At first, embryos are made up of identical cells. Then, as the embryo develops, these cells specialize into different types, such as heart and brain cells. Chemical signals sent and received by the cells are key to forming the right type of cell at the right time and place. The cellular machinery that produces and interprets these signals is exceedingly complex and difficult to understand.

In the 1950s, Conrad Waddington presented an alternative way of thinking about how an unspecialized cell progresses to one of many different fates. He suggested visualizing the developing cell as a ball rolling along a hilly landscape. As the ball travels, obstacles in its way guide it along particular paths. Eventually the ball comes to rest in a valley, with each valley in the landscape representing a different cell fate. Although this “landscape model” is an appealing metaphor for how signaling events guide cell specialization, it was not clear whether it could be put to productive use.

The egg-laying organ in the worm species Caenorhabditis elegans is called the vulva, and is often studied by researchers who want to learn more about how organs develop. The vulva develops from a small number of identical cells that adopt one of three possible cell fates. Two chemical signals, called epidermal growth factor (EGF) and Notch, control this specialization process.

Corson and Siggia have now constructed a simple landscape model that can reproduce the normal arrangement of cell types in the vulva. When adjusted to describe the effect of genetic mutations that affect either EGF or Notch, the model could predict the outcome of mutations that affect both signals at once. The twists and turns of cell paths in the landscape could also account for several non-intuitive cell fate outcomes that had been assumed to result from subtle regulation of EGF and Notch signals.

Landscape models should be easy to apply to other developing tissues and organs. By providing an intuitive picture of how signals shape cellular decisions, the models could help researchers to learn how to control cell and tissue development. This could lead to new treatments to repair or replace failing organs, making regenerative medicine a reality.

Development is a dynamical process, so models that purport to be comprehensive must explicitly describe dynamics. Typically models report changes in protein levels and use them to predict phenotypic outcomes. However, the number of parameters involved makes implementation cumbersome and predictions non-intuitive. Classical embryology emerged in the absence of genetics and describes development in terms of overall cell and tissue phenotype. Such studies allow the inference of cell states that must exist even before any overt differentiation or morphogenesis is visible. For example, a cell is competent to respond to signals during a temporal window; it is committed or specified when those signals are no longer required, and determined when other signals cannot deviate it from its normal/assigned fate. Our aim here is to retain the conceptual clarity of classical embryology in models that made novel and quantitative predictions.

Developmental states admit an intuitive topographical representation, as proposed by Waddington and later formalized mathematically (Waddington, 1957; Slack, 1991). The development of a cell is conceived as a downhill path in a shifting landscape controlled by cell signaling. Between two outcomes, or valleys, is always a ridge, and cells poised on the ridge can descend into either valley with equal probability. Once pushed off a ridge, cell fates are determined irrespective of subsequent twists and turns of the valleys. The Waddington picture suggests that cell fate decisions can be separated from the complexity inherent in specification and morphogenesis, which inherently simplifies any model.

Vulval development in Caenorhabditis elegans (Sternberg, 2005) is an appealing setting in which to quantify Waddington's landscape metaphor. Here, six vulval precursor cells (VPCs), P3.p-P8.p, which are developmentally equivalent (P3.p is less competent and ignored in the model), receive an EGF signal from the anchor cell (AC), and interact through Notch signaling, to eventually assume one of three different terminal fates (Sternberg and Horvitz, 1989). Cells P6.p and P5/7.p assume the 1° and 2° fates, respectively, and after three divisions form the vulva. The remaining cells, P3/4.p and P8.p, are assigned fate 3°. They divide once and fuse with the hypodermis.

These basic facts suggested a model in which each cell travels in a landscape with three valleys (fates) that we represent in two dimensions to allow EGF and Notch to tilt the landscape independently as development proceeds (Corson and Siggia, 2012). The movement of a cell in the landscape depends on parameters that quantify the influence of each signal on the direction of motion in the landscape. Values for these parameters were obtained from known terminal VPC fate patterns of animals defective in the two signaling pathways, as well as from limited time-specific perturbations (ablation of the AC at different stages) (Greenwald et al., 1983; Ferguson and Horvitz, 1985; Sternberg, 1988; Sundaram and Greenwald, 1993; Koga and Ohshima, 1995; Simske et al., 1995; Shaye and Greenwald, 2002; Félix, 2007; Milloz et al., 2008; Komatsu et al., 2008; Hoyos et al., 2011). Partially penetrant phenotypes are ideal for parameter fitting as they define the locations of ridges. From this data alone, focusing on the competence period (Ambros, 1999; Wang and Sternberg, 1999), we built a quantitative model for how EGF and Notch signals control fates, without considering the underlying complex genetic networks (Corson and Siggia, 2012). Our model has no fitting parameters that would couple the EGF and Notch pathways, implying that that if we fit two alleles, one in each pathway, then the outcome of the cross is defined with no additional freedom. Still, the model is sufficient to explain experiments showing epistasis, including non-intuitive interactions that were previously attributed to a context-dependent action of the signals, for example low EGF can promote 2° fate acquisition, or biochemical interactions between the pathways in a single cell. The model can predict context-dependent signals since it applies a nonlinear function to a linear combination of the two vectors representing the pathways.

Recent work (Barkoulas et al., 2013) quantified EGF levels in a series of perturbation lines and examined the cell fate outcomes in multiple combinations with Notch perturbation lines. The new data allows further quantitative refinement of the model, and an extensive test of its predictions. Using the refined model, we now generate a phase diagram that displays the terminal vulval cell fate pattern as a function of the EGF and Notch signal strengths. The diagram predicts the existence of special transition points where epistasis is particularly evident. These special points organize the entire phase diagram and define allowed and forbidden phenotypic transitions in response to continuous changes in signal strength. Verification of these transition points is an important goal of future experiments. Our data demonstrate that taking a more abstract, yet mathematically rigorous, geometric approach can reveal underlying principles of development, and can provide immediate quantitative predictions more difficult to obtain through kinetic modeling of the complex underlying gene and protein networks.

It is common in development to find a proliferating pool of progenitor cells that can be diverted into two mutually exclusive states by different signals. An example from early mammalian development is the split of the pluripotent epiblast cells into mesendoderm or neural fates at the time of gastrulation. Thus, our fate plane model with three possible fates is generic. The modeling proceeds from a general mathematical embodiment of the basic principles of development: cell fates are discrete, cells are competent to respond to signals over a limited period, and cells are committed when they assume their normal fates without additional signals. ‘Fates’ in the model are very schematic, they say nothing about the gene networks or morphogenetic events that ultimately define the fates. This simplification is plausible if the competence period ends before overt fate specification or differentiation occurs. The ‘fates’ in the model are then just a mathematical device to describe with minimal parameters all ways that two signals can divert cells towards three fates. Vulval patterning in C. elegans conforms to our assumptions (Ambros, 1999; Wang and Sternberg, 1999), and adds the simplification that the fates are terminal, so we can ignore subsequent bifurcations of the Waddington valleys.

Our fate plane represents the state of a cell. The motion of a cell in the plane will depend on signals impinging on that cell at the moment in question. Logically, the production of Notch ligand will depend only on where the cell is, not on the signals. Thus, we parameterized the onset of Notch ligand production by a line in the fate plane. Its extension far into the 3° fate domain is probably unreasonable, but since there is no situation that samples that region, it’s immaterial.

The geometry that accompanies the Waddington landscapes informs the interpretation of genetic experiments. Since the model has three distinct states represented as valleys, there must be ridges between adjacent fates, with the property that with a little noise half the cells will go to each state. The aptly named saddle point is the low point on such a ridge. The topographic analogy makes it very evident that for cells positioned anywhere near the ridge that feeds into the saddle, a small initial perturbation can lead to large changes in outcomes that are generally impossible to predict without a model. A ‘sensitized background’ in genetics is precisely an allele that flows near a saddle point. Thus to infer the activity of a gene by crossing it into a sensitized background may require a model, and conversely such experiments are very informative for model fitting since they expose the boundaries between fates. Figure 5 presented examples for vulva where small increments of Notch or EGF signaling in a sensitized background elicited non-intuitive changes in fates.

Our modeling approach yields the most parsimonious parameterization of how two signals can control three fates. The flow diagram is two dimensional to accommodate the two signals, there are three fixed points, one for each fate, and then mathematics requires the presence of two saddles. It is informative to scan the list of 12 parameters in Table 2 that appear in the model. They all plausibly can be independently varied, for example, the time scale, noise level, EGF gradient, and autocrine signal all describe different phenomena. The other parameters define various vectors, also independent. The other models for vulva we discuss below have yet more parameters.

Several other models have been constructed for vulval patterning. Reference (Fisher et al., 2007) constructs a discrete model for the vulva system incorporating virtually all relevant genes. It was shown to be consistent by logical programming techniques. However, from our perspective it does not deal with partial penetrance or treat continuously variable morphogens. A differential equation built from selected genes with Michaelis-Menten dynamics was implemented in Hoyos et al., (2011) and required over 40 parameters. Because of the plethora of parameters both papers are reduced to making model inferences by assuming that the relative volumes of parameter space correctly weight embryo phenotypes. Closest to our approach is Reference (Giurumescu et al., 2009), who describe a cell with two coordinates (for Notch and EGF). They also do not fit partial penetrance data, or autocrine signaling, ignore the anchor cell ablation data that we found crucial for model selection, and also do not build in multistability. So formally at the end of competence all cells would revert to 3° fate. Nevertheless they use 13 dimensionful parameters, one more than we do.

We were able to describe the Notch and EGF signals as two vectors combined by vector addition since they appear inside of nonlinear functions that build in the basic landscape of the VPC dynamics, namely three fixed points and two saddles (Methods). This is why we assert that the inference of interactions is contingent on how we chose to represent the data, and the bulk of this paper is devoted to showing that what appeared as interactions when counting VPC fates, disappear when fit to our model as shown in Figure 3–5. An analogy with the physics of thermal equilibrium illustrates the point. If the data consists of concentrations, then to discover interaction energies between species one has to fit the logarithm of the concentrations, not the concentrations themselves. While there are no exact results in dynamics analogous to the Arrhenius formula, imposing the basic topology of the flow field on a linear model for the signals is a plausible place to begin.

Multiple steps in a signaling pathway are collapsed into one parameter; nevertheless, the model can capture intra-pathway interactions. In (Corson and Siggia, 2012), we described the cross between a loss-of-function mutant in the MAPK phosphatase, LIP-1 and ectopic EGF from lin-15, by first adding the lin-15 contribution to the EGF from the anchor cell and then multiplying the sum by a parameter larger than one to account for loss of the phosphatase. The order of composition is dictated by the genetics. Our model that takes a linear combination of the vectors parameterizing each pathway and passes the vector sum through a nonlinear function, fits the genetic data in (Berset et al., 2001) even though LIP-1 is a Notch target and biochemically inhibits EGF signaling. We do not question the biochemistry, but it is not needed to explain the genetic result. We do not have to introduce a new fitting parameter for the down regulation of the MAPK cascade by the LIP-1 phosphatase to explain the genetic data.

In the absence of fitting parameters explicitly coupling the two pathways, our model has reproduced effects previously attributed to pathway epistasis simply by including the basic genetic facts of vulval patterning. The most visible manifestations of intrinsic pathway interactions we found are correlations between the fates of adjacent cells. We were able to account for them schematically in Figure 7, without additional parameters, merely by assuming that the effects of Notch on 1° fated cells disappears when that cell produces Notch ligands. Mutual pathway inhibition builds determination into our model, that is, a cell that gets close enough to its terminal state cannot be deflected from that state by ectopic signals.

To go further, separate parameters would be needed for the down regulation of Notch by EGF and the inhibition of MAPK by Notch signaling (Shaye and Greenwald, 2002; Berset et al., 2001; Yoo et al., 2004), which we have not pursued since the necessary data are still very sparse, and this would have obscured effects that are independent of pathway interactions.

Intuitively, the most informative experiments to fit a dynamic model manipulate signaling during the competence period when the cell is poised among its fates. The timed anchor cell ablations were very instructive for the vulva model. With recent advances in microfluidics (Keil et al., 2017) one can follow larval development for up to three days and image the worms every 8 min. All the vulva divisions can be scored with DIC optics, and it is possible to control signals with temperature and ts-signaling alleles. In Figure 6, we presented instances where a Notch or EGF pulse delivered by a ts-mutant would have very different outcomes depending on the timing of the pulse within the competence period. These are appealing experiments since the intensity and duration of the pulse (provided it is much shorter than the competence window) will be adjusted to induce an appreciable phenotype, and then fit with one parameter. Then one quantifies the outcomes as a function of the phase of the pulse within the competence window with no additional parameters. The competence window itself can be redetermined by applying a temperature step to a ts-allele and the predictions are insensitive to details such as the shape of the pulse due to induction and persistence of the proteins (van Zon et al., 2015) that are impossible to control; only the relative timing matters.

The phase diagram is computed directly from the model, but its general structure can be deduced from qualitative features of the fate plane. Our vulva model in Figure 1E allowed direct contact between any pair of fates. If instead we assumed a fate plane with the 2° fate wedged between and separating the 1° and 3° fates (as might correspond to a graded action of EGF [Katz et al., 1995]), the WT pattern is preserved but the phase diagram is very different, Figure 2—figure supplement 1. There are no longer any triple points since three fates can never coexist in the fate plane. But ablation experiments (Félix, 2007; Milloz et al., 2008) required that the three valleys be adjacent in all combinations (Corson and Siggia, 2012), implying that phase boundaries meet at triple points. This and a few quantitative phenotypes largely determine the full diagram Figure 2—figure supplements 2–4. A phase diagram forces the recognition that individual phases have boundaries and that boundaries will in general meet at triple points. A model will assign coordinates to all these features and allow experiments to be targeted to interesting points. Independent of a specific model, consideration of geometry unifies seemingly disparate outcomes.

Models that define the fate plane geometrically and then fit to genes can unify what appear to be different signaling modalities. The relative importance of induction of 2° vs. ‘lateral inhibition’ of 1° (Sternberg, 1988) by Notch has been debated. While biochemically the distinction may be very clear (e.g. when Notch signaling deactivates the MAPK pathway [Berset et al., 2001; Yoo et al., 2004]), once we insist fates 1° and 2° are distinct (bistability) there is no distinction between favoring one or inhibiting the other. While the extensive data in (Barkoulas et al., 2013) requires the authors to admit context-dependent signals (e.g. lateral inhibition vs. induction depends on EGF), our model accounts for all contexts with one set of parameters.

Similarly, vulval patterning is often conceptualized as an interplay of ‘graded’ (EGF level defines 1° vs. 2° fates [Katz et al., 1995; Sternberg and Horvitz, 1986]) and 'sequential' (Notch induces 2° fates [Koga and Ohshima, 1995; Simske et al., 1995]) signaling, which are proposed to be partially redundant (Kenyon, 1995). Our quantitative model suggests a specific instance of this: EGF reaching P5/7.p is not required for 2° fate induction but reduces loss of 2° fates under reduced Notch activity (Corson and Siggia, 2012).

Genes and their interactions are the ultimate building blocks of development, so how can a phenomenological model that eschews any one to one correspondence with genes contribute to our knowledge of development? For any signaling pathway, there are upwards of five components required from receptor to transcription and already 10’s of parameters. There are solid grounds for believing most of these parameters do not matter for the behavior of the system (http://www.lassp.cornell.edu/sethna/Sloppy/). Signal transmission between cells is less understood, particularly for hydrophobic ligands such as Hedgehog. Though pathway components and their biochemical properties have been elucidated in cell culture systems, it is far from evident that actual parameters fit to those systems have any relevance to the embryo. If the goal is to be more quantitative about how an egg turns into an embryo, then it is not clear to us, how denser lists of genes and their interactions will help. Rather pieces of the problem will be parameterized and the components joined in a structure defined by phenomenology.

Evolution has built many levels of redundancy into development that insure a robust outcome but complicate efforts to reconstruct the process from its genetic components. Geometric models take a step back from genetic reductionism and impose basic embryological phenomenology (competence, commitment, determination etc.). They begin from a parsimonious representation of how signals define fates that largely eliminates redundancy among parameters, and leads to sharper fits and more believable predictions. Geometric models should be the method of choice when confronted with sparse in-vivo data in developmental biology (Corson et al., 2017).

1. 1. Ambros V

   (1999)

   Cell cycle-dependent sequencing of cell fate decisions in Caenorhabditis elegans vulva precursor cells

   Development 126:1947–1956.

   • Google Scholar
2. 1. Barkoulas M
   2. van Zon JS
   3. Milloz J
   4. van Oudenaarden A
   5. Félix MA

   (2013) Robustness and epistasis in the C. elegans vulval signaling network revealed by pathway dosage modulation

   Developmental Cell 24:64–75.

   https://doi.org/10.1016/j.devcel.2012.12.001

   • PubMed
   • Google Scholar
3. 1. Berset T
   2. Hoier EF
   3. Battu G
   4. Canevascini S
   5. Hajnal A

   (2001) Notch inhibition of RAS signaling through MAP kinase phosphatase LIP-1 during C. elegans vulval development

   Science 291:1055–1058.

   https://doi.org/10.1126/science.1055642

   • PubMed
   • Google Scholar
4. 1. Corson F
   2. Couturier L
   3. Rouault H
   4. Mazouni K
   5. Schweisguth F

   (2017) Self-organized Notch dynamics generate stereotyped sensory organ patterns in Drosophila

   Science 356:eaai7407.

   https://doi.org/10.1126/science.aai7407

   • Google Scholar
5. 1. Corson F
   2. Siggia ED

   (2012) Geometry, epistasis, and developmental patterning

   PNAS 109:5568–5575.

   https://doi.org/10.1073/pnas.1201505109

   • PubMed
   • Google Scholar
6. 1. Ferguson EL
   2. Horvitz HR

   (1985)

   Identification and characterization of 22 genes that affect the vulval cell lineages of the nematode Caenorhabditis elegans

   Genetics 110:17–72.

   • PubMed
   • Google Scholar
7. 1. Félix MA

   (2007) Cryptic quantitative evolution of the vulva intercellular signaling network in Caenorhabditis

   Current Biology 17:103–114.

   https://doi.org/10.1016/j.cub.2006.12.024

   • PubMed
   • Google Scholar
8. 1. Fisher J
   2. Piterman N
   3. Hajnal A
   4. Henzinger TA

   (2007) Predictive Modeling of Signaling Crosstalk during C. elegans Vulval Development

   PLoS Computational Biology 3:e92.

   https://doi.org/10.1371/journal.pcbi.0030092

   • Google Scholar
9. 1. Giurumescu CA
   2. Sternberg PW
   3. Asthagiri AR

   (2009) Predicting phenotypic diversity and the underlying quantitative molecular transitions

   PLoS Computational Biology 5:e1000354.

   https://doi.org/10.1371/journal.pcbi.1000354

   • PubMed
   • Google Scholar
10. 1. Greenwald IS
    2. Sternberg PW
    3. Horvitz HR

    (1983) The lin-12 locus specifies cell fates in Caenorhabditis elegans

    Cell 34:435–444.

    https://doi.org/10.1016/0092-8674(83)90377-X

    • PubMed
    • Google Scholar
11. 1. Hoyos E
    2. Kim K
    3. Milloz J
    4. Barkoulas M
    5. Pénigault JB
    6. Munro E
    7. Félix MA

    (2011) Quantitative variation in autocrine signaling and pathway crosstalk in the Caenorhabditis vulval network

    Current Biology 21:527–538.

    https://doi.org/10.1016/j.cub.2011.02.040

    • PubMed
    • Google Scholar
12. 1. Katz WS
    2. Hill RJ
    3. Clandinin TR
    4. Sternberg PW

    (1995) Different levels of the C. elegans growth factor LIN-3 promote distinct vulval precursor fates

    Cell 82:297–307.

    https://doi.org/10.1016/0092-8674(95)90317-8

    • PubMed
    • Google Scholar
13. 1. Keil W
    2. Kutscher LM
    3. Shaham S
    4. Siggia ED

    (2017) Long-Term High-Resolution Imaging of Developing C. elegans Larvae with Microfluidics

    Developmental Cell 40:202–214.

    https://doi.org/10.1016/j.devcel.2016.11.022

    • PubMed
    • Google Scholar
14. 1. Kenyon C

    (1995) A perfect vulva every time: gradients and signaling cascades in C. elegans

    Cell 82:171–174.

    https://doi.org/10.1016/0092-8674(95)90302-X

    • PubMed
    • Google Scholar
15. 1. Koga M
    2. Ohshima Y

    (1995)

    Mosaic analysis of the let-23 gene function in vulval induction of Caenorhabditis elegans

    Development 121:2655–2666.

    • PubMed
    • Google Scholar
16. 1. Komatsu H
    2. Chao MY
    3. Larkins-Ford J
    4. Corkins ME
    5. Somers GA
    6. Tucey T
    7. Dionne HM
    8. White JQ
    9. Wani K
    10. Boxem M
    11. Hart AC

    (2008) OSM-11 facilitates LIN-12 Notch signaling during Caenorhabditis elegans vulval development

    PLoS Biology 6:e196.

    https://doi.org/10.1371/journal.pbio.0060196

    • PubMed
    • Google Scholar
17. 1. Milloz J
    2. Duveau F
    3. Nuez I
    4. Felix M-A

    (2008) Intraspecific evolution of the intercellular signaling network underlying a robust developmental system

    Genes & Development 22:3064–3075.

    https://doi.org/10.1101/gad.495308

    • Google Scholar
18. 1. Saffer AM
    2. Kim DH
    3. van Oudenaarden A
    4. Horvitz HR

    (2011) The Caenorhabditis elegans synthetic multivulva genes prevent ras pathway activation by tightly repressing global ectopic expression of lin-3 EGF

    PLoS Genetics 7:e1002418.

    https://doi.org/10.1371/journal.pgen.1002418

    • PubMed
    • Google Scholar
19. 1. Shaye DD
    2. Greenwald I

    (2002) Endocytosis-mediated downregulation of LIN-12/Notch upon Ras activation in Caenorhabditis elegans

    Nature 420:686–690.

    https://doi.org/10.1038/nature01234

    • PubMed
    • Google Scholar
20. 1. Simske JS
    2. Kirn SK
    3. Kim SK

    (1995) Sequential signalling during Caenorhabditis elegans vulval induction

    Nature 375:142–146.

    https://doi.org/10.1038/375142a0

    • Google Scholar
21. Book

    1. Slack JMW

    (1991) From Egg to Embryo (Second)

    Cambridge University Press.

    https://doi.org/10.1017/CBO9780511525322

    • Google Scholar
22. 1. Sternberg PW
    2. Horvitz HR

    (1986) Pattern formation during vulval development in C. elegans

    Cell 44:761–772.

    https://doi.org/10.1016/0092-8674(86)90842-1

    • Google Scholar
23. 1. Sternberg PW
    2. Horvitz HR

    (1989) The combined action of two intercellular signaling pathways specifies three cell fates during vulval induction in C. elegans

    Cell 58:679–693.

    https://doi.org/10.1016/0092-8674(89)90103-7

    • Google Scholar
24. 1. Sternberg PW

    (1988) Lateral inhibition during vulval induction in Caenorhabditis elegans

    Nature 335:551–554.

    https://doi.org/10.1038/335551a0

    • Google Scholar
25. 1. Sternberg PW

    (2005) Vulval development

    WormBook.

    https://doi.org/10.1895/wormbook.1.6.1

    • PubMed
    • Google Scholar
26. 1. Sundaram M
    2. Greenwald I

    (1993)

    Genetic and phenotypic studies of hypomorphic lin-12 mutants in Caenorhabditis elegans

    Genetics 135:755–763.

    • Google Scholar
27. 1. van Zon JS
    2. Kienle S
    3. Huelsz-Prince G
    4. Barkoulas M
    5. van Oudenaarden A

    (2015) Cells change their sensitivity to an EGF morphogen gradient to control EGF-induced gene expression

    Nature Communications 6:7053.

    https://doi.org/10.1038/ncomms8053

    • Google Scholar
28. Book

    1. Waddington CH

    (1957)

    The Strategy of the Genes

    Allen & Unwin.

    • Google Scholar
29. 1. Wang M
    2. Sternberg PW

    (1999) Competence and Commitment of Caenorhabditis elegans Vulval Precursor Cells

    Developmental Biology 212:12–24.

    https://doi.org/10.1006/dbio.1999.9357

    • Google Scholar
30. 1. Yoo AS
    2. Bais C
    3. Greenwald I

    (2004) Crosstalk Between the EGFR and LIN-12/Notch Pathways in C. elegans Vulval Development

    Science 303:663–666.

    https://doi.org/10.1126/science.1091639

    • Google Scholar
31. 1. Zand TP
    2. Reiner DJ
    3. Der CJ

    (2011) Ras Effector Switching Promotes Divergent Cell Fates in C. elegans Vulval Patterning

    Developmental Cell 20:84–96.

    https://doi.org/10.1016/j.devcel.2010.12.004

    • Google Scholar

Author details

1. Francis Corson

   Laboratoire de Physique Statistique, CNRS / Ecole Normale Supérieure, Paris, France

   Contribution

   Conceptualization, Software, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   corson@lps.ens.fr

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7230-137X

2. Eric D Siggia

   Center for Studies in Physics and Biology, Rockefeller University, New York, United States

   Contribution

   Conceptualization, Methodology, Writing—original draft, Writing—review and editing

   For correspondence

   siggiae@mail.rockefeller.edu

   Competing interests

   No competing interests declared

   "This ORCID iD identifies the author of this article:" 0000-0001-7482-1854

• 3,196

  views

• 518

  downloads

• 45

  citations

Views, downloads and citations are aggregated across all versions of this paper published by eLife.