones, polarity arises sequentially. A secondpolarity axis is formed after cell growth, and its orientation is “mir-rored” off the first. A putative symmetry breaking circuit is presentedadjacently [69], considering a membrane-protein (MP) activator cou-pled to modulators of endocytosis (ME), representing the effects ofsmall GTPases. (b) Gastruloids polarise and elongate only wheninitialised with a critical number of cells [72]. For seed numbersbeyond this initial bifurcation value, gastruloids can self-organisemore axes. T/Brachyury expression is localised to the protrusion inmonopolarised gastruloids, and is speculated to also be localised tofurther protrusions in multipolar variants. A potential feedback cir-cuit is drawn adjacently, which remains to be investigated. (c) Anactivator-substrate model for lung branching [25], based on autocat-alytic production of the signalling molecule Shh (activator) at thelung bud tip, via consumption of the substrate molecule Fgf10. (d)A dot-stripe mechanism is proposed to pattern the joints of develop-ing digits: a Turing-like dot-forming system specifies the positionsof bones and orients through repression a Turing-like stripe-formingsystem to specify joints. Modelled on a growing domain, sequentialjoint specification emerges, with joints forming near the developingtip. A coupled-Turing scheme is described adjacent, considering adot-forming substrate-depletion module (A,S) coupled to a stripe-forming activator-inhibitor module (B,I). emerging after growth, protruding from the opposite edge ofthe structure.
C. Sequential patterning of phalanges in developing digits iscoordinated by coupling patterning to growth
An analogous mechanism may explain the sequential spec-ification of joints in developing digits of tetrapod limbs. Digitpatterning and growth are concomitant, with joints being laiddown sequentially as progenitor cells are added to the distaltip. Guided by gene expression patterns, as well as mutantphenotypes, joint patterning has been proposed to be gov-erned by a coupled Turing system [80] (Fig. 4d). An activator-substrate system specifies the positions of bones (phalanges)by prescribing a series of ’dots’ of gene-expression; which re-presses a second activator-inhibitor system to specify jointsas ’stripes’ of gene expression at alternate positions. Simula-tions on a static domain recapitulate both wild-type and mu-tant expression patterns, but patterning occurs simultaneouslyacross the entire digit. However, simulated on a growing do-main, adding new cells distally, leads to a shift in dynamics infavour of sequential patterning. Hence here too, the coordina-tion of developmental timing and growth may be an emergentproperty of patterning by collective decision-making.
VI. REGULATING PATTERN SIZE AND LIFETIME INGROWING SYSTEMS
Pattern forming systems that align with activator-substrateor activator-inhibitor motifs are able to undergo sequentialtransitions in pattern concomitant with domain growth (Fig. 3-4). As domain growth continues, patterns undergo further bi-furcations to establish periodicity. While these motifs allowirreversible transitions in pattern, it may be desirable for sys-tems to sense intermediate sizes, and for these transitions to bereversible. One can conceive of a timer-like set-up in a bipha-sic scheme: in the assembly stage, symmetry is broken at acritical size; and in the proceeding dissolution stage, patternsare lost at some larger size. To investigate the emergence oftimer-like behavior we couple an activator-substrate system toa growing domain (Fig. 5a).Specifically, we consider isotropic growth of the domain[81] (all parts of the domain grow in a similar fashion) and thesubunit pool density P grows homogeneously with a constantrate. Due to domain growth, both the system size L ( t ) and thetotal amount of building block pool, N ( t ) = (cid:82) ( S + P ) dx ,are time-dependent. The growth of the system introduces lo-cal flow and dilution of both S and P [81]. The coupled dy-namics of S and P are given by (Fig. 5a): ∂ t S + ˙ rr ( x∂ x S + S ) = D s ∂ x S + k on P f ( S ) − k off S , (4) ∂ t P + ˙ rr ( x∂ x P + P ) = D p ∂ x P − k on P f ( S ) + k off S + G , (5)where G is the growth rate of the subunit pool and the do-main growth function r ( t ) is defined as: L ( t ) = L (0) r ( t ) ,where L (0) is the initial system size. We write the assemblyrate function as f ( S ) = κ + S n / ( S n + S n ) , where κ de-fines the size-independent rate of assembly of S . Motivated byexponentially growing cells and tissues, we specifically con-sider the case of exponential growth in system size, such that r ( t ) = e αt with α the growth rate. Since the macromolecularcomposition of cells scales with the cell size, we assume thatthe total amount of building blocks, (cid:82) P dx , grows at a rateproportional to system size L , resulting in a constant rate ofgrowth G . While the formation of patterns occurs beyond acritical system size L > L ∗ , the stability of the pattern de-pends on the interplay between the rates of growth-induceddilution, synthesis of the subunit pool P , and autocatalysis of S . Case 1: Transient polarity pattern due to growth-induceddilution
When the subunit pool grows at a rate much slower thanthe rate of system growth ˜ G (cid:28) α (where ˜ G = GL (0) ), thestructure is formed transiently and dissolves after a criticaltime T c . The initial slow growth of the system size allows theformation of pattern beyond a critical size L ∗ . However, as dL/dt increases rapidly (due to exponential growth) and be-comes much faster than the rate of pool synthesis, the subunitdensity starts decreasing. Below a critical density of subunits,the pattern dissolves and the system reaches a homogeneousstate (Fig. 5b, left). Transient structure formation has beenobserved in slime mold [82] and during mammalian develop-ment [83], and modelled using stochastic RD systems [84].Here we argue that system growth can also induce such tran-sient polarity formation.The patterned state makes a transition to a homogeneousstate at a time T c when the system size reaches L c . The crit-ical time for the transition to the homogeneous state, T c , isdetermined by the parameters of the feedback motif ( k on /k off , κ ) and the growth rates α and ˜ G (Fig. 5c). The lifetime ofthe pattern, T c , and the system size at transition to the homo-geneous state, L c , can be tuned independently of each otherby modulating κ , α and ˜ G (Fig. 5d). Controlling the lifetimeof polarity patterns is essential for regulating developmentaltransitions, and further experimental studies are essential touncover such control mechanisms. Case 2: Pattern scaling due to proportional growth of systemsize and the subunit pool
When the subunit pool grows at a rate comparable to therate of growth of system size, ˜ G ∼ α , the subunits can reach ahomeostatic density in time. As a result, the patterns formedduring growth do not dissolve. Strong autocatalytic growthprevents delocalization of the early pattern and prevents thepossibility of period doubling as seen in Schnakenberg ki-netics and Gierer and Meinhardt model [81]. This leads toa dynamic pattern scaling behaviour where the size of thepattern scales with the size of the system (Fig. 5b, middle).This mechanism of scaling is notably different from the mor-phogen gradient scaling [85]. Here, pattern scaling is a conse-quence of system growth where the polarity pattern that does t i m e x S (x,t) S (x,t) x x t i m e H i ghLo w L L L L ab Pattern SplittingPattern Scaling symmetric division c asymmetric division Transient Pattern x t i m e Pool S Growth dilutionsynthesis
A B CL C L>L C T C T B C increased k on /k off homogeneoussplittingscaling transient c de k on / k off S (x,t) G: pool growth rate α: domain growth rate ˜ G ~ α ˜ G α ˜ G α ˜ G / α FIG. 5. Pattern scaling, splitting and transient pattern formation ingrowing domains. (a) Feedback motif for an activator-substrate sys-tem coupled to a growing domain. (b) (Left) When subunits areproduced 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 tosystem size growth, the pattern formed grows in proportion to sys-tem size, exhibiting a dynamic scaling behaviour. This is differentfrom sequential pattern formation as the polarity is preserved duringgrowth. (Right) In the case of strong autocatalysis of S , the pat-tern spontaneously splits. (c) Phase diagram for pattern formationas functions of pool growth rate relative to the system, ˜ G/α , and k on /k off . Colormap denotes the inverse of the pattern lifetime, /T c .(d) Time evolution of structure size S tot = (cid:82) L Sdx (blue) and systemsize L (red) for the case of transient pattern formation. The lifetimeof the pattern T c , is coupled to the system size at transition to the ho-mogeneous state, L c . They can be tuned independently of each other,for example, by changing growth rate α (case B) where only the tran-sition 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 mecha-nism for symmetric and asymmetric cell division (in terms of polarityprotein content). When the pattern is transient (left) the dissolutionof the structure will make the daughter cells symmetric in fate, con-taining the same amount of polarity proteins. If the pattern persists(right) then the division will lead to asymmetric fate inheritance. Pa-rameter values: D P = 1 , D S = 0 . , κ = 0 . , S = 10 , α = 0 . , L (0) = 1 , and total pool density (cid:82) ( P + S ) dx/L = 2 ,with G and k on /k off variable. not change qualitatively with system size. Case 3: Pattern splitting When the rate of autocatalysis is sufficiently high, κ (cid:54) = 0 ,and the subunit pool and system size grow at similar rates, ˜ G (cid:38) α , dynamic pattern splitting can emerge in the context ofgrowth (Fig. 5b, right). Here, slower positive feedback weak-ens the long-range inhibition arising as the consequence ofpool depletion, allowing new peaks to emerge, in contrast tocase 2. Adaptive benefits of pattern splitting may be multiple,for example allowing for the emergence of sequential pattern-ing, and in maintaining relative stasis in local patterns upondomain extension. VII. USING GROWTH AS A TIMER: TRANSIENTSYMMETRY BREAKING AT INTERMEDIATE SIZE Analysis of our minimal model shows that a timer-like con-trol of pattern is an emergent property of symmetry-breakingsystems that couple pool size to system volume. We speculatethat transient symmetry breaking may serve as a control strat-egy to mediate shifts between symmetric and asymmetric celldivision in stem cell homeostasis. In particular, the biphasicnature of transient symmetry breaking scheme can help ra-tionalise the equal distribution of determinants in the face ofunequal nature of cell division. Suppose this system under-lies the establishment of cell polarity required for asymmetricdivision. A critical cell size would allow polarity to be estab-lished, preventing precocious cell division. After the cell-sizetimer has elapsed, and the cell enters the dissolution phase, po-larity proteins will return to the fast-mixing pool (Fig. 5e, left).If cell polarity can have some effect on differential daughtercell fate, such a scheme would dissolve any bias prior to divi-sion.The significance of a mechanism like this can be under-stood in cases where cell lineages undergo switches betweensymmetric and asymmetric cell division. Cases 1 and 2 of ourmodel are identical besides the relative rates of pool synthesis,system size growth and autocatalysis. Hence tuning the cou-pling between pool production rate and system size can regu-late transitions between reversible polarisation (case 1) andirreversible polarisation (case 2), wherein polarity is main-tained through a cell division event, maintaining a bias in de-terminants and seeding differential daughter-cell fate (Fig. 5e,right). In situations where such switches between symmetricand asymmetric division are dynamic, regulating the extentof growth-induced dilution to effect switches between tran-sient versus irreversible polarisation may be an optimal con-trol strategy.Given the switch between reversible and irreversible sym-metry breaking in our minimal model is governed by a singleparameter change, it is tempting to speculate that such a mech-anism may be responsible for the mixed modes of stem cellproliferation, which in many systems are seemingly stochas-tic. Under such a scenario, the switch between these divi- sion modes may be noisy at the level of individual cell de-cisions, given the requirement of being poised near the tran-sition point. However, given fate decisions and patterns ofcell division are known to be influenced by signals emanat-ing from stem cells or their progeny in many systems, sucha model would confer plasticity and robustness in stem cellhomeostasis at the level of the population. We stress that thismechanism remains a theoretical prediction and are intriguedas to whether such a control strategy is indeed utilised in na-ture. VIII. OVERCOMING SIZE CONSTRAINTS: SCALINGPATTERNS IN GROWING SYSTEMS Not all biological systems that display symmetry breakingalso show size-dependent pattern formation. Indeed, it maybe adaptive for systems to canalise their patterning mode ir-respective of size. This is a feature of case 2 of our mini-mal model (Fig. 5b, middle), which features commensurategrowth of pool size and system size, leading to scale invari-ance upon domain growth: if the system breaks symmetryto form a single structure at a smaller size, upon isotropicgrowth, the system maintains a single structure which growsin proportion to the domain as a whole. This motif utilisesthe symmetry breaking capacity of reaction-diffusion systemsbut subverts the feature of intrinsic wavelengths characteristicof traditional Turing circuits. In this section, we delineate twopotential modes of scale-invariant symmetry breaking systemsone with history dependence, and one without and argue thatsuch systems display adaptive features in certain contexts. A. Autocatalysis as a mechanism to preserve patterns in theface of growth Patterning in developing systems is almost invariably pro-ceeded by growth, which is often proportional to the initialpattern. Traditional morphogen gradient hypotheses [86] haveimplicitly assumed that the tissue is initially patterned whenit is small and subsequently undergoes growth, facilitatingproportional extension of the pattern. This two-phase modelof patterning, where cell fates are assigned during an initialpatterning phase, face the challenge of noise: while growthcan lock in the lower positional error entailed by patterningin small fields of cells, this error cannot be reduced throughgrowth. Accordingly, small errors in boundary position can beamplified upon growth, demanding the read-out of positionalinformation at early stages is exquisitely tuned. If howeverfates are assigned in a self-organised manner, as in symmetry-breaking the systems we overview in our minimal model, thishard limit on noise in boundary positioning can be surpassed.Provided the symmetry breaking system scales with domainsize, absolute noise in boundary position if anything reduceswith growth; self-organised systems such as these continuallyrefine boundaries throughout growth, rather than amplifyingnoise in initial specification.Case 2 of our minimal model allows for pattern scaling viaproportional growth of pool and domain size, and autocataly-sis, which in effect instills history-dependence in pattern for-mation, thus helping to preserve proportions. Hence the pat-tern generated at small domain sizes is preserved upon elon-gation. Given the diffusion length scale shortens with respectto relative domain size upon growth, boundaries sharpen overtime. In the context of a developing field of cells, this au-tocatalysis could represent positive feedback in master tran-scription factors or indeed epigenetic changes, which allowcells and their progeny to remember past states. Hence sucha model may help provide alternative mechanisms for scalingof patterns with growth, whereby initial stages establish thecrude pattern (e.g. number and position of structures), whichis in turn refined over time. The hallmark of actively scalingprocesses such as these is the reduction of noise in boundarypositioning, i.e. violating the data-processing inequality [87]. B. Expander-coupled systems can scale patterns to domainsize irrespective of history While symmetry-breaking schemes incorporating auto-catalysis show benefits of maintaining patterns with growth,certain systems may require patterning to be scale-invariantwithout the requirement of time-dependence. This is exem-plified in regenerating systems, which are able to regrow or-gans or entire organisms in the correct proportions, in spiteof drastically different starting sizes. Recent theoretical workhas advanced understandings of how scale-invariant symme-try breaking could operate. Werner et al. [88] proposed that athird component is required, analogous to expanders in mor-phogen gradient scaling, which dynamically modulates pat-terning wavelength as a function of system size by tuning lev-els of pattern forming molecules. This model demonstratedtime-independent scaling across several orders of magnitudedifferences in domain size. While the model is based on a tra-ditional activator-inhibitor model, the scheme is generalisableto other modes of expander-mediated modulation and othersymmetry breaking motifs such as substrate depletion. IX. DISCUSSION In this Perspective, we presented a minimal model for sym-metry breaking to serve as a unifying framework to understandpattern formation in the context of timing and growth. We ar-gue that systems that display positive feedback in activator re-cruitment, drawn from a limiting pool, can yield spontaneoussymmetry breaking. This basic scheme is mathematicallyakin to other RD mechanisms including activator-inhibitor orsubstrate-depletion motifs, all relying on a common logic oflocal activation and long-range inhibition. Thus the insightsgleaned from the phenomenological behaviour of this systemis applicable to diverse systems.Across the cases of the minimal model we consider, we ob-serve a hard size limit on pattern formation: below a criticalsize, diffusive dispersion overwhelms the capacity to breaksymmetry. Given developing systems across scales typically display patterning and growth occurring in unison, if this crit-ical size is within biologically meaningful length scales, suchbehaviour can elicit qualitative changes in patterning: growthabove a critical size leads to a bifurcation, whereby the systemtransitions from unpolarised to polarised. Viewing growth asa control parameter of the system that increases system size ata predictable rate, developmental systems can utilise this bi-furcation to enact developmental transitions at the right placeand time. As a generic by-product of symmetry breaking sys-tems, we predict that this time-keeping mechanism may bemore abundant than anticipated. We note that this feature isthe most generic among RD models of pattern formation, andamong the different cases of our minimal model: the diffusionlength-scale sets a physical limit on pattern formation.Beyond this first bifurcation, our mass-conserved RDmodel predicts different dynamic behaviors depending on theregulatory motifs. These include sequential pattern formation,transient pattern formation, pattern scaling, and pattern split-ting in growing systems. In line with the well-established lit-erature on domain size in Turing patterns [23], our model pre-dicts clock-like sequential pattern formation (Fig. 3-4): as thesystem grows larger, given patterning wavelength is intrinsicto the system, the domain can accommodate multiple struc-tures. An important dynamical consequence of this is tempo-ral ordering : growth elicits consecutive bifurcations, resultingin sequential patterning, shown to be instrumental in neuronalcell (bi)polarity [69], and joint patterning in digits [80]. Alter-natively, growth-induced pool dilution can drive systems backtowards an unpolarised state, allowing for transient patternformation at intermediate size (Fig. 5). Thus an alteration ingrowth regulation can yield timer-like dynamics, which wehypothesise may be important in orchestrating switches be-tween asymmetric and symmetric stem cell division modes.A qualitatively different behaviour upon continued growth is scale-invariance , whereby the proportions of the pattern aremaintained upon domain elongation. Scale-invariant systemsshow switch-like dynamics, becoming time-independent af-ter the first bifurcation. We argue that such behaviour couldallow patterned tissues to maintain proportions upon prolif-eration, where self-organisation continually refines boundaryposition instead of stretching noise in initial specification.Our reaction-diffusion framework for understanding devel-opmental time in terms of size-dependent symmetry break-ing is generalisable beyond the systems that couple increasesin size to developmental transitions via biochemical circuits.Firstly, decreases in system size can also be utilised by de-velopmental systems to temporal transitions. For example,the transition from asymmetric to symmetric division in theP-lineage of C. elegans can be understood in terms of se-quential reductions in cell volume pushing the system overthe critical cell size threshold for polarisation. Secondly, theorganising principle of Turing-like pattern formation local-activation and long-range inhibition extends beyond systemsbased solely on chemical cross-talk [38]: pattern formationcan emerge from cell-cell interactions or mechanical insta-bilities [89–91]. While we restricted our focus in this paperto biochemical systems, future work should attempt to unifythese results with mechanically driven size-dependent sym-0metry breaking. Indeed, we may see strong parallels in hownature utilises chemical or mechanical instabilities to regulatethe timing of developmental transitions. We hope that our pro-posed strategies for time-keeping in natural living systems canalso provide inspirations for engineering of synthetic circuitswith tunable dynamics. ACKNOWLEDGMENTS The authors thank Andrew Goryachev for many usefulcomments. SB acknowledges funding from Royal Soci-ety University Research Fellowship URF/R1/180187, andHuman Frontiers Science Program (HFSP) grant numberRGY0073/2018. 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