Mechanistic models of cell-fate transitions from single-cell data
MMechanistic models of cell-fate transitions from single-cell data
Gabriel Torregrosa ∗ and Jordi Garcia-Ojalvo † Department of Experimental and Health Sciences, Universitat Pompeu Fabra,Barcelona Biomedical Research Park, Dr. Aiguader 88, 08003 Barcelona, Spain
Our knowledge of how individual cells self-organize to form complex multicellular systems is being revolu-tionized by a data outburst, coming from high-throughput experimental breakthroughs such as single-cell RNAsequencing and spatially resolved single-molecule FISH. This information is starting to be leveraged by machinelearning approaches that are helping us establish a census and timeline of cell types in developing organisms,shedding light on how biochemistry regulates cell-fate decisions. In parallel, imaging tools such as light-sheetmicroscopy are revealing how cells self-assemble in space and time as the organism forms, thereby elucidatingthe role of cell mechanics in development. Here we argue that mathematical modeling can bring together thesetwo perspectives, by enabling us to test hypotheses about specific mechanisms, which can be further validatedexperimentally. We review the recent literature on this subject, focusing on representative examples that usemodeling to better understand how single-cell behavior shapes multicellular organisms.
I. INTRODUCTION
Single-cell technologies are producing large amounts ofdata about multicellular development, which have createdan urge to develop tools to extract information from thosedatasets. In that respect, machine learning has proven to bevery useful in many applications, where it has exhibited sub-stantial predictive power. This raises the question of whybother trying to make mechanistic models. However, as ithas already been debated [1, 2], mechanistic and machine-learning approaches are not mutually exclusive. The power ofmachine learning is the ability to find patterns in vast amountsof data in an inductive way, while mechanistic approacheswork at the level of human intuition, making their strengthdeductive. Both perspectives can complement each other tomake better theories and predictions [3]. This mutual inter-action has been present for decades: the first (mechanistic)neuron model [4] served as inspiration for the generation ofthe network approach that has culminated in the developmentof deep neural networks. There is renewed interest in this syn-ergistic point of view, as evidenced by the efforts towards thedevelopment of tools ranging from the selection of simplerand more informative models [5] to the analysis of complexsystems using machine-learning strategies [6], and includingthe proposal of benchmarks for modeling and data analysis[7, 8].
II. EXPERIMENTAL SINGLE-CELL BREAKTHROUGHS
Two main breakthroughs are providing an unprecedentedpush to study the impact of fate decisions at the single-celllevel on the development of multicellular organisms. The firstone is the fast development of high-throughput techniques tomeasure gene expression at single-cell resolution, which areleading to new insights into the cell-fate transitions underly-ing multicellular development [9, 10]. The data is increasing ∗ [email protected] † [email protected] in detail, as shown by the addition of key information suchas spatial resolution [11–13]. While these techniques are notperfect and many challenges are still open [14], the genera-tion of richer data with continuously improving quality of-fers promising avenues of research in developmental biology.The second breakthrough is the establishment of gastruloidsas novel experimental models of development [15]. These de-velopmental models have shown the ability to reproduce withunprecedented detail the main features of embryo formation,from 3D organization [16] to somitogenesis [17]. The benefitsof these systems are their increased experimental controllabil-ity, as well as the scalability of the number of samples that canbe analyzed under different perturbations. III. BUILDING MECHANISTIC MODELS AT THESINGLE-CELL LEVEL
In spite of the increase in the amount of data available,mechanistic models suffer from what has been called the“curse of parameter space” [18], namely a lack of knowl-edge of the in vivo parameter values of most biochemical andmechanical processes within live cells. This problem can becircumvented by (i) using dynamical behavior as a constraintof the models [19], (ii) applying the tools of dynamical sys-tems (in particular bifurcation theory) to explore systemati-cally how the different biological processes impact qualita-tively the behavior of the system in both model and exper-iments [20], and (iii) using minimal models to describe thespatiotemporal self-organization of tissues [21, 22]. In whatfollows, we describe several case examples illustrating the useof those methods, in particular of minimal models that cancapture the fundamental principles of the cell-fate transitionsthat underlie the formation of organisms.
A. Geometric description of cell-fate dynamics
The simplest paradigm of organism development comesfrom the epigenetic landscape picture introduced by Wadding-ton, in which cells start from a pluripotent state and roll down a r X i v : . [ q - b i o . CB ] F e b through developmental valleys towards one of several alter-native committed states (Fig. 1) [23]. This qualitative de-scription is amenable to be interpreted with the formalism ofdynamical systems [24, 25]. Starting from this conceptual“landscape model”, Corson and Siggia used the geometricalperspective of dynamical systems to explain the characteris-tic fate patterning of vulval development in C. elegans [26].Fate commitment in these cells involves both positional EGFcues from an anchor cell and lateral cell-cell communicationamong the vulval precursor cells via Notch (Fig. 2A). To un-
FIG. 1. A cell undergoing a stochastic trajectory down Waddington’slandscape. derstand how the system integrates the two types of signals,the authors ignored all biochemical details of the system, fo-cusing on the fate patterning across a range of conditions.With this information, they created the simplest model thatcaptures the shape of the landscape over which the cells evolvetowards the committed fates (Fig. 2B). Using the effective de-grees of freedom of the system to direct behaviors of the dataallowed the authors to explore systematically the properties ofthe model, and study in particular the epistatic interactions be-tween the exogenous EGF cue and the Notch-based cell-cellcommunication (Fig. 2C).Despite the reduced dimensionality of the model, whichonly consists of eight cells with two effective degrees of free-dom each, we can extract several lessons. First, the modelexploits the qualitative description of dynamical systems togain insights at multiple levels of complexity, suggesting inthe process alternative interpretations to our current under-standing and proposing insightful experiments to test the cur-rent state of knowledge. The second lesson is the ability ofthis approach to discern among one of several theories, usu-ally considered mutually exclusive, when interpreting experi-mental observations. In this case, the model captures the jointeffect that extrinsic cues and mutual cell-cell communicationhave on the system, conciliating an apparent contradiction be-tween the two mechanisms. The simultaneous dependence ondiverse factors is a natural assumption given the complexityof biology. Other studies have reached similar conclusionsfor phenomena ranging from the refinement of cell-fate pat-terning [27] to robustness inactivating mechanisms [28, 29].Being able to select or integrate multiple models points at thebenefits that single-cell resolution provides.
ABC
FIG. 2. Geometric approach to cell-fate transitions. A: The vulvalprecursor cells (VPCs) in
C. elegans take fates based on the posi-tional EGF cue from the anchor cell (AC, red arrows). Addition-ally, VPCs communicate with each other through the Notch pathway(green arrows). B: Both the EGF cue and the cell-cell communi-cation modulate the landscape felt by each cell according to theirposition (left to right) and across time (top to bottom), reaching anattractor by the time the process finishes (at time t = B. A single-cell dynamical landscape
From a cell-centric perspective, the genome contains allthe necessary information describing the cell-fate landscape.However, cells are not isolated systems. They feel externalstimuli and gradients, and are sensitive to temperature and tomechanical interactions, both with the environment and be-tween them. All these factors impact the individual cells, cre-ating complex landscapes. In principle, we could imagine theglobal landscape of the full dynamical system containing allits degrees of freedom. However, we do not use this picture forseveral reasons. First, time-dependent extrinsic factors willkeep the landscape dynamic. Besides, such an image com-pletely obscures the elegant geometric understanding that wehave of dynamical systems, and hence prevents us from ex-tracting any possible intuition from it. Finally, and more prac-tically, individual cells inside an organism are amenable tobe interpreted as identical cell blocks, conferring the complexcollective space with a valuable assumption of symmetry.Considering each cell’s dynamical system as a buildingblock of the population, we find two common approachesto study cell-fate decisions at the collective level: continu-ous models (CM) and agent-based models (ABM). Continu-ous models exist since the inception of the mathematical studyof morphogenesis with the pioneering work of Turing [30]. Insuch an approach, we take the limit of the cells to be a contin-uum. This modeling approach is very amenable to analyticalwork, and was for years the only approach used. However,complexities arise when introducing cell division and inho-mogeneities in the system. As an alternative, with the rise ofcomputational power, it has become possible to simulate cellsas individual entities following simple rules. This approachhas the benefit that it naturally accounts for the individualityof cells, describing complex intrinsic properties flexibly. Themain drawback of this simulation framework is that it is hardto reach an analytical understanding that goes beyond specificparameter values. Careful considerations are required to avoidover-complications of the model and extract reliable conclu-sions from their use. The two techniques are not exclusive andcan be complementary. For example, Manukyan et al [31] usea CM to explain the patterning of reptile scales, while provingthat the approach maps to an ABM depending on the lengthscale at play. The use of one method or the other will dependon the modeling purpose, with the question that we want toassess fitting naturally one or the other.As an illustrative example of this combined approach, Saizet al [32] investigated the robustness of cell fate proportionsduring the early development of the mouse embryo. The cur-rent knowledge of this process points to a complex interactionof different interconnected pathways combined with intercel-lular communication (Figure 3A). Despite the complexity ofthe genetic network, the dimensionality of the biochemicalmodel can be reduced substantially through adiabatic consid-erations, leading to a single degree of freedom per cell. Thissimple biochemical model is enough to explain the balanceddistribution of fates between epiblast and primitive endodermin terms of only cell-cell communication via FGF/ERK sig-naling, which leads to an effect akin to lateral inhibition inneighboring cells without the need of intracellular mutual in-hibition between the master regulators of the two cell fates(Figure 3B). To include the effects of cell mechanics and pro-liferation, the authors used an ABM where the cells obey theminimal biochemical model described above (Figure 3C). Themodel reproduced the experimentally observed robustness toperturbations in the initial embryo size, using both scaling ex-periments in which half-sized and double-sized embryos were A NANOGGATA6 FGF4
FGFRs x a x b B ERK time E % o f I C M F Embryo (by final number of ESCs, after 48h culture)
1x 2x 4x0.5x 8xFinal number of ESC (as xEPI)
Identity
PrE DP EPI ESC
N=135 % o f I C M H In silico embryo (by final number of ESCs, after 48h culture)
Identity
PrE DP EPI ESC
A BCDE experimentsmodel
FIG. 3. Using a minimal model to study the robustness of collectivecell-fate transitions. A: Biochemical network describing the inter-action between the master regulators of the epiblast and primitiveendoderm (Nanog and Gata6, red and blue, respectively) in the earlymouse embryo. The two transcription factors mutually inhibit eachother via FGF/ERK signaling. B: Phase plane analysis of a minimalone-dimensional version of the network depicted in A. The black cir-cles depict the two fates into which all cells alternatively differentiateas the embryo develops. C: Filmstrip showing the results of an agent-based model incorporating the minimal biochemical network shownin A,B. D,E: Cell-fate distribution at the late blastocyst stage whenembryonic stem cells (light green) are added in different stages ofthe blastocyst development, in both experiments (D) and model (E).Adapted from [32]. considered, and perturbation experiments in which the rela-tive sizes of the two fates were varied, by either addition ofembryonic stem cells (Figures 3D,E) or targeted removal ofcells of one type or the other at different times. The agree-ment between the model and the experiments sheds light intothe collective nature of cell-fate decisions in the early embryo.
C. The heterogeneity of life: towards an non-equilibriumstatistical mechanics of cell-fate decisions
As we have discussed in the preceding section, the cell-fate transition landscape changes between cells under the ef-fect of external signals and cell-cell interactions. But thereis an additional detail left: the stochastic nature of biology.Noise is ubiquitous in cells, from the sensing of external sig-nals to gene expression [33, 34]. The effects of noise at thesingle-cell level have been studied for more than two decades[33, 35–38]. A standard way to approach this issue so farhas been to include noise sources as part of the modeling ap-proach, usually working on small gene and protein circuitsdescribed by low-dimensional stochastic dynamical systems.However, a fundamental understanding of the large-scale in-terplay between the noise and the behavior of cell-fate de-cisions is still missing. The unique character of single-cellexperiments should help us reach deeper insights into thisproblem. In particular, the recent deluge of data is challeng-ing established concepts such as that of cell identity [39, 40],and provides unprecedented insight into the dynamics of cell-fate decisions in commitment [37, 41, 42] and reprogramming[42, 43].Previous proposals highlight the potential benefit of usingthe formalism of non-equilibrium statistical mechanics to ac-count for this inherent stochasticity [44, 45]. This route cannevertheless be challenging. A step in this direction wastaken by Stumpf et al [46], who used single-cell data anda combination of machine learning and mechanistic model-ing to examine the in vitro differentiation of stem cells asthey evolve towards the neuroectoderm fate (Figure 4A). Celltrajectories towards the committed state, as inferred fromsingle-cell transcriptome analysis, are highly stochastic (Fig-ure 4B). The single-cell data also allowed the authors to in-fer a regulatory network whose components change along thedevelopmental path, activating and deactivating three differ-ent regulatory modules (Figure 4C, see labels 1-2-3 in theplot). According to this analysis, cells evolve through threemacrostates, which contain several microstate transitions. Acareful comparison between different alternative models re-veals that the transitions are not consistent with Markoviandynamics (Figure 4D), but they rather correspond to a non-Markovian stochastic process (Figure 4E). This combinationof stochastic modeling and statistical data analysis shows away forward in our quest to understanding cell-fate transitionsin development using high-throughput single-cell experimen-tal methods.
IV. FINAL REMARKS
Single-cell methods, particularly those that offer spatialresolution [11–13], will be of great relevance to disentanglemany open questions on the dynamics of the cell-fate deci-sion landscapes. However, we need to be mindful of the mosteffective way in which we can leverage those techniques toadvance our understanding of the fundamental mechanismsbehind multicellular self-organization. Looking for the most F E A B
ABD CE
FIG. 4. Combining stochastic modeling and statistical data analy-sis to analyze cell-fate transitions. A: Experimental scheme for the in vitro differentiation of stem cells to the neuroectoderm fate. B:Single-cell transcriptomics data projected in the plane of the first twoprincipal components. The color code from yellow to blue repre-sents increasing time. C: Gene regulatory network inferred from thesingle-cell transcriptomics data. D,E: Fitting the data to a stochasticfirst-order model without memory (D) and to a hidden Markov model(E). Adapted from [46]. minimalistic approach to explain current knowledge is at theheart of any modeling approach. However, although simplic-ity is readily appreciated [26, 27, 32], the continuous increaseof detailed knowledge and the rising amount of data generatedat single-cell resolution makes it very tempting to constructover-complicated models, which might not be the best wayforward.The reasons for keeping things simple are twofold. First,over-detailed attitudes may obscure our understanding andcurtail our ability to both search for trustful explanatory mech-anisms and open up further avenues of research. Second, de-spite the amount of data that single-cell techniques providefor us, we should always assess their explanatory capacity;overwhelmingly complex models can be impossible to vali-date even with extensive experimental efforts. Many studieshave focused on the assessment of both issues, both lookingsystematically for simpler models compatible with the data[5, 7, 47] and assessing the limits of the information that wecan obtain from those models and from the available data[48]. These are the approaches that will generate understand-ing from the current revolution in data gathering.
ACKNOWLEDGMENTS
This work was supported by the Spanish Ministry of Sci-ence, Innovation and Universities and FEDER (under projectsFIS2017-92551-EXP and PGC2018-101251-B-I00, and bythe “Maria de Maeztu” Programme for Units of Excellence in R&D, grant CEX2018-000792-M), and by the General-itat de Catalunya (ICREA Academia programme and grant2017 SGR 1054). G.T. is funded by PhD grant FPU18/05091from the Spanish Ministry of Science, Innovation and Univer-sities. [1] Huang, S.. The tension between big data and theory in the"omics" era of biomedical research. Perspectives in biology andmedicine 2018;61(4):472–488. doi:10.1353/pbm.2018.0058.[2] del Sol, A., Jung, S.. The importance of computationalmodeling in stem cell research. Trends in Biotechnology2021;39(2):126–136. doi:10.1016/j.tibtech.2020.07.006.[3] Baker, R.E., Peña, J.M., Jayamohan, J., Jérusalem, A..Mechanistic models versus machine learning, a fight worthfighting for the biological community? Biology Letters2018;14(5):20170660. doi:10.1098/rsbl.2017.0660.[4] McCulloch, W.S., Pitts, W.. A logical calculus of the ideasimmanent in nervous activity. The bulletin of mathematical bio-physics 1943;5(4):115–133. doi:10.1007/BF02478259.[5] Proulx-Giraldeau, F., Rademaker, T.J., François, P.. Untan-gling the hairball: Fitness-based asymptotic reduction of bio-logical networks. Biophysical Journal 2017;113(8):1893–1906.doi:10.1016/j.bpj.2017.08.036.[6] Wang, S., Fan, K., Luo, N., Cao, Y., Wu, F., Zhang, C.,et al. Massive computational acceleration by using neural net-works to emulate mechanism-based biological models. NatureCommunications 2019;10(1):4354. doi:10.1038/s41467-019-12342-y.[7] Croydon Veleslavov, I.A., Stumpf, M.P.H.. Repeated decisionstumping distils simple rules from single cell data. bioRxiv2020;:2020.09.08.288662doi:10.1101/2020.09.08.288662.[8] Pratapa, A., Jalihal, A.P., Law, J.N., Bharadwaj, A., Murali,T.M.. Benchmarking algorithms for gene regulatory networkinference from single-cell transcriptomic data. Nature Methods2020;17(2):147–154. doi:10.1038/s41592-019-0690-6.[9] Briggs, J.A., Weinreb, C., Wagner, D.E., Megason, S.,Peshkin, L., Kirschner, M.W., et al. The dynamics of gene ex-pression in vertebrate embryogenesis at single-cell resolution.Science 2018;360(6392). doi:10.1126/science.aar5780.[10] Nowotschin, S., Setty, M., Kuo, Y.Y., Liu, V., Garg, V.,Sharma, R., et al. The emergent landscape of the mouse gut en-doderm at single-cell resolution. Nature 2019;569(7756):361–367. doi:10.1038/s41586-019-1127-1.[11] Vickovic, S., Eraslan, G., Salmén, F., Klughammer, J.,Stenbeck, L., Schapiro, D., et al. High-definition spatialtranscriptomics for in situ tissue profiling. Nature Methods2019;16(10):987–990. doi:10.1038/s41592-019-0548-y.[12] Rodriques, S.G., Stickels, R.R., Goeva, A., Martin, C.A.,Murray, E., Vanderburg, C.R., et al. Slide-seq: A scal-able technology for measuring genome-wide expression athigh spatial resolution. Science 2019;363(6434):1463. doi:10.1126/science.aaw1219.[13] Eng, C.H.L., Lawson, M., Zhu, Q., Dries, R., Koulena, N.,Takei, Y., et al. Transcriptome-scale super-resolved imaging intissues by RNA seqFISH+. Nature 2019;568(7751):235–239.doi:10.1038/s41586-019-1049-y.[14] Lähnemann, D., Köster, J., Szczurek, E., McCarthy, D.J.,Hicks, S.C., Robinson, M.D., et al. Eleven grand challengesin single-cell data science. Genome Biology 2020;21(1):31. doi:10.1186/s13059-020-1926-6.[15] van den Brink, S.C., Baillie-Johnson, P., Balayo, T., Had-jantonakis, A.K., Nowotschin, S., Turner, D.A., et al. Sym-metry breaking, germ layer specification and axial organisationin aggregates of mouse embryonic stem cells. Development2014;141(22):4231. doi:10.1242/dev.113001.[16] Beccari, L., Moris, N., Girgin, M., Turner, D.A., Baillie-Johnson, P., Cossy, A.C., et al. Multi-axial self-organizationproperties of mouse embryonic stem cells into gastruloids. Na-ture 2018;562(7726):272–276. doi:10.1038/s41586-018-0578-0.[17] van den Brink, S.C., Alemany, A., van Batenburg, V., Moris,N., Blotenburg, M., Vivié, J., et al. Single-cell and spa-tial transcriptomics reveal somitogenesis in gastruloids. Nature2020;582(7812):405–409. doi:10.1038/s41586-020-2024-3.[18] Tyson, J.J., Novak, B.. A dynamical paradigm for molecularcell biology. Trends in Cell Biology 2020;30(7):504–515. doi:10.1016/j.tcb.2020.04.002.[19] Kirk, P.D.W., Toni, T., Stumpf, M.P.H.. Parameterinference for biochemical systems that undergo a Hopf bi-furcation. Biophysical Journal 2008;95(2):540–549. doi:10.1529/biophysj.107.126086.[20] Espinar, L., Dies, M., Ça˘gatay, T., Süel, G.M., Garcia-Ojalvo, J.. Circuit-level input integration in bacterial generegulation. Proceedings of the National Academy of Sciences2013;110(17):7091. doi:10.1073/pnas.1216091110.[21] Hiscock, T.W., Megason, S.G.. Mathematically guided ap-proaches to distinguish models of periodic patterning. Devel-opment 2015;142(3):409. doi:10.1242/dev.107441.[22] Schweisguth, F., Corson, F.. Self-organization in patternformation. Developmental Cell 2019;49(5):659–677. doi:10.1016/j.devcel.2019.05.019.[23] Minelli, A., Pradeu, T.. Towards a theory of development.Oxford University Press; 2014.[24] Ferrell, J.E.. Bistability, bifurcations, and Waddington’s epi-genetic landscape. Current Biology 2012;22(11):R458–R466.doi:10.1016/j.cub.2012.03.045.[25] Moris, N., Pina, C., Arias, A.M.. Transition states and cell fatedecisions in epigenetic landscapes. Nature Reviews Genetics2016;17(11):693–703. doi:10.1038/nrg.2016.98.[26] Corson, F., Siggia, E.D.. Gene-free methodology for cellfate dynamics during development. eLife 2017;6:e30743. doi:10.7554/eLife.30743.[27] Corson, F., Couturier, L., Rouault, H., Mazouni, K.,Schweisguth, F.. Self-organized notch dynamics generatestereotyped sensory organ patterns in Drosophila. Science2017;356(6337):eaai7407. doi:10.1126/science.aai7407.[28] Gross, P., Kumar, K.V., Goehring, N.W., Bois, J.S., Hoege,C., Jülicher, F., et al. Guiding self-organized pattern formationin cell polarity establishment. Nature Physics 2019;15(3):293–300. doi:10.1038/s41567-018-0358-7.[29] Green, J.B.A., Sharpe, J.. Positional information and reaction-diffusion: two big ideas in developmental biology combine. De-[1] Huang, S.. The tension between big data and theory in the"omics" era of biomedical research. Perspectives in biology andmedicine 2018;61(4):472–488. doi:10.1353/pbm.2018.0058.[2] del Sol, A., Jung, S.. The importance of computationalmodeling in stem cell research. Trends in Biotechnology2021;39(2):126–136. doi:10.1016/j.tibtech.2020.07.006.[3] Baker, R.E., Peña, J.M., Jayamohan, J., Jérusalem, A..Mechanistic models versus machine learning, a fight worthfighting for the biological community? Biology Letters2018;14(5):20170660. doi:10.1098/rsbl.2017.0660.[4] McCulloch, W.S., Pitts, W.. A logical calculus of the ideasimmanent in nervous activity. The bulletin of mathematical bio-physics 1943;5(4):115–133. doi:10.1007/BF02478259.[5] Proulx-Giraldeau, F., Rademaker, T.J., François, P.. Untan-gling the hairball: Fitness-based asymptotic reduction of bio-logical networks. Biophysical Journal 2017;113(8):1893–1906.doi:10.1016/j.bpj.2017.08.036.[6] Wang, S., Fan, K., Luo, N., Cao, Y., Wu, F., Zhang, C.,et al. Massive computational acceleration by using neural net-works to emulate mechanism-based biological models. NatureCommunications 2019;10(1):4354. doi:10.1038/s41467-019-12342-y.[7] Croydon Veleslavov, I.A., Stumpf, M.P.H.. Repeated decisionstumping distils simple rules from single cell data. bioRxiv2020;:2020.09.08.288662doi:10.1101/2020.09.08.288662.[8] Pratapa, A., Jalihal, A.P., Law, J.N., Bharadwaj, A., Murali,T.M.. Benchmarking algorithms for gene regulatory networkinference from single-cell transcriptomic data. Nature Methods2020;17(2):147–154. doi:10.1038/s41592-019-0690-6.[9] Briggs, J.A., Weinreb, C., Wagner, D.E., Megason, S.,Peshkin, L., Kirschner, M.W., et al. The dynamics of gene ex-pression in vertebrate embryogenesis at single-cell resolution.Science 2018;360(6392). doi:10.1126/science.aar5780.[10] Nowotschin, S., Setty, M., Kuo, Y.Y., Liu, V., Garg, V.,Sharma, R., et al. The emergent landscape of the mouse gut en-doderm at single-cell resolution. Nature 2019;569(7756):361–367. doi:10.1038/s41586-019-1127-1.[11] Vickovic, S., Eraslan, G., Salmén, F., Klughammer, J.,Stenbeck, L., Schapiro, D., et al. High-definition spatialtranscriptomics for in situ tissue profiling. Nature Methods2019;16(10):987–990. doi:10.1038/s41592-019-0548-y.[12] Rodriques, S.G., Stickels, R.R., Goeva, A., Martin, C.A.,Murray, E., Vanderburg, C.R., et al. Slide-seq: A scal-able technology for measuring genome-wide expression athigh spatial resolution. Science 2019;363(6434):1463. doi:10.1126/science.aaw1219.[13] Eng, C.H.L., Lawson, M., Zhu, Q., Dries, R., Koulena, N.,Takei, Y., et al. Transcriptome-scale super-resolved imaging intissues by RNA seqFISH+. Nature 2019;568(7751):235–239.doi:10.1038/s41586-019-1049-y.[14] Lähnemann, D., Köster, J., Szczurek, E., McCarthy, D.J.,Hicks, S.C., Robinson, M.D., et al. Eleven grand challengesin single-cell data science. Genome Biology 2020;21(1):31. doi:10.1186/s13059-020-1926-6.[15] van den Brink, S.C., Baillie-Johnson, P., Balayo, T., Had-jantonakis, A.K., Nowotschin, S., Turner, D.A., et al. Sym-metry breaking, germ layer specification and axial organisationin aggregates of mouse embryonic stem cells. Development2014;141(22):4231. doi:10.1242/dev.113001.[16] Beccari, L., Moris, N., Girgin, M., Turner, D.A., Baillie-Johnson, P., Cossy, A.C., et al. Multi-axial self-organizationproperties of mouse embryonic stem cells into gastruloids. Na-ture 2018;562(7726):272–276. doi:10.1038/s41586-018-0578-0.[17] van den Brink, S.C., Alemany, A., van Batenburg, V., Moris,N., Blotenburg, M., Vivié, J., et al. Single-cell and spa-tial transcriptomics reveal somitogenesis in gastruloids. Nature2020;582(7812):405–409. doi:10.1038/s41586-020-2024-3.[18] Tyson, J.J., Novak, B.. A dynamical paradigm for molecularcell biology. Trends in Cell Biology 2020;30(7):504–515. doi:10.1016/j.tcb.2020.04.002.[19] Kirk, P.D.W., Toni, T., Stumpf, M.P.H.. Parameterinference for biochemical systems that undergo a Hopf bi-furcation. Biophysical Journal 2008;95(2):540–549. doi:10.1529/biophysj.107.126086.[20] Espinar, L., Dies, M., Ça˘gatay, T., Süel, G.M., Garcia-Ojalvo, J.. Circuit-level input integration in bacterial generegulation. Proceedings of the National Academy of Sciences2013;110(17):7091. doi:10.1073/pnas.1216091110.[21] Hiscock, T.W., Megason, S.G.. Mathematically guided ap-proaches to distinguish models of periodic patterning. Devel-opment 2015;142(3):409. doi:10.1242/dev.107441.[22] Schweisguth, F., Corson, F.. Self-organization in patternformation. Developmental Cell 2019;49(5):659–677. doi:10.1016/j.devcel.2019.05.019.[23] Minelli, A., Pradeu, T.. Towards a theory of development.Oxford University Press; 2014.[24] Ferrell, J.E.. Bistability, bifurcations, and Waddington’s epi-genetic landscape. Current Biology 2012;22(11):R458–R466.doi:10.1016/j.cub.2012.03.045.[25] Moris, N., Pina, C., Arias, A.M.. Transition states and cell fatedecisions in epigenetic landscapes. Nature Reviews Genetics2016;17(11):693–703. doi:10.1038/nrg.2016.98.[26] Corson, F., Siggia, E.D.. Gene-free methodology for cellfate dynamics during development. eLife 2017;6:e30743. doi:10.7554/eLife.30743.[27] Corson, F., Couturier, L., Rouault, H., Mazouni, K.,Schweisguth, F.. Self-organized notch dynamics generatestereotyped sensory organ patterns in Drosophila. Science2017;356(6337):eaai7407. doi:10.1126/science.aai7407.[28] Gross, P., Kumar, K.V., Goehring, N.W., Bois, J.S., Hoege,C., Jülicher, F., et al. Guiding self-organized pattern formationin cell polarity establishment. Nature Physics 2019;15(3):293–300. doi:10.1038/s41567-018-0358-7.[29] Green, J.B.A., Sharpe, J.. Positional information and reaction-diffusion: two big ideas in developmental biology combine. De-