Significance
Cells have to reliably process biochemical signals in the presence of many different sources of information. For instance, immune cells have to discriminate foreign peptides from many potentially similar self-peptides, which sometimes lead to the absence of active decision even in the presence of infection (a process called antagonism). In this article, we mathematically formalize this chemodetection problem. By contrasting two types of models for ligand detection (independent or coupled receptors) we show that if environmental noise is not statistically negligible (e.g., “long-tailed”), antagonism can be advantageous to buffer undesirable fluctuations in environmental composition.
Keywords: chemodetection, fluctuating environment, nonspecific interactions, statistical decision, immune recognition
Abstract
Variability in the chemical composition of the extracellular environment can significantly degrade the ability of cells to detect rare cognate ligands. Using concepts from statistical detection theory, we formalize the generic problem of detection of small concentrations of ligands in a fluctuating background of biochemically similar ligands binding to the same receptors. We discover that in contrast with expectations arising from considerations of signal amplification, inhibitory interactions between receptors can improve detection performance in the presence of substantial environmental variability, providing an adaptive interpretation to the phenomenon of ligand antagonism. Our results suggest that the structure of signaling pathways responsible for chemodetection in fluctuating and heterogeneous environments might be optimized with respect to the statistics and dynamics of environmental composition. The developed formalism stresses the importance of characterizing nonspecific interactions to understand function in signaling pathways.
Information transmission within biological networks has recently been subject to intense scrutiny. Quantities such as mutual information appear to be biologically relevant and optimized in many contexts (1), such as neural coding (2) and development (3, 4). In these situations, the nature of the signal is unambiguous: only uncertainty in the value of the input limits information content. In other realistic biological problems, the input consists of a complex mixture, with the signal of interest buried in a sea of nonspecific interactions. For instance, multiple different molecules could bind to a cellular receptor, in which case the signaling pathway downstream needs to find a way to discriminate between correct and spurious signals. An example is immune ligand detection: T cells need to detect foreign ligands at the surface of antigen presenting cells (APC) but there are many other nonagonist ligands interacting with receptors in charge of detection (5).
The consideration of heterogeneous environments, with multiple ligand types binding to a receptor, corresponds to a departure from past theoretical analyses of bounds on the performance of cellular measurements (6, 7) and requires specific treatment (8). Predefined mixtures of ligands were previously considered in ref. 9 with an emphasis on optimum decision time for given ligand compositions. Here, we use statistical detection theory to formalize the general problem of detection of a chemical signal in a time-varying mixture. We compare the detection performance of optimal proofreading networks of independent receptors to receptors with inhibitory coupling whose strength increases with the input. We find that despite antagonism and signal attenuation, coupled receptors outperform independent receptors in strongly varying environments. Intuitively, a negative feedback growing with the input size damps rare large fluctuations in the spurious ligand concentration, while retaining sensitivity to situations where few correct ligands dominate. Our results are relevant for signaling pathways having to sort weak signals from a fluctuating background of biochemically similar nonspecific interactions.
Materials and Methods
We consider a situation (Fig. 1A) where two types of ligands (“correct” and “incorrect” ligands, number of molecules in the environment, respectively, and ) can be present in the environment. Correct ligands correspond to the signal to be detected by the cell. Incorrect ligands, which interact with the same cellular receptors as correct ligands, constitute noise in the environmental composition (10). Over time, cells are assumed to be exposed to mixtures with different and . Possible values for are drawn without correlations from a predefined distribution , which characterizes environmental variability. Practically, we simulated a fluctuating environment where ligand numbers and randomly change with a typical timescale s, consistent with the duration of contacts between APC and T cells (11) (see details in SI Appendix, section S4).
Fig. 1.
Chemodetection in a fluctuating, heterogeneous environment. (A) Two types of ligands in the environment fluctuate in time. Correct ligands (red) need to be detected, whereas incorrect ligands (blue) constitute a form of environmental noise. (B) Network of transition between receptor’s internal states (example). Numbers indicate internal receptor states (here internal states) and red–blue indicate binding to different ligand types. Filled symbols correspond to the active (output) receptor state. Diffusible molecules (such as mediator M here) can be shared and modified by all receptors (), effectively coupling them (green symbols). κ is the on rate (assumed identical for correct–incorrect ligands), τ the inverse off rate, and ϕ the forward rate from 1 to 2. See the main text and SI Appendix, section S1 for details.
We take the distinguishing feature of ligand types to be their binding time (inverse off rate) , to the receptor, in accordance with the “lifetime dogma” for immune detection (12). Following ref. 13, we introduce the parameter , which quantifies the similarity of ligand types, with (correct ligands taken to have higher affinity for the receptor). We fix s to be the typical timescale for the inverse off rate, similar to agonist ligands of T-cell receptors (TCRs) (14). Ligands are biochemically indistinguishable from the perspective of the receptor for . We will be interested in the limit where similarity between ligand types makes detection difficult, i.e., (we take for concreteness). Assuming a substantial affinity of incorrect ligands to the receptor naturally precludes the consideration of a “perfect instrument” (that would measure only correct ligands) à la Berg and Purcell (15), and demands that more attention be given to events downstream of the receptor–ligand binding.
We assume that many receptors are present at the surface of a cell [ TCRs on the surface of immune cells (14)], which ensures that receptors are unsaturated for any realistic ligand concentration. In addition to possibly being bound or unbound to ligands, we suppose that each cellular receptor can be in one of N internal states and that transitions between these states are possible (see, e.g., Fig. 1B, where ). The number of receptors in state j bound to ligand type s will subsequently be denoted . For immune detection, the internal states could represent the number of phosphorylated sites on the intracellular tail of the TCRs (ITAM).
Transition rates between internal states can be potentially catalyzed by diffusible mediator molecules (M in Fig. 1) whose activities could in principle themselves depend on the receptor states (e.g., transition from to M in Fig. 1B depending on the number of receptors in state 2). These interactions effectively couple the receptors. Models considered in this article are based on the classical kinetic proofreading (KPR) scheme (16, 17) as well as subsequent applications to the immune context by McKeithan (18) and ourselves (19, 20); see Results for details. We make a stochastic description of those networks of transition between receptors’ internal states using a chemical master equation formalism. For example, the master equation for the model of Fig. 3B [adaptive sorting (20)] in the presence of only one type of L ligands, with association constant κ and binding time τ, is
[1] |
Fig. 3.
Performance of three different detection networks. We display informative quantities following conventions of Fig. 2. Network structure, input–output relationships, and ROC are on the same level for a given network. See discussion in main text for signification of special symbols. (A–C) Network of transitions among a receptor’s internal states. The open squares correspond to an unbound receptor whereas the open circles correspond to a bound receptor. The filled circles are the last internal state whose time average is the output. Green arrows correspond to transitions among internal receptor states catalyzed by diffusible chemicals. Starred arrows indicate that all receptors contribute to a catalysis. (A) KPR (). (B) AS (with ). (C) Simplified immune detection model including phosphatase SHP-1 (S). All parameters are given in SI Appendix, section S1.4. (D–F) Average output value as a function of for different values of , for the three different models (color code for is indicated at the bottom). Intrinsic fluctuations (SD) with s are shown as error bars for . x-axis scale is logarithmic. Dashed lines indicate response to pure correct ligands as a function of for comparison. (G) Distribution of used for ROCs computation. (H–J) Convolution of output from D–F with from G. Distributions for different s correspond to the theoretical bound for KPR (H), but to binning of full stochastic simulations for the AS (I) and SHP-1 (J) models. Overlaid pale curves in I and J correspond to theoretical curves, showing close to perfect agreement; see SI Appendix, section S1.6 for details. Insets for distributions correspond to logarithmic x axis for . (K–M) Corresponding ROCs obtained by varying for the three different networks. (Inset) Semilog plot for .
where is the probability for one receptor to be in internal state j and corresponds to the unbound receptor. ϕ is the forward rate in the cascade and is the fraction of active mediator molecules. Receptors in the above model are coupled (in a mean field) because the transition from state to N depends on which depends on . See SI Appendix, section S1.4 for details on the master equations of the models.
The total number of receptors in internal state N, , will be taken as the “output” of the system, subsequently called O. In an immune context, would thus be the fully phosphorylated ITAMs of the TCR. This output is assumed to be time-averaged, to decrease intrinsic variability. Such time averaging sets a typical timescale for integration of all signals leading decision, as observed in the immune context and discussed in refs. 19, 20. Time averaging is taken to be performed biochemically via activation by of a chemical species with a degradation time of . We take in the present work. Our results are unaffected if (SI Appendix, section S1.5 and Fig. S14).
Taking into account all receptors in the cell, a given model is then characterized by (the signaling pathways’ “input–output” relationship), the stationary output distribution at fixed and . The index reminds us that even when environmental composition is fixed, the output will fluctuate due to the intrinsic stochasticity of biochemical reactions. Finally we assume the cell uses a thresholding procedure to make decisions: if , where is a predefined threshold, correct ligands are deemed present.
As we are concerned with environments where the detection of correct ligands is challenging, we perform our analysis for heavy-tailed distributions (e.g., power law, log-normal) for as these correspond to extreme cases of variability in . Our conclusions are valid for distributions with substantial weight at low and high values of . The magnitude of is kept as a parameter in our analysis to evaluate how detection performance improves as the size of the signal of interest increases. All our simulations are performed using Gillespie’s stochastic simulation algorithm (21).
Results
Ligand Detection as Hypothesis Testing.
To formalize the process of chemodetection, a fruitful perspective is to consider the cell as effectively testing for hypotheses from the observations of O, as discussed in ref. 9. Only two situations are possible at a given time: no correct ligands are present in the environment (hypothesis ), or correct ligands are present in the environment (hypothesis H1).
We can assess detection performance by first determining how the output concentration is distributed in the presence and absence of correct ligands, denoted, respectively, by and . In the limit of slow environmental switching (), these distributions reduce to the weighted sum [weight ] of stationary distributions at fixed ligand concentration, (see SI Appendix, sections S4 and S5 for a discussion of transients):
[2] |
[3] |
The resulting output distributions for H0 and H1 could overlap considerably even for vanishing intrinsic noise as the output distributions are obtained by marginalizing over the variability in environmental composition (10). Hence, as opposed to the problems of measuring the fixed concentration of a single compound (15) or to discriminating between predetermined mixtures (9), a long averaging time does not necessarily lead to perfect decision making in time-varying mixtures. The essence of our work is to discuss how cells can make decisions despite such adverse conditions.
Quantifying Detection Performance.
With our description of the environment [, ], and of the detection network [], we can quantify detection performance using statistical detection theory (22). Two types of errors are possible: calling H1 when H0 is true and vice versa. Define the detection and false positive (conditional) probabilities to be, respectively,
Perfect detection corresponds to and . By our thresholding assumption, H1 is called when the output concentration exceeds a threshold . We therefore have
[4] |
[5] |
and are equal to 1 for and vanish as . Whenever and overlap, there is a tradeoff between having a large detection probability but few false positives. We do not know a priori how tolerant various systems are of the two types of errors. A conservative way to analyze detection performance is then to display the whole range of possible and (corresponding to varying from 0 to ) for a given detection network in a given environment. A plot of versus , the receiver operating characteristic (ROC), will be our main tool of analysis of detection performance. The shape of the ROC quantifies the stringency of the tradeoff between and . An ROC with quickly rising detection probability with increasing false positive probability corresponds to high detection performance. Mathematical quantities required to assess detection performance are shown in Fig. 2 for hypothetical and .
Fig. 2.
Illustration of the main quantities involved in evaluating detection performance in a given environment for hypothetical detection network and environmental statistics. (A) Sketch of the input–output relationship, [mean of ] as a function of for a hypothetical network. The black curve corresponds to and the red one to . Error bars correspond to intrinsic noise [SD of ]. (B) Example of statistics of incorrect ligands in a given environment . (C) Convolving distribution with gives the output distribution for the specific detection network (A) and environment (B) corresponding to the two hypotheses (Eqs. 2 and 3). Filled areas identify and for a given threshold . (D) ROC for this situation, obtained by varying threshold and evaluating and for each threshold value. High detection performance corresponds to the top left corner ().
The above formalism allows us to evaluate and compare the detection performance of the different networks introduced in Fig. 3. Our focus is on comparing optimal networks of independent receptors (Fig. 3A) to networks of coupled receptors (e.g., Fig. 3 B and C) to investigate whether and how coupling between receptors can facilitate chemodetection.
Independent Receptors.
We first consider independent receptors. McKeithan (18) proposed a KPR model explaining how a weak signal could be detected in the presence of a fixed large background of substantially dissimilar () ligands. In this scheme, the receptor sequentially visits internal states upon binding of a ligand. At each internal state, the receptor–ligand complex can unbind, following which the internal state of the receptor is assumed to quickly return to its first “unmodified” state; see Fig. 3A corresponding to Eq. 1 with . At fixed and , for receptors that are not saturated, the mean number of receptors bound to type s ligands in the last internal state N is (SI Appendix, section S1.2)
[6] |
with an increasing function of τ amplifying ligand differences. The mean output value is equal to the total number of receptors in state . As a result of receptor independence, does not appear in and vice versa, in contrast with networks of interacting receptors (e.g., Eq. 9 below). This input–output relationship is shown in Fig. 3D for pure (solid), pure (dashed), and mixtures (color) of ligands, respectively.
The ratio can be thought of as the specificity of the network toward the ligand types. For KPR cascades with N internal states, the specificity satisfies the bounds . This is to be compared with , the maximal ratio of bound correct ligands to bound incorrect ligands. Notably, it was shown in ref. 13 that for fixed N, the attainable specificity for any network of transitions among a receptor’s internal state is smaller than . In particular, any transition constituting a “short-circuit” along the sequential path through the internal states decreases the number of discard pathways, which are responsible for generating the exponential enhancement of specificity (see SI Appendix, section S1.3 for explicit calculations pertaining to our formalism). However, it must be emphasized that to attain the performance bound , the rate of progression down the cascade needs to be much smaller than , which slows down any realistic network dynamics considerably, especially for large N.
Nevertheless, the idealized performance of KPR constitutes an illuminating theoretical bound for the detection performance of independent receptors. Throughout our analysis of independent receptors, the optimal specificity bound is assumed and in our comparison between networks in Fig. 3, degradation in performance due to the slow dynamics of the KPR network is neglected.
Independent Receptors Cannot Reliably Detect Correct Ligands in Variable Environments with Low .
Despite the enhancement of specificity arising from KPR steps, detection performance in the presence of large environmental fluctuations is poor with low as can be seen on the ROC in Fig. 3K [corresponding to a log-normal ].
Intuitively, large and rare environmental fluctuations in should not be detected as correct ligands to get a low . The problem is the mean output is linear in (Eq. 6), which actually implies that large fluctuations will yield correspondingly large output values. Threshold of detection thus has to be higher than those large outputs to avoid false positive, but this in turn considerably decreases detection performance.
More quantitatively, let us first define the minimum size of fluctuations in the environment occurring in less than a fraction f of possible environments, i.e., . In the limit where intrinsic variability is neglected compared with environmental fluctuations (which is the case if environmental variability is large, see error bars in Fig. 3D), linearity of Eq. 6 carries for the minimum threshold of detection , corresponding to a false positive probability and we get (see the full derivation in SI Appendix, section S2)
[7] |
In environments where very large values of can be observed, diverges as [for example, for the log-normal distribution of Fig. 3G]. Moreover, the slower the decay of at large , the faster the divergence.
In the presence of correct ligands, the detection probability will be 1 irrespective of the environment provided the corresponding output is above the threshold (from Eqs. 4 and 6). Using our specificity bound for KPR, the minimum correct ligand concentration (“size” of the signal of interest) to have is thus
[8] |
For small and reasonably small N proofreading steps, this condition will lead to large , as the specificity enhancement is insufficient to decrease significantly the factor of for heavy-tailed . Thus, low correct ligand concentrations cannot reliably be detected with low false positive probabilities. For given s, s corresponding to condition [8] are identified by in Fig. 3K for of Fig. 3G.
As an alternative explanation for the poor performance of networks of independent receptors at low , note that as , we have just shown that threshold diverges. One therefore needs to integrate the rightmost tail of distribution to compute . Due to the linearity of the output on and the heavy tail of , asymptotically tends to at large enough output values, irrespective of ( in Fig. 3H, Inset). This is because incorrect ligand concentrations higher than dominate any realistic concentration of for small . As a result, will tend linearly to as ( in Fig. 3K). Thus, independent receptors cannot reach the favorable top left region of the ROC.
Inhibitory Coupling Between Receptors.
Linearity of the mean output with ligand concentrations makes detection networks vulnerable to variability in the environment. If interactions between receptors could somehow lead to a plateauing output value, then fluctuations in output resulting from variability in the environment would be considerably suppressed and detection improved, as was alluded to in ref. 9.
To quantify this point, we consider a minimal modification to the KPR backbone where a negative feed-forward interaction (Fig. 3B and Eq. 1) is added. We assume the transition from internal state to N (earlier transition could be considered) in the cascade of reactions requires the presence of a diffusible catalyst . We suppose that can be “deactivated” into M, which is unable to catalyze the transition . We take the total number of receptors in a given internal state to catalyze the deactivation of ( in Eq. 1). We further take to be rapidly diffusing in the intracellular region. In this regime, whereas each receptor occupancy is low on average, overall, there are still enough occupied receptors to significantly alter the activity of , thereby effectively mediating a mean-field negative interaction between all receptors. Such a network was recently uncovered by in silico evolution as a motif maximizing mutual information between output and ligand type and was termed adaptive sorting (AS) (20).
In this case, following the same convention as before, the average bound to type-s ligands is (see SI Appendix, section S1.4.1)
[9] |
where above and . Again, . For pure inputs (one ligand type, i.e., in Eq. 9 above), the output plateaus to value for Ls higher than which can be much smaller than the ligand concentration required to saturate the receptors (20). This behavior is shown in the black solid line (incorrect ligands only) and dashed line (correct ligands only) curves of Fig. 3E. Flattening of the output is due to the negative feed-forward loop via , which leads to a form of biochemical adaptation, making the output O independent from the (pure) ligand concentration . However, because of the KPR backbone, the plateau value is still an increasing function of the binding time τ, which allows for a discrimination between ligand types (20).
In the presence of mixtures, the inhibitory interactions between receptors damp the output arising from correct ligands (exemplified by the decrease of the colored curves of Fig. 3E), a well-known phenomenon in the immune context termed antagonism (14). In Eq. 9, function quantifies the strength of this antagonistic interaction. Various tradeoffs exist between the above quantities, discussed in SI Appendix, section S1.4.2. For instance, earlier deactivation of in the cascade (smaller m) increases the strength of antagonism––which is deleterious for detection within mixture––although this leads to a smaller ––which lowers the ligand concentration required for output flattening.
The specific form of inhibitory interaction was chosen for simplicity. We also present results for recent models of the early T-cell signaling network, where the activation of phosphatase SHP-1 (denoted S) is responsible for increased backward rate down the KPR (as opposed to deactivation of enzyme required for progression down the cascade) (14, 19); Fig. 3C. Because the activation of S saturates at large ligand concentrations, the output flattens only over a limited range of ligand concentration (Fig. 3F). Details for this detection network can be found in SI Appendix, section S1.4.3.
As we see in the next section, nonlinearity (i.e., flattening) in the input–output relationship arising from inhibitory receptor coupling can significantly improve detection performance.
Output Flattening Allows for Detection Even at Low .
The detection performance of interacting receptors is drastically different from that of KPR. Qualitatively, increases much faster close to (compare Fig. 3 K and L).
The main reason is that linear relationship (Eq. 6) between ligand and output does not hold for AS. Instead saturates at for moderately high (black curve in Fig. 3E). Hence, contrary to KPR, the heavy tail of the distribution does not lead to a heavy tail in the output distribution , which instead is concentrated around the value (see correspondence between the plateau from Fig. 3E and peak in distributions below the indicated threshold in Fig. 3I). It then follows that output fluctuations are dominated by intrinsic noise for AS, in contrast with KPR where environmental fluctuations dominate. Because intrinsic fluctuations are normally distributed, they decay rapidly for large output value (Fig. 3I, Inset).
Intuitively, negative coupling between receptors buffers the large environmental fluctuations that were responsible for the failure of detection for the KPR. Buffering however comes at the price of antagonism. If a large concentration of incorrect ligands coincides with the presence of correct ligands , the response is attenuated as seen in the colored curve at large in Fig. 3E. As a result, the output distribution will have a peak below threshold (Fig. 3I) for which detection is impossible. Nevertheless, for the difficult case of small , environmental conditions with small will have lower antagonism, leading to output values of the order of (colored curve at small in Fig. 3E and corresponding peaks in distribution in Fig. 3I). Hence, detection is statistically improved if one chooses a threshold so that , which translates into a rapidly increasing ROC at low . In SI Appendix, section S3, we make these intuitions quantitative by analytically deriving an explicit bound of given a .
Finally, for AS, we can estimate detection probability at a given and . From Eq. 9, we can compute the incorrect ligand concentration required to decrease the mean output value below the threshold, i.e., , where is given by the bound derived in SI Appendix, section S3.1 (shown for in Fig. 3 E and I, where ). In environments where , the mean output is not damped below and correct ligands can be detected. Conversely, for , antagonism decreases response below and detection is impossible. then allows us to determine the fraction of environmental conditions for which antagonism is weak enough to allow for detection. A crude estimate of the detection probability is therefore . This estimate is shown for as in Fig. 3L, Inset, in good agreement with numerical simulations.
For the model of Fig. 3C inspired by the immune system with phosphatase SHP-1, the fact that output flattening occurs only over a finite range implies that, like KPR, distribution displays a heavy tail ( in Fig. 3J, Inset). As a result, at very low remains true ( in Fig. 3M, Inset), similarly to what happened for KPR. Nevertheless, the partial flattening of the output does prevent false positive responses at intermediate values of , corresponding to an improvement over KPR. Finally, the superiority of interacting receptors is illustrated for a time course of a fluctuating environment in SI Appendix, Fig. S18 (AS) and Fig. S20 (SHP-1).
Discussion
Cells have to perform signal processing and make decisions in the presence of fluctuations in environmental composition, where nonspecific interactions potentially interfere with the input signal of interest. The paradigmatic example is immune recognition, where the self/not-self discrimination is first done at the receptor levels. In this work, we formalized the chemodetection problem based on statistical decision theory. We demonstrated and quantified how a negative feedback increasing with ligand concentration improves decision by filtering out large fluctuations of nonspecific ligands.
Earlier attempts have considered probabilistic models of encounters aiming at distinguishing self from not-self, exploiting proofreading to detect long tails in detection events (23). Detection of mixture of ligands is a more challenging problem, which has so far only been considered in refs. 9, 18 for mixtures of fixed compositions. Negative feedback at the level of the receptors had been shown to be a crucial element of the signaling pathway of early T-cell activation (14). Our main contribution is to quantify how such negative feedback enhances detection performance in fluctuating environments, in the sense that the detection probability at low false positive is higher for AS than for any model of independent receptors (Fig. 3). Because a low probability of having correct ligands in the environment exacerbates the detrimental effect of false positives (see Precision and Recall in SI Appendix, section S6), inhibitory receptor coupling could be essential for the proper performance of systems where the signal to be detected is rare, e.g., in an immune context.
Our results contrast recent work (24) that argues that independent receptors are optimal for the cellular sensing problem in environments with one ligand type. We find that in heterogeneous environments, inhibitory coupling is beneficial to detection if fluctuations are sufficiently large, such as for heavy-tailed distribution of biochemically similar incorrect ligand . Given the multiplicity of possible self-ligands and of TCRs (25), it is plausible that the effective nonagonist distribution is rather broad. Indeed, such distributions are the norm rather than the exception in many biological contexts (26). For simplicity, we have focused here on the worst case of incorrect and correct ligands close biochemically ( of order 1). Considering more complex mixtures with other ligand types with larger would not alter our conclusions.
It is often suggested (8) that biochemical networks have been “optimized” by evolution, with respect to various constraints. We propose that properties of signaling pathways responsible for chemodetection might be tuned to buffer nonspecific interactions with statistical and biochemical properties characteristic of their environment. In the AS and SHP-1 network presented here, negative feedback acts to buffer an uncorrelated environmental noise. This explains counterintuitive effects such as nonmonotonicity of response (see ref. 19 and Fig. 3F), which appears to be common in complex pathways such as endocrine signaling (27). Possible future work would be to extend our analysis to environments with correlations in the s. There, a cell could modulate its threshold of response based on measured . Such an adaptive mechanism (effectively a high-pass filter) would then filter the long timescale fluctuations in , thereby facilitating detection of few correct ligands. Such mechanism has actually been suggested to be present to some extent in immune recognition (28, 29). These examples demonstrate that quantifying the statistics of nonspecific interactions in the context of chemodetection might be crucial to understand the potential adaptive value of observed complex feedback mechanisms.
Supplementary Material
Acknowledgments
We acknowledge discussion with Arthur Charpentier. We thank members of the François group and two anonymous referees for useful comments on the manuscript. J.-B.L. is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Fonds de recherche du Québec–Nature et technologies (FRQNT). P.F. is supported by NSERC, FRQNT, and partially supported by the Human Frontier Science Program and a Simons Foundation Investigator Award in the Mathematical Modeling of Living Systems.
Footnotes
The authors declare no conflict of interest.
This article is a PNAS Direct Submission.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1420903112/-/DCSupplemental.
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