Abstract
Models of cell function that assign a variable to each gene frequently lead to systems of equations with many parameters whose behavior is obscure. Geometric models reduce dynamics to intuitive pictorial elements that provide compact representations for sparse in vivo data and transparent descriptions of developmental transitions. To illustrate, a geometric model fit to vulval development in Caenorhabditis elegans, implies a phase diagram where cellfate choices are displayed in a plane defined by EGF and Notch signaling levels. This diagram defines allowable and forbidden cellfate transitions as EGF or Notch levels change, and explains surprising observations previously attributed to contextdependent action of these signals. The diagram also reveals the existence of special points at which minor changes in signal levels lead to strong epistatic interactions between EGF and Notch. Our model correctly predicts experiments near these points and suggests specific timed perturbations in signals that can lead to additional unexpected outcomes.
https://doi.org/10.7554/eLife.30743.001eLife digest
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 egglaying 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 nonintuitive 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.
https://doi.org/10.7554/eLife.30743.002Introduction
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 nonintuitive. 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.pP8.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 timespecific 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 nonintuitive interactions that were previously attributed to a contextdependent 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 contextdependent 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.
Results
A geometric model for C. elegans vulval development
To quantify the Waddington topography yet retain its intuitive appeal requires a geometrical representation of dynamics that we illustrate in two dimensions, Figure 1A,B. The axes of this socalled fate plane are abstract and only acquire meaning when we fit to actual data. Dynamics is represented in the fate plane by arrows (a flow) defining the velocity of a point representing the state of a cell. The developmental history of a cell is defined by tracing its trajectory in the fate plane. The very bottom of a valley is defined by a point with only inward pointing arrows. The low point on a ridge separating two valleys, a socalled saddle point, has two inward directions along the ridge and two outward sectors representing flow into the valleys.
Turning to C. elegans vulval patterning, we adopt a twodimensional fate plane to accommodate the two signaling pathways, Figure 1C (Corson and Siggia, 2012). The VPCs initially form an equivalence group, thus the same model suffices for each cell, and the three possible fates require three distinct points at which cell trajectories stably terminate. Each of the VPCs starts from the same location at the beginning of competence, but their fate planes distort in different ways in response to EGF and Notch signals, yielding divergent trajectories (Figure 1D and Video 1). The VPCs receive graded EGF levels that are constant in time, consistent with observed mRNA levels in the AC (Barkoulas et al., 2013). Notch ligand production, on the other hand, is turned on in cells approaching the 1° fate (see the response in P7.p at t = 0.5 after P6.p has reached the dotted line in Figure 1D). The model also incorporates fluctuations to account for partially penetrant phenotypes. The states of a VPC for a group of animals is depicted as a cloud of points traveling in the fate plane (Figure 1E). With wildtype parameters, each of these clouds lies well within a valley following competence, corresponding to an invariant fate pattern.
New data from Barkoulas et al. (2013) provide VPC fates for a set of lines with defined EGF levels crossed into strains with perturbed Notch levels to generate a grid of data. This invites comparison with the outcome of the model as signal levels are continuously varied that is, a phase diagram. While our previous model is consistent with experimental outcomes, we can use some of the new data to more finely estimate its parameters. The new fit is fully described under Methods (see also Tables 1 and 2, Figure 1—figure supplement 1) and its implications explored here.
A phase diagram predicts novel, quantitative, features of cell fate determination
We ran our model for a range of Notch and EGF signaling levels, and determined the collective fate, or pattern, of the VPCs for each condition. The phase diagram in Figure 2 displays the outcome of our simulations and makes some surprising facts intuitive. Its power derives from the empirical observation that VPC fates, hence vulval patterns, are generally discrete and retain their identity under mutations in the signaling pathways. We highlight four notable features that are common to any system with two signals regulating three fates.
There exist continuous domains in the NotchEGF plane where the fate pattern is unique. The wildtype domain reflects the range of EGF and Notch perturbations that still lead to normal development, assigning a quantitative measure to the ‘robustness’ of the system.
Domains are separated by fuzzy lines, reflecting partial penetrance, that mark transitions between patterns. Transitions from one pattern to another are generally restricted to fate changes of a single VPC (or a symmetric pair). As shown in Figure 2, at transition points, the cloud of cell states is stretched out across the boundary between two fates, corresponding to a variable outcome (Figure 2B1).
Domain boundaries are not parallel to the coordinate axes. Thus, cell fate is not uniquely specified by the levels of either EGF or Notch. Rather, a combination of these signals defines the terminally differentiated state.
Boundary lines typically meet at threeway junctions, which we term triple points to follow the thermodynamic analogy. Near these points, small changes in EGF, Notch, or both can produce very different fate patterns, providing a strong test of our model. At a triple point, a single VPC can assume one of three fates (Figures 2B2 and 2B3).
Triple points evidently require that any pair of fates have a coexistence boundary in the fate plane as is evident in Figure 1E. A cloud of points induced by noise that lands near the point where the three phases meet, will populate all three fates. In the limit of a pure graded induction model for the vulva, Figure 2—figure supplement 1, the 2° fated domain sits between the 1° and 3° fates. The fate plane is effectively onedimensional (flow onto a line with two saddles separating three fates) and triple points cannot occur. This configuration is ruled out by the anchor cell ablation data (Corson and Siggia, 2012; Félix, 2007; Milloz et al., 2008). As the ablation time increases, P6.p transits from the 3° fate to equal fractions of 1° and 2°, implying boundaries between all three domains.
Aspects of the phase diagram are specific to vulva but obvious without a model. Under low Notch, the pattern transitions from 33333, to 33133, 31113, and 111111 as EGF levels increase and become sufficient to induce more distal cells. Under low EGF, strong ectopic Notch activity yields all 2° fates. Some quantitative features are straightforward consequences of the experimental data, Figure 2C. EGF and Notch null mutations are recessive, that is, a halfdosage is sufficient for patterning (+/− labels); this and an EGF overexpression line with known EGF levels (JU1107 label; see Table 3 for a list of the experimental lines referred in this study) delimit the central wildtype region. Similarly, data on animals with impaired Notch function and 1 or 2 ACs position the boundary between 33133 and 31113.
Elsewhere, the structure of the phase diagram is less intuitive and requires explanation. Around 0.3 × WT EGF and WT+ 0.2 Notch, small changes in EGF and/or Notch lead to five different fate patterns. Removing the molecular noise, Figure 2—figure supplement 2B, reveals that the triple point where P6.p assumes all three fates lies very close to where the other VPCs convert to fate 2°. This degeneracy is caused by the simultaneous conversion of all five VPCs to 2° fates for EGF = 0.
A second confluence of multiple phases near the JU1100 point (~4 × WT EGF), as well as the very narrow domain of 33133 for Notch >1, both are due to how data positions the dashed line in Figure 1E for the onset of Notch ligand production: near to the boundary between fates 1° and 2°. The JU1100 fate proportions (P4/8.p partially converted to 1° and 2° fates, and P5/7.p partially converted to 1° fate) were used to fit and are represented as a triple point (P4/8.p assume all three fates) nearly coincident with the boundary between WT and 31113, Figure 2—figure supplement 2A. Similarly, the phenotype of the CB1417 line implies that loss of the 1° and 2° fates occurs around the same EGF level. If we shift the signaling threshold away from the 1° domain, these fate confluences disappear and different patterns become accessible, Figure 2—figure supplements 3 and 4. (It is perfectly possible to observe with Notch = 1 the patterns 32223, 32123, and 22122 as EGF increases from below one to above, while retaining a large domain for the WT pattern signifying its robustness).
Importantly, our model makes precise predictions about the activity levels of the EGF and Notch pathways at interesting points in the phase diagram. Figure 2D overlays the new experiments from (Barkoulas et al., 2013) on our phase diagram. In all but one case (JU1100), the EGF levels were measured while Notch levels are not and their values constitute a prediction. For the lowest expressing EGF hypomorph, lin3(e1417), (denoted as CB1417 in (Barkoulas et al., 2013)), our model could not readily accommodate the very low EGF mRNA level in (Barkoulas et al., 2013) and the reported phenotype (Table 1). If, however, we use the model to predict the EGF level from the phenotype (Table 2), it agrees quite well with another measurement of the same allele reported in (Saffer et al., 2011) (Table 3). Elsewhere the correspondence between EGF level and VPC fates agrees well with experiments (Figure 2—figure supplement 5B).
Taken together, these examples reveal unexpected structure in the phase diagram that could not be gleaned from the available data without a model.
Triple points reveal abrupt fate changes
The phase diagram in Figure 2A predicts that at certain Notch and EGF signal strengths, strong genetic interactions can be observed, for example the triple point T1, where P5/7.p adopt equal proportions of all three fates. Our fit places this point close to WT EGF levels, and 0.3 × WT Notch activity, thus a moderate reduction in Notch creates a novel type of ‘sensitized background’ in which modest increases or decreases in EGF both lead to loss of 2° fates (see path P1 in Figure 3A).
This prediction is born out by experiment (Barkoulas et al., 2013). Whereas animals subjected to RNAi against lin12/Notch (fit as 0.4 × WT Notch) show occasional conversion of P5/7.p to the 3° fate, elevating EGF to only 1.25 × WT results in a substantial fraction of P5/7.p cells adopting a 1° fate, Figure 3B–C and Table 4. We predict that in the same RNAi background, removing one EGF copy (50% EGF activity) would result in substantial conversion of P5/7.p to the 3° fates. The different outcomes of Notch reduction under different EGF levels could be taken to suggest a switch between two functions of Notch, induction of the 2° fate and inhibition of the 1° fate (Barkoulas et al., 2013). However, our model allows an apparently simpler explanation: EGF and Notch function independently, and because the path P1 cuts through a corner of the WT domain, changes up or down in EGF can result in conversion of 2° to 1° or 3° fates.
Transient signaling explains how EGF favors adjacent 2°
The confluence of several patterns at the end of the path P2 in Figure 4A and the new data from (Barkoulas et al., 2013) allow a test of our assertion that the phenotype of a genetic cross can be predicted by fitting only the single mutants. The strain JU2092 (Figure 4B,C2, Table 4) combines modest ectopic Notch activity (whose level we fit from an independent experiment, Figure 2C), with a defined level of EGF overexpression. In these animals, P4/8.p cells are partially converted to the 2° fate while leaving P57.p unaffected. This is quite surprising, as increasing EGF leads to a 2° fate without an adjacent 1° fate cell. Our model provides a simple, yet unexpected, explanation (Figure 4C2, Video 2). When EGF levels are sufficiently large, we predict that P5/7.p transit through the 1° fate domain on their way to the 2° fate, and thus transiently signal to P4/8.p. This transient signal synergizes with ectopic Notch activity to induce 2° fates in P4/8.p. This prediction depends critically on how we model Notch signaling in the fate plane (dashed line in Figure 1E), a fit obtained from different experiments. The same position of the line for Notch signaling is independently required to explain the outcome of AC ablation experiments (Corson and Siggia, 2012). Thus, our geometric model remarkably ties together disparate experimental facts.
Cell intrinsic dynamics reveal how EGF/Notch can favor 2°/1° fates
Yet another observation affirming a contextdependent association of signals and fates is the observation that ectopic Notch activity favors the 1° fate in an EGF hypomorph (Barkoulas et al., 2013) (path P3 in Figure 5A,B). This fact appears in the phase diagram as a slight inclination of the boundary separating the 33333 territory from 33133, Figure 5A. This counterintuitive behavior of Notch is easily explicable from the trajectories in the fate plane of the model, Figure 5C. The EGF hypomorph leaves P6.p poised between fates 3° and 1° and stretched out along the line connecting the saddle or decision point to the associated fates. But the Notch signal has no reason to be parallel to the 'ridge' leading into the saddle and it in fact favors fate 1°, explaining the increased 1° fates with ectopic Notch. More generally as a function of increasing Notch, we predict a peak in the fraction of 1° fated P6.p followed by their conversion to entirely 2° fates for highest Notch as observed (Barkoulas et al., 2013), Figure 5B.
A converse experiment (Zand et al., 2011) found that mild EGF signaling could induce the 2° fate in a sensitized background of weak ectopic Notch that alone was not strong enough to induce appreciable 2° fates. The fate plane of the model, Figure 5D, shows that the sensitized state is poised near the saddle point between fates 2° and 3°. Again because the vector representing EGF is not parallel to the flow into the decision point, EGF can tip the balance and favors 2°. We could explain the enhanced fraction of 2° fates induced by EGF observed in Zand et al. (2011), merely from the flow in the fate plane around the saddle point, with no special assumptions. By contrast, Ref. (Zand et al., 2011) related this outcome to a switch between different Ras effectors, suggesting (if we translate to our language) that the EGF vector, whose direction remains fixed in our model, rotated and pointed upward toward 2° and away from 1° for low ligand levels.
Dynamical perturbations are sensitive tests of the model
Timedependent perturbations during the competence period, as exemplified by anchor cell ablation (Félix, 2007; Milloz et al., 2008), are more informative for model fitting and validation than end point data. However, other than these ablations, there is relatively little dynamic data to compare with. Timed temperature perturbations in temperature sensitive backgrounds clearly supply dynamic data and our model can predict the conditions under which dramatic effects are expected, Figure 6, Videos 3–4.
We showed in Figure 5 that our model could explain the rescue of 1° fate induction by Notch in an EGF hypomorph. The predicted effect is very sensitive to timing, and can be realized by a temperature sensitive Notch mutant, Figure 6A. A Notch pulse early in induction diverts P6.p further away from the ridge between 3° and 1° than a late one, and thus induces more P6.p cells to the 1° fate.
The same holds for the converse effect, induction of 2° fates by weakly activating lin15(ts) in a background with weak ectopic Notch activity and no AC (Zand et al., 2011). The outcome is more dramatic if we take the same integrated level of EGF and concentrate it in the first quarter of the competence period, Figure 6B. This provides a direct test of our interpretation of pathway epistasis vs. Ras effector switching as described in Zand et al. (2011).
Finally, we predict dramatic effects in the pure lin15(ts) background, if we include P3.p in the model and then extend the duration of the heat pulse, Figure 6—figure supplement 1. Because the induced EGF is uniform across the VPCs, P3.p gets induced and provides an additional lateral signal to P4.p. Thus, there is a marked asymmetry between P4/8.p in the degree to which they are induced to 2° fates. To realize this state may require some tuning of the intensity of the temperature step, but the predicted 22123 pattern is noteworthy since it is induced by EGF and involves adjacent 2° fates.
Our fits imply that the EGF pathway is saturated in P6.p under WT levels. As a result, EGF activity and the rate at which P6.p progresses toward the 1° fate should not be too sensitive to moderate changes in EGF dosage. This prediction is consistent with the observed dynamics of Notch ligand expression in P6.p (an indicator of EGF activity and/or progression toward the 1° fate) under EGF overexpression, which is similar to WT (van Zon et al., 2015). The saturation of the EGF pathway can be tested by ablating the anchor cell in an EGF +/ background. We predict little change from the outcome for WT animals, in contrast with predictions for ablations in Notch perturbation lines (Figure 6—figure supplement 2).
Fate correlations and multistability
Until this point, we have used a geometric model with no coupling between the two signaling pathways to explain epistasis in a variety of experiments. However, it is well known that Notch signaling inhibits the MAPK pathway and Ras activation inhibits Notch signaling (Shaye and Greenwald, 2002; Berset et al., 2001; Yoo et al., 2004). In what way could the predictions of the model be improved by adding terms to represent the known biochemical interactions within a cell?
A simple way to incorporate one such coupling in the model, with no additional parameters, is to downregulate Notch signaling in 1° fated cells along the same threshold that defines the production of Notch ligands, cf. (Shaye and Greenwald, 2002). Such a term could schematically at least represent the down regulation of Notch receptors in 1° cells, and thus capture intracell pathway interactions that were not present in the base model that only described the (nonlinear) production of Notch ligands in 1° fated cells. Fitting this modified model to the same data, we find only minor adjustments to parameter values (Table 2), and the predictions are largely unchanged. The resulting phase diagram, Figure 7B, is very similar to Figure 2A. The most noticeable difference is in the regime of reduced Notch and excess EGF: the 21112 domain extends to lower EGF levels, forming a longer boundary with the WT domain.
Why the two models differ in this regime is clear from the corresponding cell trajectories, Figure 7C–F. In the absence of coupling, moderate EGF overexpression (1.5 × WT) partially converts P5/7.p to the 1° fate, but the cells are clustered along the 1°−2° boundary and 1° fated cells do not generate sufficient Notch signal to divert P4/8.p from the 3° fate. Downregulation of Notch in 1° fated cells restores a saddle point on the 1°−2° boundary that partitions the cells into distinct 1° and 2° groups, allowing the 1° cells to signal strongly to their neighbors. The resulting correlation between 1° fates in P5/7.p and 2° fates in P4/8.p is consistent with observations in the JU2113 line, with a slightly lower measured EGF level (1.25 × WT). The same bistability in P5/7.p implies a direct transition from a WT pattern to 21112, which is a strong prediction of our geometric modeling scheme, Figure 7B.
Discussion
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 nonintuitive 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 MichaelisMenten 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 intrapathway interactions. In (Corson and Siggia, 2012), we described the cross between a lossoffunction mutant in the MAPK phosphatase, LIP1 and ectopic EGF from lin15, by first adding the lin15 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 LIP1 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 LIP1 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 tssignaling alleles. In Figure 6, we presented instances where a Notch or EGF pulse delivered by a tsmutant 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 tsallele 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 contextdependent 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 invivo data in developmental biology (Corson et al., 2017).
Materials and methods
Model
Request a detailed protocolThe model (Corson and Siggia, 2012) describes the state of each VPC by a vector $\overrightarrow{r}$ in twodimensional space and its dynamics by the stochastic differential equation
where
is a polynomial vector field with threefold symmetry and the nonlinear function
ensures that the dynamics is bounded. The term
integrates a default bias towards the default fate 3°, ${\overrightarrow{m}}_{0}$, and the effect of EGF and Notch signaling, parameterized by two vectors, ${\overrightarrow{m}}_{1}$ and ${\overrightarrow{m}}_{2}$ (red and green arrows in Figure 1C), which are combined linearly according to the levels of EGF and Notch ligands, ${l}_{1}$ and ${l}_{2},$ on the cell.
Variability in the dynamics is described by the stochastic term $\overrightarrow{\eta}\left(t\right)$, parameterized by a coefficient of diffusion D in phase space:
A fixed exponential gradient of EGF is assumed
while the level of Notch ligands a cell exposes to its neighbors is a function of its current state
with
and varies continuously from 0 to 1 across a line parameterized by ${n}_{0}$ and ${\overrightarrow{n}}_{1}$ (dashed line in Figure 1E). The level of Notch ligands received by a cell integrates contributions from its neighbors and from autocrine signaling, e.g. for P6.p
where $\alpha $ is the relative strength of autocrine signaling. Modulations in signaling activity are represented as additive or multiplicative changes in the ligand levels. For example, EGF overexpression is described by
where $\lambda >1$ denotes the dosage of EGF relative to WT, and ectopic EGF expression in a lin15 mutant (assumed uniform) by
where $\delta l$ denotes the level of ectopic EGF.
Simulations proceed as follows. The VPCs are initially equivalent and their state is drawn from the (Gaussian) steady state distribution obtained in the model when the term $\overrightarrow{f}$ that makes the dynamics multistable is removed and there is no signaling (an ansatz for dynamics prior to VPC fate specification). The full dynamics are then computed for one unit of time, representing the period during which VPCs respond to EGF and Notch signaling, after which the fate of each cell is scored according to its final position. For this purpose, the dynamics is run for one more unit of time in the absence of signaling to allow convergence toward an attractor, then fractional fate assignments are computed according to the distance to the three attractors. With this fractional fate assignment, fate proportions output by the model under fixed realizations of the noise are continuous functions of model parameters, which is numerically convenient for parameter fitting.
It may be noted that the form of our equation for VPC dynamics bears a resemblance to standard models for gene expression, comprised of a sigmoidal production term and a linear degradation term. However, the equation is vectorial, and the two coordinates in phase space do not stand for the levels of particular molecular species, providing instead an effective representation of the progression of a cell toward its eventual fate.
Parameter fitting
Request a detailed protocolModel parameters are fit to a set of experimental data, that is, the proportions of the different fates, 1°−3°, adopted by each cell, P48.p, in different conditions (Table 1). In some backgrounds in which EGF signaling is perturbed, lin3 mRNA levels have been measured and are used as a proxy for EGF levels. Elsewhere, ligand levels are treated as unknown parameters and fit to the data.
Following the principles of Bayesian inference, we compute the posterior probability of a parameter set according to the deviation between fate proportions in the model (computed from multiple simulation runs) and in experiments, and to priors that disfavor values that are deemed biologically unreasonable, for example very large pathway sensitivities (${\overrightarrow{m}}_{1}$, ${\overrightarrow{m}}_{2}$) or a very short response time (τ) (Corson and Siggia, 2012).
Numerically, local maxima of the posterior probability are computed using the LevenbergMarquardt algorithm. Several runs initialized from different, random parameter sets drawn from the prior distribution identify a unique global maximum. We then sample from the posterior using a Markov chain Monte Carlo algorithm initialized at the global maximum (Corson and Siggia, 2012).
The posterior probability of a parameter set $\Theta $ is given by
where $P\left(\Theta \right)$ is the prior probability of $\Theta $ and $P\left(D\right\Theta )$ is the likelihood of the experimental data $D$ given the outcome of the model with parameters $\Theta $.
In Corson and Siggia (2012), we used the following approximation for the likelihood of the data,
where
In this equation, ${p}_{e,c,f}^{\text{exp}}$ and ${p}_{e,c,f}^{\text{num}}$ denote the fraction of animals where cell c adopts fate f in experiment e, in experiments and in the model, respectively. N is the typical number of animals and simulation runs per condition, set to 100. This inference procedure is intended to yield an approximate confidence interval for parameter values, and we use a uniform value of N rather than actual animal counts, in order to give all phenotypes equal weight in the fit.
With the addition of new experiments to the data set (see below), we found that some experiments could be improperly fit. Specifically, for the JU1100 line where P4/8.p adopt the 1° fate in a small fraction of animals, the parameter ensemble obtained by Monte Carlo sampling included both parameter sets that match the phenotype, and parameter sets such that P4/8.p never adopt the 1° fate. Indeed, the ‘cost’ imposed by Equation 14 for not fitting a small but nonzero ${p}_{e,c,f}^{\text{exp}}$ is small, which is an artifact of that approximate expression: the likelihood of the data, $P\left(D\right\Theta )$, should be very small when this occurs. Here, we thus replaced Equation 14 with
With the original data set from (Corson and Siggia, 2012), Equation 15 yields a parameter distribution (means and standard deviations) that is very similar to that obtained using Equation 14. For the augmented data set used here, we obtain a better fit of the JU1100 phenotype.
Augmented data set and comparison with original fit
Request a detailed protocolOur model (Corson and Siggia, 2012) was fit to the vulval patterns observed in various mutants. But to be quantitative, it was essential that we used partially penetrant alleles and fit the observed variable outcomes. However, several of the relevant mutants did not have welldefined EGF levels. These could in principle be fit along with the phenotype, but as such only loosely constrained the model. Reference (Barkoulas et al., 2013) now has provided an abundance of quantitative data for defined EGF levels. While there are no qualitative discrepancies with our prior model, we can make a much more stringent test of our modeling framework against the new data set if we add one of the newly quantified lines (JU1107) to the training data (Table 1) and then predict all the others. Doing this increases the parameter controlling the range of the EGF gradient ($\gamma $; cf. Equation 6) by about 50%. With the resulting flatter gradient, it takes a smaller increase in EGF dosage to obtain a phenotype. Our fit for the EGF level in the JU1100 overexpression line (Hoyos et al., 2011) (relative to WT) is now 4.2 ± 0.5 instead of 7.4 ± 1.4 This level was not measured and is a testable prediction of the model.
The second new piece of data we add to the training set is for epistasis between low EGF and Notch. We previously showed that model outcomes in this regime are sensitive to the directions of EGF and Notch signaling and can be synergistic (Corson and Siggia, 2012). The new data provides evidence for such a synergy (Barkoulas et al., 2013), and is accommodated by a slight adjustment of the directions, which differ by less than 20° from the default values we took in Corson and Siggia (2012) (Table 2 gives all the parameter values and SD for the old and new fit).
Incorporation of coupling between EGF and Notch
Request a detailed protocolTo incorporate downregulation of Notch signaling in 1° fated cells (Shaye and Greenwald, 2002) without introducing new parameters, Notch sensitivity is simply taken to vary inversely with a cell's production of Notch ligands. That is, we replace the term $\overrightarrow{m}$ describing the response to signals, cf. Equations 1 and 3, by
The sensitivity of the cell to Notch signaling thus varies continuously from 1 to 0 as it crosses the dashed line in Figure 1E.
With this modification, the parameters of the model must be adjusted, and they are fit in the same way as for our main model.
References

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Cell cycledependent sequencing of cell fate decisions in Caenorhabditis elegans vulva precursor cellsDevelopment 126:1947–1956.
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Identification and characterization of 22 genes that affect the vulval cell lineages of the nematode Caenorhabditis elegansGenetics 110:17–72.
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Predictive Modeling of Signaling Crosstalk during C. elegans Vulval DevelopmentPLoS Computational Biology 3:e92.https://doi.org/10.1371/journal.pcbi.0030092

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Predicting phenotypic diversity and the underlying quantitative molecular transitionsPLoS Computational Biology 5:e1000354.https://doi.org/10.1371/journal.pcbi.1000354
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Mosaic analysis of the let23 gene function in vulval induction of Caenorhabditis elegansDevelopment 121:2655–2666.
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Genetic and phenotypic studies of hypomorphic lin12 mutants in Caenorhabditis elegansGenetics 135:755–763.
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Competence and Commitment of Caenorhabditis elegans Vulval Precursor CellsDevelopmental Biology 212:12–24.https://doi.org/10.1006/dbio.1999.9357
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Decision letter

Sean R EddyReviewing Editor; Howard Hughes Medical Institute, Harvard University, United States
In the interests of transparency, eLife includes the editorial decision letter and accompanying author responses. A lightly edited version of the letter sent to the authors after peer review is shown, indicating the most substantive concerns; minor comments are not usually included.
Thank you for submitting your article "Gene free methodology for cell fate dynamics during development" for consideration by eLife. Your article has been favorably evaluated by Arup Chakraborty (Senior Editor) and three reviewers, one of whom is a member of our Board of Reviewing Editors. The reviewers have opted to remain anonymous.
The reviewers have discussed the reviews with one another and the Reviewing Editor has drafted this decision to help you prepare a revised submission.
The authors previously introduced a "geometric" dynamical model for C. elegans vulval development (Corson and Siggia 2012). A subsequent paper (Barkoulas et al. 2013) provided quantitative data for mutants perturbing the EGF and Notch pathways. In this manuscript, Corson and Siggia update their model, show that it accounts quantitatively for almost all observed phenotypes, and that it makes additional testable, nonobvious predictions.
The reviewers feel that this is strong and interesting work that merits publication in eLife, but we do have several issues and questions that we would ask you to address in a revised manuscript, listed in the following 9 points:
1) A main assertion is that this "geometric model with no coupling between the two signaling pathways (EGF and Notch) [is used] to explain epistasis in a variety of experiments". However, the model does couple EGF and Notch signaling pathways in several ways, including:
◦ While it is true that the two signals are parameterized by two vectors m→1and m→2, which are combined in a linear manner in Equation 4, the linear combination (vector m) is an argument to nonlinear functions, including the hyperbolic tangent and the norm (Equation 3). This nonlinear mapping will, in effect, introduce interaction terms involving products of m→1 and m→2. For example, consider the secondorder term of the Taylor series expansion of the function tanh, or expand out the norm of l1m→1 + l2m→2, and one sees that there are products of components of vectors m→1 and m→2 in numerous places in the model equations.
◦ Making the above more complex to interpret, the interaction terms have no basis in mechanism.
◦ Next, the weighting factor (l2) for Notch signaling in Equation 4 depends on the level of EGF and Notch signaling, according to Equation 7 (note the dependence on the vector r, which depends on EGF and Notch signaling). Thus, the magnitude of Notch signaling depends on EGF signaling – a form of coupling.
◦ Finally, the threshold line for Notch expression (dotted line in Figure 1D) cuts diagonally through the fate plane. This line represents yet another form of coupling, since the level of Notch ligand (l2) depends on passing this threshold line (which depends on EGF and Notch signaling).
2) It is asserted that this model framework is the "most parsimonious parameterization of how two signals can control three fates". This statement is debatable, and readily disproven by drawing examples from literature. Based on Table 2, there appears to be 12 parameters in the model. The footnote to the table indicates an additional one that was not fit (norm of vector 1→1), but is a parameter. In total, we have at least 13. There are many examples of developmental patterning models, including but not limited to this C. elegans system, with fewer parameters. Some of these other models even include molecular mechanism (coarsegrained), and are capable of predicting phase diagrams of multicellular phenotype. Therefore, parsimonious parameterization is probably not a major feature of the approach presented here. Rather, it seems more accurate to describe the model as taking a complex fate plane structure (Equations 111) and using numerous parameters to fit the dynamicallyevolving landscape to experimentallyobserved VPC fate choices. In this respect, the approach opens itself to a classical critique of phenomenological modeling in biology: given a sufficiently complex model structure, one can fit observed behavior to it.
3) There are a number of predictions about the dynamical behavior of the system (Figures 5–6, subsection “Dynamical perturbations are sensitive tests of the model”). However, it is unclear what confidence to have in these predictions because there is so little temporal data for this system with which to parameterize dynamics (as pointed out in the manuscript). For example, the dataset in [Barkoulas et al., 2012] is extensive, but appears to be only endpoint measurements. According to Table 1, it seems that the only data for intermediate time points came from anchor cell ablations in the wildtype worm ([S12]). While this data provides VPC fate choices (as shown in the table), there is no data on the level of Notch signal. Additional detail is needed on what temporal data is used to parameterize the model, how, and what degree of confidence one can have on the parameterization of dynamics.
4) This modeling approach provides a quantitative framework for predicting phenotypes, but is largely phenomenological. One may be left wondering what new things we learned about mechanisms involved in C. elegans vulval development. We are left with a blackbox(*) of modulating EGF/Notch signal on one end, and post or predicting phenotypes on the other end. Whether this is the most insightful avenue for modeling and drawing mechanistic insight into developmental systems is unclear. (*) perhaps a greybox since the mathematical construction is precisely known to us. But, the construction is phenomenological and has no mechanistic basis.
5) Related #4, the notion that "geometric models should be the method of choice when confronted with sparse invivo data" seems an overstatement. Even in light of sparse invivo data and nascent mechanistic knowledge, significant progress has been made in many developmental model systems by using mechanistic models to predict phenotypes, and revising these models accordingly when experiments disprove model predictions.
6) There is little, if any, comparison to other models that predict phase diagrams for this system. Does this modeling approach match experimentally observed phenotypes better than other models? Do other models predict "sensitization" by quantitatively modulating specific signaling mechanisms? What are the pros and cons of the approach presented here relative to other methods?
7) Figure 1D and 1E show a dashed line in the fate plane, with an explanation in the legend that "cells positioned below the dashed line express Notch ligands." The dashed line appears invariable across the different panels. Does this mean that cells express Notch ligands regardless of their fate types (t=0.5 for P5/7.p for secondary fate, and several panels for tertiary fate)? If so, it will be good to cite experimental evidence for this.
8) In Figure 4C2, where authors predicts a transient primary fate for the green cloud (P5/7.p), the path is above the dashed line. How would these cells produce Notch ligands to transiently signal P4/8.p to induce secondary fate? I'm assuming that the paths are simulated results and not schematic drawings. If the latter, one may argue that the ectopic Notch is epistatic to the EGF, in which case one does not need to invoke a transition through the primary fate.
9) P6/P7/P8 are supposed to be an equivalence group, but Figure 1D shows P6.p with a different fate plane from P7.p/P8.p. Doesn't this beg the question? The model needs to explain how P6 becomes different from P7/P8. Figure 1D appears to be trying to illustrate the notion of competence windows (that P6.p has an early competence window to respond the EGF signal, and P7.p has a middle competence window to receive the Notch signal from P6.p). But it's unclear how the model treats competence windows. It seems the authors are trying to say that P6.p sees the highest/earliest EGF signaling from the AC, so P6.p starts "moving" in its fate plane first; and when P6.p starts expressing Notch, that starts moving P7.p. But it's not clear why the fate planes themselves change. If the EGF signal causes the fate plane of P6.p to change to an all1* fate plane, doesn't this become tautological? (That is: the geometric model explains how P6.p adopts 1* fate by saying that P6.p moves toward 1* fate.)
[Editors' note: further revisions were requested prior to acceptance, as described below.]
Thank you for resubmitting your work entitled "Gene free methodology for cell fate dynamics during development" for further consideration at eLife. We apologize for the delay in examining your revised manuscript. Your revised article has been favorably evaluated by Arup Chakraborty (Senior Editor), and a Reviewing Editor.
The manuscript has been greatly improved, and the referees agree that most concerns have been satisfactorily addressed. However, there is one aspect (1a, b below) that needs to be addressed before a decision to accept the paper can be made, because a central claim of the manuscript remains unclear. There is also one other aspect (2 below) that the referees note for you to consider at your discretion.
1a) Regarding whether the model includes coupling between EGF and Notch, your response acknowledges "couplings induced between the N and EGF pathways by the nonlinear functions that are used". However, the manuscript still states in several places that E and N are not coupled in the model, including:
"Our model has no overt coupling between the EGF and Notch pathways, implying that if we fit two alleles,[…]".
"Until this point, we have used a geometric model with no coupling between the two signaling pathways[…]"
"Our model with no pathway interactions then fits the genetic data in…"
You are trying to make a distinction between "overt" versus implicit coupling. Whether the coupling is overt or implicit, however, the end result is that the signals are coupled with each other, and the manuscript should not claim that they are not.
1b) In addition to coupling introduced by nonlinear functions, the model includes coupling in its geometric construction. This point was in the original review (see final bullet under point 1) but left unaddressed. Consider Figure 7 and subsection “Fate correlations and multistability”, where it says "a simple way to incorporate one such coupling in the model, with no additional parameters, is to downregulate Notch signaling in 1º fated cells along the same threshold [the dotted line in Figure 1D] that defines the production of Notch ligands". If downregulating Notch signaling along a line in the fate plane is a way to introduce EGFNotch coupling, would not upregulating/increasing the production of Notch ligands as cells cross the same line also be a form of coupling? The latter is part of the model from the start.
This coupling has nothing to do with the shape/steepness/nonlinearity of the threshold – it is a coupling introduced by a demarcation in the fate plane at which Notch ligands are unregulated. To cross that demarcation, cells must possess a combination of EGF and Notch signaling that lands them in that place in the phase plane. Thus, the level of EGF and Notch signaling affects expression of Notch ligand, which in turn affects EGF and Notch signaling: geometric coupling.
This is a second reason to question the manuscript's central claim to predict epistasis without invoking coupling; please clarify.
2) The comparisons to other models were useful. Regarding model complexity as quantified by numbers of parameters – the manuscript still speaks of a vague "plethora of parameters", but you provided useful parameter numbers for the other models in your response (over 40 parameters (Hoyos) and 13 parameters (Giurumescu)). It would clarify the scale of the disparity if you included these numbers in the manuscript.
https://doi.org/10.7554/eLife.30743.027Author response
[…] The reviewers feel that this is strong and interesting work that merits publication in eLife, but we do have several issues and questions that we would ask you to address in a revised manuscript, listed in the following 9 points:
1) A main assertion is that this "geometric model with no coupling between the two signaling pathways (EGF and Notch) [is used] to explain epistasis in a variety of experiments". However, the model does couple EGF and Notch signaling pathways in several ways, including:
◦ While it is true that the two signals are parameterized by two vectors m→1 and m→2, which are combined in a linear manner in Equation 4, the linear combination (vector m) is an argument to nonlinear functions, including the hyperbolic tangent and the norm (Equation 3). This nonlinear mapping will, in effect, introduce interaction terms involving products of m→1 and m→2. For example, consider the secondorder term of the Taylor series expansion of the function tanh, or expand out the norm of l1m→1 + l2m→2, and one sees that there are products of components of vectors m→1 and m→2 in numerous places in the model equations.
◦ Making the above more complex to interpret, the interaction terms have no basis in mechanism.
◦ Next, the weighting factor (l2) for Notch signaling in Equation 4 depends on the level of EGF and Notch signaling, according to Equation 7 (note the dependence on the vector r, which depends on EGF and Notch signaling). Thus, the magnitude of Notch signaling depends on EGF signaling – a form of coupling.
◦ Finally, the threshold line for Notch expression (dotted line in Figure 1D) cuts diagonally through the fate plane. This line represents yet another form of coupling, since the level of Notch ligand (l2) depends on passing this threshold line (which depends on EGF and Notch signaling).
We fully agree with the bulleted remarks of the referees pointing to couplings induced between the N and EGF pathways by the nonlinear functions that are used to saturate the response and create the correct topology (i.e., saddles and fixed points) of the model. This is a feature not a bug, as one says in the software field!
Our intention is best expressed by an analogy. If one is given data for the concentrations of various components in a system in thermal equilibrium, then it is most sensible to fit the log of the concentrations to free parameters not the concentrations themselves. This is because the Arrhenius formula shows that the log is necessary to extract interaction energies that are the most compact representation of statistical mechanics. Thus we remark in the Discussion regarding Berset et al., 2001, that their genetic data does not require pathway interaction (according to our model, but contrary to their assertions), though they supplied biochemical evidence that N inhibits EGF signaling, which of course we cannot question. A most important consequence of our paper, and why we hope it reaches a wide biological audience, is that many assertions about genetic interactions in the literature are not well founded, precisely because of the choice of variables used to make the comparison.
Although it’s very unlikely that some exact decomposition of nonequilibrium systems exists, one’s first guess as to what the analogue of Arrhenius would be is to build in the correct topology of the developmental system in question and then ‘tilt’ the landscape with linear functions. This is certainly a welldefined procedure and we show it suffices to fit the available data. It is universal in neural nets for machine learning, to subject a linear combination of inputs to a nonlinear thresholding function with one output. However in biological problems we claim the nonlinear function should encapsulate the topology of system. In our 2012 PNAS paper we showed that a topology of three fixed points and two saddles in one dimension was qualitatively wrong for the vulva and our model is the next more complex one. Topologies are discrete and just as one does in fitting data with a polynomial, the simplest one that works wins.
Evidently we have not been clear about what to us is the general message of our paper. We thus added a paragraph to the Discussion where we pose the question of induced interactions via the nonlinear functions (raised by the referees), make the Arrhenius analogy, and give some intuition why our procedure for a linear representation of the signaling pathways subject to a nonlinear thresholding function dictated by the topology is the optimal way to model developmental transitions.
2) It is asserted that this model framework is the "most parsimonious parameterization of how two signals can control three fates". This statement is debatable, and readily disproven by drawing examples from literature. Based on Table 2, there appears to be 12 parameters in the model. The footnote to the table indicates an additional one that was not fit (norm of vector n→1), but is a parameter. In total, we have at least 13. There are many examples of developmental patterning models, including but not limited to this C. elegans system, with fewer parameters. Some of these other models even include molecular mechanism (coarsegrained), and are capable of predicting phase diagrams of multicellular phenotype. Therefore, parsimonious parameterization is probably not a major feature of the approach presented here. Rather, it seems more accurate to describe the model as taking a complex fate plane structure (Equations 111) and using numerous parameters to fit the dynamicallyevolving landscape to experimentallyobserved VPC fate choices. In this respect, the approach opens itself to a classical critique of phenomenological modeling in biology: given a sufficiently complex model structure, one can fit observed behavior to it.
We intended our statement to be provocative but it is close to the truth.
The first four parameters in Table 2 are the two dimensional vectors representing the N and EGF influence on the state of a cell. Representing a signaling pathway (from receptor to fate and thus implicitly transcriptional output) with two parameters, is certainly parsimonious and given our two dimensional phase plane, nothing less is possible. A gene level description would have MichaelisMenten expression for ligands to receptors, receptors phosphorylate effectors, and perhaps the MAPK pathway for EGF etc.
The fifth parameter is a time scale, that determines the time needed for induction relative to the duration of the competence period, as probed by AC ablation experiments.
The diffusion constant largely controls the extent of the partial penetrance of many of the phenotypes we fit, hence manifestly necessary.
Next is how the EGF gradient from the Anchor Cell falls off in space. Clearly independent of other parameters.
Autocrine N signaling, is the one parameter that allows P6.p to assume some 2 fate under early AC ablation.
The remaining two parameters (or three if the referees insist) define the domain in the fate plane where the cell turns on N ligand to signal to itself and its neighbors. A line is the simplest way of dividing a plane. Notch turns on smoothly (via a tanh function) as cells approach the dashed line in Figure 1. The width of the onset is not constrained by the data and not fit. This is because cells will continue to approach this line, till they are pushed upwards by the Notch signal; changing the width of the transition will alter slightly when they reflect off the line, but not their ultimate fate. Hence 11 parameters in Table 2 in all, (Note none of the data constrains the angle of m_{0}, points in that region always relax towards the fixed point, so we just left it at a default value). Thus we claim each parameter represents a nonredundant piece of any model that will describe comparable phenomena. We give error bars on the parameters in Table 2 from Monte Carlo simulations and all are modest, hence all parameters are constrained by the data.
The referees correctly ask us in #6 to mention other models of vulval patterning, so we can count parameters in these papers:
Hoyos..Felix 2011. An ODE model with 11 species, and over 40 parameters, as one would expect for MichaelisMenten dynamics.
Closer to our work is Giurumescu..Sternberg 2009 in that they represent each cell with two variables for EGF and N taken to define the coordinate axes. They have 13 dimensional parameters, 9 dimensionless (see below their Equation 6) but they do not treat partial penetrance, hence omit our D, and do not consider experiments revealing autocrine signaling. Hence we are more parsimonious 11 – 2 < 9 + 1 (counting one absolute time scale for each model). More important their model is not multistable hence there is no specification/commitment, remove the external signal and the cells collapse to fate 3.
Fisher..Hajnal 2007 build a discrete model with virtually all genes known. They have’ 92,000 possible reachable states’, but no real numbers.
Evidently we succeeded in being provocative without being cogent. So we have qualified this statement that our parsimony is relative to other models (real and imaginable) of vulval patterning, and then explain briefly why the various fitting parameters are nonredundant. Clearly a model with relaxation in a 1D potential with 3 minima would have fewer parameters, but would not encode the salient facts of the vulva system (a point made in our 2012 PNAS paper and mentioned again here).
3) There are a number of predictions about the dynamical behavior of the system (Figures 5–6, subsection “Dynamical perturbations are sensitive tests of the model”). However, it is unclear what confidence to have in these predictions because there is so little temporal data for this system with which to parameterize dynamics (as pointed out in the manuscript). For example, the dataset in [Barkoulas et al., 2012] is extensive, but appears to be only endpoint measurements. According to Table 1, it seems that the only data for intermediate time points came from anchor cell ablations in the wildtype worm ([S12]). While this data provides VPC fate choices (as shown in the table), there is no data on the level of Notch signal. Additional detail is needed on what temporal data is used to parameterize the model, how, and what degree of confidence one can have on the parameterization of dynamics.
We agree fully with the referee’s remarks that the only dynamical data are the anchor cell ablations from Felix that were exceedingly tedious to obtain but very valuable (see our 2012 PNAS paper). There are also no good fluorescent reporters for cell fate, e.g., egl17 does not correlate well with primary fate in mutant backgrounds. Endpoint data is typically very insensitive to the nonlinear dynamics that led there, a point to be made forcefully about genetic screens that assay the mutated embryos well after the genes under study were active. However our model, after fixing an overall time scale, makes predictions for dynamics with no free parameters, hence this becomes a strong test of the model.
Our strategy to address these problems begins with a new microfluidic device built by a postdoc working with Siggia and Shaham (Keil, Dev Cell 2016), that permits multiday larval imaging. (This device is unprecedented in the field as evidenced by 3 collaborations Keil has entered with prominent worm labs outside of Rockefeller.) We are using ts alleles to deliver pulses of N or EGF to a population of worms in parallel. We have complete temporal control of the temperature. The worms are not precisely synchronized in the device, but we can define their developmental phase relative to competence by recording the divisions (which we can do with 8 min accuracy down to the last one).
We emphasized potential experiments in Figure 6 where there were dramatic changes in outcome depending where within the competence periods a stereotyped pulse of N or EGF was applied. Our apparatus will manifestly achieve these conditions. We will tune the temperature to see an effect, and the N or EGF level realized becomes a fitting parameter in the model. But the prediction is how the outcome changes with the phase of the pulse relative to the competence window. (The competence window itself can be redetermined with temperature steps followed by a record of when and how the cell subsequently divides.) The shape of the pulse does not matter and is folded into the fit of the amplitude. One pulse is applied to worms randomly scattered over the competence window. There are no free parameters once the pulse amplitude is fit.
We have substantially increased the paragraph in the Discussion that deals with future dynamic experiments to address these questions.
4) This modeling approach provides a quantitative framework for predicting phenotypes, but is largely phenomenological. One may be left wondering what new things we learned about mechanisms involved in C. elegans vulval development. We are left with a blackbox(*) of modulating EGF/Notch signal on one end, and post or predicting phenotypes on the other end. Whether this is the most insightful avenue for modeling and drawing mechanistic insight into developmental systems is unclear. (*) perhaps a greybox since the mathematical construction is precisely known to us. But, the construction is phenomenological and has no mechanistic basis.
5) Related #4, the notion that "geometric models should be the method of choice when confronted with sparse invivo data" seems an overstatement. Even in light of sparse invivo data and nascent mechanistic knowledge, significant progress has been made in many developmental model systems by using mechanistic models to predict phenotypes, and revising these models accordingly when experiments disprove model predictions.
We believe the community has been overly sanguine about what has/can be achieved by ‘mechanistic models’ in a developmental context, where we interpret ‘mechanistic’ to mean models where the variables correspond to the proteins, messages, regulatory DNA etc. in question. Although our remark has most force in development i.e., an embryo, it’s still relevant to single cell systems.
The basis for our pessimism about mechanistic models is illustrated by the Wnt pathway homepage that R. Nusse maintains. For just Wnt reception in a cell, there are about 8 components in a minimal model and perhaps another 40 have known interactions with the core 8, and this merely for signal reception. Any ‘mechanistic’ model represents composite variables.
The problem is actually more severe since a colleague Jim Sethna (http://www.lassp.cornell.edu/sethna/Sloppy/) has shown in great detail that models fit to pathway data for a single cells really only have a few well defined parameters, the others are ‘sloppy’, completely unconstrained by the data. This is shown mathematically from the eigenvalues of the Hessian of the error function being minimized.
We are interested in embryos. We do not accept that rates measured in a random cell line have any relevance to an embryo, hence there is exceedingly little quantitative embryo data, and even for the extraordinarily wellstudied vulva we have to resort to phenomenological geometric models. If some mechanism is retained, it should be in the context of a phenomenological model to cover the rest of the system.
We have now incorporated the above discussion, and particularly the sloppy model connection, into a new penultimate paragraph exposing more fully issues living below the surface of mechanistic models. Our final sentence about “geometric models method of choice” will be restricted to developmental problems.
The referees under #4 also raise an interesting epistemological question, of whether a phenomenological model, can possibly add to knowledge since the atoms of knowledge in development (we infer from those remarks) must be phrased in terms of genes and their interactions. We would argue (following the Hegelian synthesisantithesis path), that the very success of developmental genetics for the reasons above requires we reconceptualize developmental biology. A vast spreadsheet of all genes and their interactions is information rather than knowledge. More modestly if ‘knowledge’ implies some degree of quantitative prediction, that with stem cells will allow one to build embryos and organs, then we maintain that phenomenology is a prerequisite to understanding.
The penultimate paragraph of the Discussion rephrases the above argument. If any of the referees or editors find it too preachy, pretentious, or opinionated, we are happy to excise the paragraph in the editorial or proofing stage of the manuscript. It may be a good idea to do so.
6) There is little, if any, comparison to other models that predict phase diagrams for this system. Does this modeling approach match experimentally observed phenotypes better than other models? Do other models predict "sensitization" by quantitatively modulating specific signaling mechanisms? What are the pros and cons of the approach presented here relative to other methods?
This was an omission, we have added a discussion of the 3 papers mentioned under item 2, since they cover the different prototypes of modeling strategies. Prior theories never had the means to fit quantitative partial penetrance data, which is for us is the most revealing since it defines the boundaries between fates. There is no discussion of sensitized backgrounds as flows near saddle points, since partial penetrance is not allowed from the start. Further details are in the new Discussion.
7) Figure 1D and 1E show a dashed line in the fate plane, with an explanation in the legend that "cells positioned below the dashed line express Notch ligands." The dashed line appears invariable across the different panels. Does this mean that cells express Notch ligands regardless of their fate types (t=0.5 for P5/7.p for secondary fate, and several panels for tertiary fate)? If so, it will be good to cite experimental evidence for this.
This is a fair criticism, since there is biochemical data showing that the signaling pathways interact at the level of signal reception e.g., N signaling induces a MAKP phosphatase or EGF inhibits N signaling. However the dashed line defines where in the fate plane of the cell N ligand is produced. Logically for us, the state of the cell i.e., the point in the fate plane, defines whether that cell produces Notch ligands. Our assumption that the fate plane is 2D has a lot of content. We do not know if a cell in the 3º fated domain were to suddenly receive a huge ectopic EGF signal, would secrete Notch ligands before it moved to the relevant region of the phase plane. The model would say no, but there is no data. (Note the large green domain at T=0.5 for P5/7.p mentioned by the referee means only that under those signaling conditions, all cells move towards fate 2, not that they are already fate 2). So as elsewhere, the dashed line is the minimal fitting parameter, with the same logical status as the two vectors representing N and EGF signaling. Where it extends deep into the 3º domain it’s probably incorrect, but there is no data in that region, and no way we can see for forcing cells there.
We have not added these clarifications to the first paragraphs of Results where we introduce the model, to keep the Introduction as simple as possible. They are picked up in Discussion when talking about the minimality of the model.
8) In Figure 4C2, where authors predicts a transient primary fate for the green cloud (P5/7.p), the path is above the dashed line. How would these cells produce Notch ligands to transiently signal P4/8.p to induce secondary fate? I'm assuming that the paths are simulated results and not schematic drawings. If the latter, one may argue that the ectopic Notch is epistatic to the EGF, in which case one does not need to invoke a transition through the primary fate.
As we mentioned under #2 above, the dashed line is the center of a smooth ramp whose width (the additional parameter n_{1} in question #2) that we did not fit. The points shown are actual simulations, and during their trajectory (more so than at the endpoint) P4/8.p have moved far enough onto the ramp representing N production that the resulting signaling together with ectopic N in that background directs the P4/8.p cells upwards into the fate 2 domain. There would be a minimal change in outcomes if we made reasonable changes in the width of that domain. The ectopic Notch in this strain, would not in isolation lead to the conversion of P4/8.p to fate 2, it required the added EGF to push P5/7.p close to the dashed line and generate additional N signal.
9) P6/P7/P8 are supposed to be an equivalence group, but Figure 1D shows P6.p with a different fate plane from P7.p/P8.p. Doesn't this beg the question? The model needs to explain how P6 becomes different from P7/P8. Figure 1D appears to be trying to illustrate the notion of competence windows (that P6.p has an early competence window to respond the EGF signal, and P7.p has a middle competence window to receive the Notch signal from P6.p). But it's unclear how the model treats competence windows. It seems the authors are trying to say that P6.p sees the highest/earliest EGF signaling from the AC, so P6.p starts "moving" in its fate plane first; and when P6.p starts expressing Notch, that starts moving P7.p. But it's not clear why the fate planes themselves change. If the EGF signal causes the fate plane of P6.p to change to an all1* fate plane, doesn't this become tautological? (That is: the geometric model explains how P6.p adopts 1* fate by saying that P6.p moves toward 1* fate.)
This is a problem with our figure, and more generally using a static figure to stand in for a movie. Hopefully with the web centric eLife presentation it all will be clear. In Figure 1D we should have labeled the first row as t=0+, meaning just after the EGF signal turned on. Prior to that time the movie makes very clear that P48.p are completely equivalent (modulo the molecular noise) and bouncing around in the same potential. We model the competence window with hard boundaries, i.e. EGF reception jumps from 0 to 1 at t=0 and then off at t=1, since there is no data that is sensitive to what must be a continuous onset and decay of signaling. The last row of Figure 1D should be labeled t=1+, i.e., the signal turns off simultaneously for all cells and the flow fields are identical for P48.p though where the cells are is different and reflects their histories. Thus as expected P48.p are completely equivalent prior to competence and the same competence window applies to all cells.
We have modified the time labels on Figure 1D to add the ‘+’. Our meaning was defined in the caption, which in the review pdf was separated from the figure. Hopefully on line and with the linked video, or on a properly formatted pdf the equivalence group and common competence window all will be clear.
[Editors' note: further revisions were requested prior to acceptance, as described below.]
The manuscript has been greatly improved, and the referees agree that most concerns have been satisfactorily addressed. However, there is one aspect (1a, b below) that needs to be addressed before a decision to accept the paper can be made, because a central claim of the manuscript remains unclear. There is also one other aspect (2 below) that the referees note for you to consider at your discretion.
Our terminology is borrowed from conventional usage in C.elegans (e.g., http://wormbook.org/chapters/www_vulvaldev/vulvaldev.html, Section 8 coupling of LET23 and LIN12) where ‘coupling’ means intracellular interaction between EGF and N. For instance activation of the N receptor in a cell, induces a phosphatase that deactivates the MAPK pathway stimulated by the EGF receptor in the same cell. We adhere to an analogous use of the term in describing how our model fits data.
There are no new fitting parameters devoted to pathway interactions (until Figure 7EF where there is interaction with no new parameters). Operationally this means that if we fit allele 1 for EGF and allele 2 for N, then we have no additional freedom to fit an animal with both alleles. Hence our paper was devoted to the challenge of fitting a matrix of conditions in the Barkoulas paper with parameters that depended individually only on the two axes. More colloquially N^2 phenotypes are fit with 2N parameters.
We thus consider it an achievement to demonstrate that a model with parameters only for the two pathways separately and independently, fits data that by phenotype shows interaction between the pathways. We need make this distinction clear to the readership of eLife and not just the worm community. Thus we propose the following changes in response to the referee report:
1a) Regarding whether the model includes coupling between EGF and Notch, your response acknowledges "couplings induced between the N and EGF pathways by the nonlinear functions that are used". However, the manuscript still states in several places that E and N are not coupled in the model, including:
"Our model has no overt coupling between the EGF and Notch pathways, implying that if we fit two alleles,…".
New text:Our model has no fitting parameters that would couple the EGF and Notch pathways, implying that if we fit two alleles, one in each pathway, then the outcome of the cross is defined with no additional freedom. […] The model produces a context dependent association between signals and fates since it applies nonlinear functions to a linear combination of the two vectors representing the pathways.
"Until this point, we have used a geometric model with no coupling between the two signaling pathways[…]"
New text:Until this point, we have used a geometric model with no fitting parameters that couple the two signaling pathways to explain epistasis in a variety of experiments. However it is well known that Notch signaling inhibits the MAPK pathway and Ras activation inhibits Notch signaling [...]. In what way could the predictions of the model be improved by adding terms to represent the known biochemical interactions within a cell?
"Our model with no pathway interactions then fits the genetic data in[…]"
New text: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 LIP1 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 LIP1 phosphatase to explain the genetic data.
You are trying to make a distinction between "overt" versus implicit coupling. Whether the coupling is overt or implicit, however, the end result is that the signals are coupled with each other, and the manuscript should not claim that they are not.
1b) In addition to coupling introduced by nonlinear functions, the model includes coupling in its geometric construction. This point was in the original review (see final bullet under point 1) but left unaddressed. Consider Figure 7 and subsection “Fate correlations and multistability”, where it says "a simple way to incorporate one such coupling in the model, with no additional parameters, is to downregulate Notch signaling in 1º fated cells along the same threshold [the dotted line in Figure 1D] that defines the production of Notch ligands". If downregulating Notch signaling along a line in the fate plane is a way to introduce EGFNotch coupling, would not upregulating/increasing the production of Notch ligands as cells cross the same line also be a form of coupling? The latter is part of the model from the start.
The referee has thoroughly understood the content of the model. In the material quoted above, we were making a distinction between the production of Notch ligands in 1 fated cells, so something secreted and contained in the base model, versus a new term that down regulates Notch signaling within a 1 fated cell. We considered the new term, ‘interaction’ because it would represent biochemistry happening within a cell. We claim there is a logical distinction here, but evidently to a careful reader the term ‘interaction’ does not convey our meaning. We propose the following changes to two paragraphs in the new draft.
New text:[…] “In what way could the predictions of the model be improved by adding terms to represent the known biochemical interactions within a cell?[…] Such a term could schematically at least represent the down regulation of Notch receptors in 1º cells, and thus capture intracell pathway interactions that were not present in the base model that only described the (nonlinear) production of Notch ligands in 1º fated cells.”
This coupling has nothing to do with the shape/steepness/nonlinearity of the threshold – it is a coupling introduced by a demarcation in the fate plane at which Notch ligands are unregulated. To cross that demarcation, cells must possess a combination of EGF and Notch signaling that lands them in that place in the phase plane. Thus, the level of EGF and Notch signaling affects expression of Notch ligand, which in turn affects EGF and Notch signaling: geometric coupling.
We agree with the referee’s interpretation of the model, and thus have revised our language in various places. (e.g. see the second paragraph of the subsection “Fate correlations and multistability”).
This is a second reason to question the manuscript's central claim to predict epistasis without invoking coupling; please clarify.
We concede that our prior terminology of ‘no pathway interactions’ and ‘predict epistasis’ could be read as an oxymoron. We hopefully have clarified that by interaction we mean ‘new fit parameters describing intracell events that are specific to a particular combination of the two pathways’.
We have searched our draft for all occurrences of ‘coupling’ and ‘interaction’ and verified that all associated remarks clearly convey the above meaning, and nothing more. The authors are happy to clean up their use of language.
2) The comparisons to other models were useful. Regarding model complexity as quantified by numbers of parameters – the manuscript still speaks of a vague "plethora of parameters", but you provided useful parameter numbers for the other models in your response (over 40 parameters (Hoyos) and 13 parameters (Giurumescu)). It would clarify the scale of the disparity if you included these numbers in the manuscript.
We wanted to be less confrontational regarding prior work in our text so were vague about number of parameters. We have rephrased those sentences to state the number of parameters employed in prior work.
https://doi.org/10.7554/eLife.30743.028Article and author information
Author details
Funding
National Science Foundation (PHY1502151)
 Eric D Siggia
National Science Foundation (PHY1125915)
 Francis Corson
The funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.
Acknowledgements
EDS was supported by NSF grant PHY1502151. We thank W Keil, F Schweisguth, and S Shaham for comments. Some of this work was performed at the KITP Santa Barbara under National Science Foundation Grant No. NSF PHY1125915
Reviewing Editor
 Sean R Eddy, Howard Hughes Medical Institute, Harvard University, United States
Publication history
 Received: July 25, 2017
 Accepted: December 11, 2017
 Accepted Manuscript published: December 13, 2017 (version 1)
 Version of Record published: January 17, 2018 (version 2)
Copyright
© 2017, Corson et al.
This article is distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use and redistribution provided that the original author and source are credited.
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