Combinatorial signal perception in the BMP pathway

Summary

The Bone Morphogenetic Protein (BMP) signaling pathway comprises multiple ligands and receptors that interact promiscuously with one another, and typically appear in combinations. This feature is often explained in terms of redundancy and regulatory flexibility, but it has remained unclear what signal processing capabilities it provides. Here, we show that the BMP pathway processes multi-ligand inputs using a specific repertoire of computations, including ratiometric sensing, balance detection and imbalance detection. These computations operate on the relative levels of different ligands, and can arise directly from competitive receptor-ligand interactions. Furthermore, cells can select different computations to perform on the same ligand combination through expression of alternative sets of receptor variants. These results provide a direct signal processing role for promiscuous receptor-ligand interactions, and establish operational principles for quantitatively controlling cells with BMP ligands. Similar principles could apply to other promiscuous signaling pathways.

Graphical Abstract

By harnessing promiscuous receptor-ligand interactions in the BMP pathway a single cell can perform different computations depending on which combinations of receptor ligands are present.

Introduction

Many intercellular signaling pathways, such as Bone Morphogenetic Protein (BMP), Wnt, Notch, and JAK-STAT exhibit a curious feature. Rather than using a single ligand and receptor, these systems comprise multiple ligand and receptor variants that interact promiscuously with one another to combinatorially generate a large set of distinct signaling complexes. These complexes activate the same intracellular targets, and therefore appear to operate redundantly. Previous work has suggested that the use of redundant ligands and receptors offers regulatory flexibility (Llimargas and Lawrence, 2001), or provides robustness to genetic variation (Dudley and Robertson 1997; Edson et al. 2010). However, it is also possible that this apparent redundancy provides specific signal processing capabilities (Mueller and Nickel, 2012; Murray, 2007; Schmierer and Hill, 2007; Wodarz and Nusse, 1998).

The BMP pathway is an ideal example of a promiscuous receptor-ligand architecture (Fig. 1A). In mammalian species, it includes more than 20 distinct ligands, 4 type I receptors (Bmpr1a, Bmpr1b, Acvr1, and Alk1) and 3 type II receptors (Bmpr2, Acvr2a, and Acvr2b). These components could interact combinatorially to form hundreds of distinct receptor-ligand signaling complexes, each composed of 2 type I and 2 type II receptors binding a dimeric ligand. Active signaling complexes phosphorylate Smad1, 5 and 8, which, together with Smad4, translocate to the nucleus to regulate target gene expression (Heldin et al., 1997; Massagué, 1998).

Figure 1.

Promiscuous receptor-ligand interactions can be analyzed in terms of multi-dimensional ligand and receptor spaces (schematic). (A) In the BMP signaling pathway, multiple ligand variants (blue, green) interact promiscuously with multiple distinct type I (orange, yellow)-type II (purple, pink) receptor heterodimers. Most ligands interact with multiple receptor complexes (arrows), but all active signaling complexes phosphorylate the same second messenger, Smad1/5/8. Phosphorylated Smad1/5/8, in complex with Smad4, activates endogenous targets (white) and a stably integrated fluorescent reporter gene (yellow). (B, C) Cellular environments and expression levels can be represented as points in multi-dimensional spaces. (B) Ligand concentration space represents the possible local environments of cells. Only 3 ligands are plotted for simplicity, but the full space includes dimensions for each ligand species. Zoomed circles indicate examples of two environments with distinct concentrations of ligands. (C) Receptor space represents the space of possible receptor expression profiles. Only 3 of 7 dimensions are shown. Two example cell types with distinct receptor expression profiles are indicated (circles). These representations provoke the questions of (D) how multiple ligands combine to determine pathway activity in a given cell type, and (E) how different cell types respond to the same ligand combination.

Two features of the BMP pathway suggest the possibility of more complex signal processing. First, in most contexts, multiple BMP ligands and receptors appear in overlapping spatio-temporal distributions, and therefore appear to be utilized in combinations (Danesh et al., 2009; Faber et al., 2002; Lorda-Diez et al., 2014; Salazar et al., 2016). For example, BMP9 and BMP10 co-regulate the formation of vasculature (Chen et al., 2013; Ricard et al., 2012), while BMP2, BMP4, GDF5 and GDF6 operate together in joint development (Storm and Kingsley, 1996). Similarly, at least 6 distinct ligands and 3 receptors are involved in kidney development (Simic and Vukicevic, 2005). Second, individual ligands preferentially signal through particular receptors. For example, Alk1 is preferentially activated by BMP9 and BMP10 in endothelial cells (David et al., 2007); GDF5 signals mainly through Bmpr1b and not Bmpr1a (Nishitoh et al., 1996); and BMP2/4 and BMP6/7 signal through distinct receptors to induce mesenchymal stem cell differentiation (Lavery et al., 2008).

The ability to form many competing complexes with distinct affinity and activity preferences could in principle, allow the system to perform complex signal processing. However, we lack a general quantitative framework to understand how the BMP pathway perceives combinations of ligands, how such combinatorial perception emerges from underlying molecular interactions, and whether and how distinct cell types respond differently to the same ligand combinations.

Here, combining theoretical and experimental approaches, we show that the BMP pathway perceives ligand combinations through a specific family of multi-dimensional response profiles. These profiles allow the pathway to perceive relative, in addition to absolute, levels of multiple ligands. Mathematical modeling further reveals that these response profiles can arise from an interplay between receptor-ligand binding affinities and the quantitative activity of each complex. The former determine what complexes are formed, while the latter determine how the activities of those complexes combine to establish overall pathway activity. Critically, we find that the response profiles differ qualitatively and quantitatively depending on the expression levels of the different receptor variants. As a result, different cell types, with distinct receptor expression profiles, can respond to distinct features in the multidimensional space of ligand concentrations. Together, these results establish a general framework for analyzing the BMP signaling pathway and reveal a more general design principle for biological signaling systems containing promiscuous receptor-ligand interactions.

Results

Theoretical Framework

To analyze the way in which the BMP pathway uses multiple receptor variants to integrate signals from multiple dimeric ligand species, it is useful to consider two multi-dimensional spaces. Cellular environments, specified by the concentrations of each of the dimeric ligand species, can be represented as points in a multi-dimensional ‘ligand space’ (Fig. 1B). Similarly, individual cell types typically co-express multiple type I and type II receptors (Cheifetz, 1999) and can therefore be represented as points in a 7-dimensional ‘receptor space’ specified by the individual expression levels of each receptor (Fig. 1C). (This space is, more precisely, the combination of a 3-dimensional space for the type I receptors, and a 4-dimensional space for the type II receptors). Not every point in ligand or receptor space may be realized biologically, and other secreted and intracellular factors further modulate BMP signaling in specific contexts (Balemans and Van Hul, 2002; Zakin and De Robertis, 2010). Nevertheless, understanding signal processing by the BMP pathway requires determining how multiple ligands combine, or integrate, to control the pathway activity in a cell with a given receptor configuration (Fig. 1D), and whether distinct cells, expressing specific combinations of receptors, can integrate the same ligands in qualitatively different ways (Fig. 1E).

BMP ligands exhibit combinatorial effects

In order to address these questions experimentally, we set out to measure the dependence of BMP pathway activity on individual ligands and ligand combinations. Ligand monomers form covalent homodimers and heterodimers with distinct activities (Israel et al., 1996; Neugebauer et al., 2015; Valera et al., 2010). Here, we focused on mixtures of distinct homodimeric ligands, which have been shown to produce non-additive responses in some systems (Ying et al. 2000; Ying and Zhao 2001; Ying et al. 2001; Açil et al., 2014). Mixtures of heterodimeric ligands could be analyzed similarly.

To quantitatively measure BMP pathway activity, we constructed a reporter cell line, by stably integrating a Histone 2B (H2B)-Citrine fluorescent reporter driven by a BMP response element (BRE) specific for Smad1/5/8 (Korchynskyi and ten Dijke, 2002) into the NAMRU mouse mammary gland (NMuMG) epithelial cell line, in which the BMP pathway can be activated without inducing differentiation (Piek et al., 1999). Reporter expression correlated with phosphorylation of Smad1/5/8 and with endogenous BMP target gene expression (Fig. S1A–C), and exhibited a unimodal distribution for each ligand concentration (Fig. S1D). After an elevated transient response to BMP addition, pSmad levels reached a steady state within 90 min (Fig. S1E). The steady-state behavior was also reflected in reporter fluorescence, which accumulated at an approximately constant rate over time for up to 48h (Fig. S1F). (Since the fluorescent protein is stable and the cell cycle is greater than 24h in these conditions, linear accumulation indicates a constant rate of reporter expression.) Based on these dynamics, we selected 24h post-induction as a time-point for subsequent analysis.

As a first step to classifying ligand integration behaviors, we sought to identify candidate ligand pairs for subsequent higher resolution analysis. We performed a coarse-grained survey of 15 commercially available homodimeric ligands (Fig. 2A). We measured reporter expression in response to each ligand individually, at a specific base concentration (see Methods, Table S1); each ligand at twice its base concentration (diagonal elements); and each pair of ligands at their base concentrations (other matrix elements). To quantify pathway activity, we normalized each measurement by basal activity with no added ligand (bottom).

Figure 2.

The BMP pathway perceives ligand combinations. (A) NMuMG reporter cells were exposed to 136 different combinatorial pairings of 15 homodimeric BMP ligands, as indicated. Color scale indicates mean fluorescence level at 24h, normalized by the uninduced population (‘relative activity’). (B) From the complete interaction matrix we extracted the individual response to each ligand (top row) and compared to the response to BMP4 (bottom row) and to mixtures of each ligand with BMP4 (middle row). By comparing each vertical triplet we see that specific ligands combine with BMP4 in different ways, both synergistically and antagonistically. (C–E) Measurements of full input-output response profiles for specific ligand pairs. (C) BMP4 and BMP9 combine to increase pathway activity in an additive fashion. (D) BMP4 and GDF5 combine in a ratiometric manner. (E) BMP4 and BMP10 showed an ‘imbalance detection’ response. For each plot in C–E, the dashed outline indicates a set of ligand concentrations varying from high concentration of one ligand (top left corner) to high concentration of the other ligand through intermediate states containing both ligands (e.g. top right). In C–E, the bottom row and left column correspond to an absence of the indicated ligand. (F–H) The responses along this contour are plotted for each ligand combination in C–E. These plots show that each pair shows a different dependence on ligand ratio. The logarithmic levels of each ligand are indicated schematically by the heights of the blue/green bars along the x-axis. Error bars indicate standard deviation calculated from at least 3 experiments. See also Figure S1 and S2 and Tables S1 and S4.

Many individual ligand pairs generated stronger or weaker responses than expected given their individual effects (Fig. 2A, S1G, H). For example, BMP3 combined antagonistically with almost every other ligand. Furthermore, some individual ligands combined in qualitatively different ways with different ligands. For example, BMP7 and BMP4 each exhibited a mixture of antagonistic and synergistic interactions with other ligands. Overall, these results indicate that the effect of any given ligand on pathway activity can, in general, depend in complex ways on other ligands.

Higher resolution analysis reveals distinct multi-ligand response profiles

To gain a clearer view of multi-ligand responses, we analyzed the diverse ways in which BMP4, one of the best-studied BMP ligands, combines with other ligands (Fig. 2B), particularly BMP9, GDF5, and BMP10 (Fig. 2C–E). To quantitatively characterize these interactions in a manner independent of the choice of base concentrations, we analyzed a 2-dimensional matrix of logarithmically spaced ligand concentrations (Fig. 2C–E). The broad (3 orders of magnitude) concentration range covers the full input dynamic range for each ligand pair in the NMuMG cell line, and overlaps ligand concentrations in circulating blood (David et al. 2008), as well as those used to induce BMP dependent responses in vitro (Lavery et al. 2008; Heggebö et al. 2014; Hatsell et al. 2015).

Each of the three ligand pairs showed qualitatively distinct response profiles. BMP4 and BMP9 increased pathway activity both individually and in combination, exhibiting an additive response, with little dependence on ligand identity, as one would expect for ligands that function redundantly (Fig. 2C, S2A). By contrast, GDF5 reduced activation by BMP4 in a dose-dependent fashion, such that pathway output approximated the ratio of the concentrations of the two ligands (Fig. 2D, S2A). Similar ratiometric responses have been observed in other systems (Atkinson, 1968; Berg et al., 2009; Escalante-Chong et al., 2015; Madl and Herman, 1979). Finally, and most intriguingly, BMP4 and BMP10 were potent activators individually, but each became inhibitory in the presence of high concentrations of the other ligand, resulting in a weaker response when both ligands were present (Fig. 2E, S2A). Interestingly, in this mode, each ligand can play both activating and inhibitory roles. We termed this integration mode “imbalance detection” because it responds maximally to extreme ratios (imbalances) of the two ligand concentrations. We note that the same two-dimensional response profiles were observed using independently generated reporter cell lines, indicating that they do not reflect aspects of the chromatin configuration of a specific reporter integration site (Fig. S2C). The integration functions were also independent of the ligand supplier (Fig. S2D). Together, these results identify three distinct ways in which the BMP pathway can integrate ligand pairs.

Interestingly, these responses depend in distinct ways on the ligand composition, defined as the relative concentrations of the two ligands. To study this dependence, we plotted the response to varying relative ligand concentrations, at high total ligand concentration (Fig. 2C–E, dashed outline). These contours reveal that pathway activity is independent of ligand composition in the additive case (Fig. 2F), monotonically dependent in the ratiometric case (Fig. 2G), and non-monotonically dependent in the imbalance detection case, where pathway activity declines near a specific intermediate ligand ratio (Fig. 2H). Similar composition-dependence can be observed at lower ligand concentrations. The only exception is imbalance detection, which at ligand concentrations <10ng/ml becomes indistinguishable from the additive response (Fig. 2C). These results indicate that the BMP pathway implements a diverse set of computations on multi-ligand inputs in NMuMG cells, and can be strongly controlled by ligand composition, as well as absolute ligand concentration.

Response profiles emerge rapidly and are stable

We next asked at what level and over what timescales these response profiles emerge. First, to access an earlier and more direct readout of pathway activity we measured Smad1/5/8 phosphorylation at 20 minutes after stimulation with select ligand pairs, using immunostaining (Fig. 3A) and Western blots (Fig. S2E, F). Both measurements revealed qualitatively similar response profiles as the fluorescent reporter, indicating that computations emerge within 20 minutes, and can be observed at the level of Smad phosphorylation. We note that Erk1/2, a non-canonical output (Nohe et al. 2004), did not respond to BMP stimulation in this cell context (Fig S2G).

Figure 3.

Combinatorial ligand response profiles emerge rapidly, persist for long times, and do not require co-factors. (A) phospho-Smad immunostaining reveals responses to BMP4-BMP9, BMP4-GDF5, and BMP4-BMP10 ligand combinations 20 minutes after ligand addition. (B) The dynamical response to mixtures of BMP4 and BMP10 is plotted over 70 hours after addition of the ligands. Data is normalized at each time point to the response of cells treated with BMP4 only. (C) Expressed BMP modifiers, identified in RNAseq (see Table S2) were depleted from NMuMG using siRNA. The relative expression levels of Fst, RGMb and Twsg1 were measured using qPCR in cells transfected with the corresponding siRNA (blue) normalized to their levels in cells transfected with a random siRNA (grey). (D–G) After depletion by random siRNA (D), Fst siRNA (E), RGMb siRNA (F) or Twsg1 siRNA (G), cells were treated with varying levels of BMP4 and the indicated ligand to assess their potential effect on combinatorial ligand response profiles. See also Figure S2 and S3 and Tables S2–4.

Next, to better understand the dynamics of the BMP response, we used time-lapse imaging to track reporter expression over time in response to BMP4 and/or BMP10 (Fig. 3B, S3A). The imbalance detection response could be identified by 6h and persisted for more than 96h. However, the relative level of activation caused by the combination of BMP4 and BMP10, compared to the individual ligands, remained constant (Fig. S3B), indicating that the imbalance detection response profile is stable over extended periods.

Feedback loops and pathway modulators

We next asked whether known feedback loops in the BMP pathway were necessary for the observed computations. The negative pathway regulator Smad6 is a downstream target of BMP (Fig S1B) (Li et al. 2003; Afrakhte et al. 1998). However, knock-down of Smad6 did not qualitatively change the shape of the response profiles (Fig. S3C–E). Another reported feedback involves up-regulation of Bmpr2 in response to BMP9 stimulation (Long et al. 2015). Addition of 400ng/ml BMP9 generated a 2-fold increase in Bmpr2 expression (Fig. S3F). However, even this relatively modest effect appeared only at ~12 hours, consistent with the timescales of transcriptional regulation, and too late to explain the appearance of the computations at earlier timepoints. These results suggest that these feedback loops are not required for the computations observed here, although feedbacks may play other roles in enhancing the amplitude or dynamics of the pathway over longer timescales.

The BMP pathway utilizes many secreted and surface-bound modulators to shape the spatial distribution of available ligands. To test whether these factors play a role in ligand integration, we first determined which ones were expressed in the NMuMG cell line (Table S2). Individually depleting each of these factors using siRNAs (Fig. 3C) did not affect the type of response profile generated by BMP4 in combination with BMP9, BMP10, or GDF5 (Fig. 3D–G). In addition, BMPs could interact more generally with heparan sulfate proteoglycans (HSPGs). Enzymatically perturbing HSPGs with heparinase or inhibiting their biosynthesis with CaClO₃ showed minimal effects on the response of the pathway to BMP combinations (Fig S3G–J). Together, these results suggest that, while these modulators play key roles in other aspects of BMP signaling, they are not required for the observed multi-ligand computations. These results are, however, consistent with computations emerging directly from receptor-ligand interactions.

A minimal model of promiscuous receptor-ligand interactions

To understand how receptor-ligand interactions could generate the observed complex ligand integration modes, we constructed a simplified mathematical model that incorporates two key features of the BMP pathway: the bipartite structure of active BMP receptors complexes (i.e. the requirement for both type I and type II receptors), and promiscuous, competitive receptor-ligand interactions (Fig. 4A, B) (Heinecke et al., 2009; Massagué, 1998; Mueller and Nickel, 2012; Nickel et al., 2009; Vilar et al., 2006). The model considers a set of ligands, denoted L_j, and two types of receptors, denoted A_i and B_k, analogous to the BMP type I and type II receptor subunits, respectively. Each ligand can bind with affinity $K_{ij}^D$ to an A-type receptor to form a dimeric complex D_ij, which in turn can bind a B-type receptor with affinity $K_{\mathit{ijk}}^T$ to form an active trimeric complex, T_ijk. Because the affinity parameters can differ for each ligand-receptor combination, this model allows both receptor preferences as well as promiscuous interactions. Each trimeric complex phosphorylates Smad proteins at a distinct rate, or activity, denoted ε_ijk, to produce an overall output signal S at steady state. The model considers the experimental regime of large extracellular volume, but similar conclusions occur at finite volume (Supplementary Text). Here, we focus on the minimal case of 2 ligands, 2 A-type receptors and 2 B-type receptors, whose behavior can be specified by 16 independent biochemical parameters and 4 receptor expression levels, and which is sufficient to explain the present experimental observations.

Figure 4.

Mathematical modeling shows that combinatorial receptor-ligand interactions generate a specific repertoire of computational functions. (A) Schematic representation of ligands (top row), type A receptors (second row), type B receptors (third row), intermediate complexes (fourth row), and signaling complexes (fifth row), as described in the text. Only a subset of possible complexes is shown for simplicity. Colored lines highlight interactions involved in the formation of a single signaling complex, with corresponding parameters indicated. (B) Reactions (left) and corresponding steady-state equations (right) for the model. (C) With 2 ligands, and 2 variants of each receptor type, the model produces a variety of different signal processing behaviors. Each point represents the behavior of one randomly chosen parameter set. The x-axis represents the type and strength of interference between the ligands, from antagonism (negative values) to synergy (positive values). The y-axis represents the relative strength of the two ligands individually, as defined in Fig. S4B and Supplementary Text. Most parameter sets generate computations that fall within a triangular region, while some show more extreme phenotypes. The four archetypal computations, shown in D–G, are indicated by colored dots. (D–G) The four archetypal computations are shown (top) together with corresponding profiles showing pathway activity as a function of ligand ratio, as in Fig. 2F–H (bottom). See also Figure S4.

This simplified model omits several known features of the BMP pathway, such as variations in the sequence of binding reactions (Gilboa et al., 2000; Rosenzweig et al., 1995; Ventura et al., 1995), the hexameric nature of actual signaling complexes, as well as the roles of other BMP regulatory factors. These features likely play important biological roles, e.g. in controlling the amplitude and spatio-temporal dynamics of signaling, that should be considered in models of specific biological processes. However, incorporation of these additional features in the model does not change the types of inputoutput computations examined here (Supplementary Text).

Archetypal functions define the range of response profiles

To explore the range of integration modes produced by the model, we computed the input-output behavior of the system for 100,000 random parameter sets (Fig. S4A). The model produced a repertoire of computational response profiles, which included additive, ratiometric, and imbalance detection. To more quantitatively characterize this repertoire, we defined two features that together capture key aspects of the shape of the response profiles (Fig. S4B–D). First, we defined the Relative Ligand Strength (RLS) to quantify the asymmetry in pathway activity generated by the ligands individually. The RLS is defined as the ratio of pathway activity produced by the weaker ligand to that produced by the stronger ligand. Second, we defined the Ligand Interference Coefficient (LIC) to quantify the degree to which the two ligands positively or negatively synergize (see Supplementary Text). The LIC is defined by the deviation of pathway activity in mixed ligand environments beyond the range of the responses in single ligand environments.

When plotted in this two-parameter phenotypic space, the simulated systems occupied a continuous region that loosely conformed to an inverted triangle (Fig. 4C, S4E). Two vertices of the triangle strikingly resembled the ratiometric and imbalance detection functions observed experimentally (cf. Fig. 2C–E and Fig. 4D–F). The third vertex, occurring for ligands with a relative ligand strength of 1 and a positive ligand interference coefficient, represented a new predicted behavior, which we termed ‘balance detection’ because it shows a maximal response when both ligands are present at a specific ratio. All other functions, including the additive interaction at RLS=1, LIC=0 (top middle, Fig. 4C), interpolated between these three archetypal functions (Fig. S4E) (Hart et al., 2015; Tendler et al., 2015). The archetypal functions identified here differ from standard Boolean logic, since they depend asymptotically on ligand ratios, rather than absolute concentrations (Supplementary). These conclusions remain qualitatively similar if one considers a finite extracellular volume (Supplementary Text, Fig S5A, B). This analysis provides an intuitive way to understand the distribution of response profiles.

To better characterize the distribution of response profiles, we quantified the percentage of occurrences of each response type in regions around each of the archetypal responses (Fig. S5C, D). All archetypal behaviors occurred, whether parameters were chosen from a full range of values, or restricted to a biologically relevant range (Supplementary). However, parameters in the biological range showed an enrichment for the additive, balance detection, and imbalance detection response profiles (Fig. S5E). We further note that natural biological parameters could have been selected by evolution for functionality, including the ability to generate balance or imbalance detection. Together, these results show that this minimal model can generate the full range of observed response profiles for biologically reasonable parameter values.

Complex response profiles emerge from the interplay of receptor-ligand affinities and activities

We next asked how the archetypal ligand integration modes arise within the model. To do so, we analyzed the corresponding parameter regimes in more detail (Supplementary, Fig. S6). As expected, additive responses occur when the two ligands are approximately equivalent, forming signaling complexes with similar phosphorylation activities (ε_i1k~ε_i2k, Fig. 5A, S6A). By contrast, ratiometric behaviors occur when signaling complexes containing one ligand have higher activities than those containing the other (ε_i1k ≪ ε_i2k), such that a weaker ligand competitively inhibits activation by the other, stronger ligand (Fig. 5B, S6B). Imbalance detection occurs when each receptor preferentially binds to a distinct ligand with which it forms a less active signaling complex (Fig. 5C, S6C). When only one type of ligand is present, it can bind both receptors, forming signaling complexes with both higher and lower activity. When both ligands are present, the affinity preferences cause ligands and receptors to self-sort, and predominantly form less active signaling complexes, reducing total pathway activity (Fig. 5E). Finally, balance detection occurs through a similar mechanism, except that the relative affinities are reversed, favoring formation of more active signaling complexes when both ligands are present (Fig. 5D, S6D).

Figure 5.

The four computational archetypes (cf. Fig. 4D–G) arise through the interplay between interaction affinities and complex activity. Representative parameter regimes producing each of the four archetypes are indicated schematically. Upper and middle arrow thicknesses indicate the affinities $K_{ij}^D$ and $K_{\mathit{ijk}}^T$, respectively. Lower arrow thicknesses indicate the phosphorylation rate of each signaling complex ε_ijk. (A) When two ligands are equivalent (similar arrow thicknesses), they combine additively. (B) When different ligands generate different levels of activity in complex with the same receptors (thin vs. thick bottom arrows), the less active ligand (blue) competitively inhibits the more active ligand (green), leading to ratiometric behavior. (C, D) Imbalance and balance detection regimes occur when affinity and activity parameters enable ligands to preferentially form less active (C), or more active (D), complexes, respectively. (E) For example, in the parameter regime corresponding to the imbalance detection, cells exposed only to a single ligand species (i.e. only blue or green ligands) produce a mixture of strong and weakly active complexes (left, right), but cells exposed to mixtures of the two ligands predominantly form weakly active complexes (middle), leading to the imbalance detection behavior. See also Figure S6.

A critical feature of the model is that the overall activity of the pathway depends not only on how much of each ligand is complexed with receptors, but also on how that ligand is distributed across the range of distinct possible receptor complexes. In the model, simply changing the activities of the complexes can result in completely different response profiles (Fig. S4F, G). As a result, addition of a second ligand can change not just the amount of the first ligand that is bound to receptors, but more importantly the distribution of that ligand across different potential signaling complexes with distinct activities. This could explain how two ligands can exhibit similar receptor preferences but still combine in qualitatively different ways with a third ligand.

Taken together, these results indicate that promiscuous receptor-ligand binding interactions are sufficient to produce a diverse repertoire of specific multi-ligand response profiles, including those observed experimentally. They reveal how the full functional repertoire can be understood as interpolating among three archetypal functions (ratiometric, imbalance detection, and the predicted balance detection function). Finally, they show how these functions arise through specific relations between the affinity parameters that control what receptor complexes will form, and the activity parameters that control how the resulting signaling complexes contribute to the cellular response. Thus, as suggested experimentally, the full spectrum of observed computations require only the ability of receptors and ligands to compete to form a variety of distinct signaling complexes, and differences in the relative activities of those complexes. Despite its simplicity, this system allows for remarkable computational diversity.

Receptor expression reprograms ligand response profiles

Within an organism, different cell types generally express receptors at different levels. Changes in receptor expression could in principle alter BMP responses in similar, or different, ways compared to changes in ligand concentrations. To gain insight into the possible role of receptor expression in pathway computations, we varied receptor expression levels in the model, while holding the biochemical parameter values ( $K_{ij}^D,K_{\mathit{ijk}}^T$, ε_ijk) fixed. We repeated this analysis for different biochemical parameter sets. In these simulations, some biochemical parameter sets produced only a limited range of ligand integration modes (Fig. 6A, left), while others were more versatile, capable of generating a diverse range of computations as receptor expression levels were varied (Fig. 6A, right). The existence of such versatile parameter sets in the model suggests the hypothesis that different cell types, by expressing different receptor profiles, might compute different responses to the same ligands.

Figure 6.

Receptor expression controls computations. (A) Comparison of two simulated biochemical parameter sets (see Table S5 and Methods for parameter values). For each set, multiple receptor expression profiles are plotted (individual dots). Dot color indicates the most similar archetype (cf. Fig. 4C). For one parameter set (non-versatile, left), receptor expression only weakly affected computation. For the other parameter set (versatile, right), variation in receptor expression generates the full range of possible computations. (B) BMP receptor expression profiles for three cell lines. Bars indicate expression levels of each receptor (FPKM). Error bars represent standard deviation of three independent biological replicates. (C–E) Computation correlates with receptor expression pattern for three ligand pairs. Each column shows the response to the same pair of ligands for the indicated pair of ligands. Note the qualitative change in function between mESCs (bottom) and the other cell lines. Line colors refer to closest archetype (as in Fig. 4C). (F–H) Perturbing receptor expression level reprograms computations in NMuMG cells. Wild type cells (black points) were compared to cells with perturbed receptor expression (white points). Specific receptor perturbations are indicated next to each line, with up and down arrows indicating overexpression and siRNA, respectively. In C–H, error bars indicate standard deviation of at least 3 replicates. See also Figure S7 and Tables S3–5.

If the BMP pathway exhibits and utilizes such versatility, cell lines with different receptor expression profiles could show distinct response profiles for the same ligands. To test this hypothesis, we compared the response of NMuMG cells to E14 mouse embryonic stem (ES) cells, which express less Bmpr2 and Acvr1 and more Acvr2b (Fig. 6B). As a control, we also analyzed NIH-3T3 fibroblasts, which had similar receptor expression to NMuMG cells (Fig. 6B). The ES cells indeed exhibited different response profiles compared to NMuMG for the same ligands (Fig. 6C–E, S7A). Most strikingly, BMP4 and BMP9 integrated in an additive fashion in NMuMG and NIH-3T3, but showed the balance detection archetype in ES cells (Fig. 6C). Other ligand pairs were also integrated similarly in NIH-3T3 and NMuMG, but differently in ES cells (Fig. 6D–E). Together, these results show that cell lines differ qualitatively in their ligand integration modes, in a manner that correlates with their receptor expression profiles, as predicted by the model. Furthermore, these results also validate the model prediction of balance detection (Fig. 4G).

Reprogramming response profiles by direct manipulation of receptor expression levels

Finally, to test whether changes in receptor expression are sufficient to reprogram computations, we directly perturbed receptor expression in NMuMG cells. Depletion of the most highly expressed type II receptor in this cell type, Bmpr2, with siRNA, changed the BMP4-BMP9 response from additive to ratiometric (Fig. 6F, S7C). This indicates that BMP4 activates the pathway predominantly through Bmpr2. By contrast, BMP9 can activate through other type II receptors, but BMP4 can effectively inhibit such Bmpr2-independent BMP9 signaling.

As a second example, ectopically expressed Bmpr1b, which is known to mediate GDF5 signaling (Nishitoh et al., 1996), enabled GDF5 to activate, rather than inhibit, the pathway, and thereby reprogrammed the ratiometric BMP4-GDF5 interaction to an additive one (Fig. 6G, S7C). Furthermore, combining ectopic Bmpr1b to enable GDF5 signaling with depletion of Bmpr2 to reduce BMP4-dependent signaling inverted the ratiometric response (Fig. 6G, S7C).

Third, we asked whether we could reprogram imbalance detection between BMP4 and BMP10 (Fig. 2E). In the model, imbalance detection results from ligand competition for receptors. To alleviate this competition, we ectopically expressed the Alk1 receptor, which is known to mediate BMP9 and BMP10 signaling (David et al., 2007). This perturbation indeed removed competition, generating the predicted additive response (Fig. 6H, S7C).

Taken together, these results show that receptor expression levels directly control computations, and demonstrate that this effect enables rational manipulation of ligand integration modes using insights from the model (Fig. S7B).

Discussion

Our results show that promiscuous BMP receptor-ligand interactions enable cells to perceive information encoded in combinations of ligands (Fig. 7). They do so through a specific set of computations over the multi-dimensional space of ligand concentrations, with the computations performed on a given set of ligands depending on the repertoire of receptors the cell expresses. These computations interpolate between archetypes loosely analogous to addition (additive, Fig. 4D), subtraction (imbalance detection, Fig. 4F), multiplication (balance detection, Fig. 4G), and division (ratiometric, Fig. 4E). This indicates that cells do not, in general, perceive ligand abundance, but rather perceive specific functions of ligand combinations.

Figure 7.

Schematic illustration of computational plasticity in the BMP signaling system (cf. Fig. 1D, E). Ligand combinations represent inputs to the pathway, which processes them through receptor-ligand interactions to control the expression level of downstream target genes. In this scheme, a given receptor configuration can perform different computations on different ligand combinations (e.g. additive and imbalance, top panel), whereas cells expressing different receptor profiles can perform distinct computations on the same combination of ligands (e.g. ratiometric and additive, lower panel).

This system provides several key capabilities for cells: First, it is sensitive to both absolute concentrations of individual ligands and their relative concentrations. Encoding signals in relative ligand concentrations can increase robustness to variations in ligand accessibility, cell surface area, and other properties that affect all ligands in a correlated way. Second, computation is integrated with sensing. The system performs computations on ligand concentrations directly through competitive binding interactions, at steady state, without requiring regulatory cascades or transcriptional feedback loops. The observed computations arise because affinities among components need not correlate with the activities of the resulting signaling complexes. This allows ligands to compete for receptors to form a variety of distinct signaling complexes with distinct efficiencies. Third, and most intriguingly, this system possesses computational plasticity. By controlling the abundance of different receptor variants, a cell can control which computations it performs, and thus what features of the ligand environment it responds to. These capabilities could enable non-intuitive operative modes. For example, the use of ligand combinations may offer the ability to selectively activate a given cell type, since different cell types may respond to specific ligand combinations. Temporal changes in the concentration of a single ligand could elicit different, or opposite, changes in signal perception in distinct cell types.

These results should improve our ability to understand and manipulate natural BMP-dependent processes. For example, efficient primordial germ cell differentiation was shown to require a combination of both BMP4 and BMP8b homodimers, provoking the question of whether these ligands are integrated through balance detection (Ying et al., 2001). Conversely, BMP2 and BMP7 show opposing effects on ureter branching in developing kidneys (Piscione et al., 1997), suggesting they may operate in a ratiometric mode, and similar interactions were recently reported for BMP2 and GDF5 in multiple contexts (Klammert et al., 2015; Liu et al., 2016). The framework described here can be used to analyze these and other specific biological processes that utilize multiple BMPs (Açil et al., 2014; Bandyopadhyay et al., 2006; Chen et al., 2013). In the context of disease, many therapeutic strategies have focused on using a single ligand to treat conditions such as bone injuries and abnormalities, arthritis, diabetes, vascular conditions, obesity, and cancer (Kim and Choe, 2011; Wang et al., 2003). Similarly, directed differentiation approaches in regenerative medicine often rely on a single BMP ligand. However, ligand combinations may provide more potent, and specific, control in these contexts.

Further work on higher dimensional combinations of ligands, as well as investigation of the effects of diffusible inhibitors such as Noggin and Chordin, will help to provide an understanding of systems like kidney development that depend on many different ligands, receptors, and modulators expressed in spatially and temporally overlapping patterns (Simic and Vukicevic, 2005). Similarly, quantitative analysis of receptor expression states will help elucidate the specific combination of ligands that each cell type senses. Experimental measurements of the effective parameter values for each specific molecular component in the BMP pathway could enable a more direct, quantitative, and predictive modeling framework. At a finer level, within a single cell type or state, fluctuations, or “noise,” in receptor expression could affect how cells perceive ligand combinations. However, within the model, typical levels of receptor expression noise show only mild effects (Fig. S7D–H, and Supplementary text).

Finally, further analysis could reveal additional computations beyond those described above. Extending the model to include the TGF-β ligands could be used to understand newly discovered ligand-level competition between the two branches of this signaling pathway (Hatsell et al., 2015; Lowery et al., 2015; Olsen et al., 2015) and incorporate them into a single framework. More generally, Wnt, FGF, JAK-STAT, Eph-Ephrin, and other signaling pathways also exhibit promiscuous interactions between multiple ligand, receptor and co-receptor variants and may function according to similar principles (Jørgensen et al., 2009; Murray, 2007; Wodarz and Nusse, 1998). Future elucidation of the principles of programmable computation through promiscuous receptor-ligand interactions could be used to engineer precise multicellular behaviors for synthetic biology and tissue engineering applications.

STAR Methods

CONTACT FOR REAGENT AND RESOURCE SHARING

Further information and requests for resources and reagents should be directed to and will be fulfilled by the Lead Contact, Michael B. Elowitz (melowitz@caltech.edu).

EXPERIMENTAL MODEL AND SUBJECT DETAILS

Tissue culture and cell lines

NMuMG (NAMRU Mouse Mammary Gland cells, female) and NIH3T3 (mouse fibroblast, male) cells were acquired from ATCC (CRL-1636 and CRL-1658, respectively). E14 cells (mouse embryonic stem cells, E14Tg2a.4, male) were obtained from Bill Skarnes and Peri Tate. All cells were cultured in a humidity controlled chamber at 37°C with 5% CO₂. NMuMG cells were cultured in DMEM supplemented with 10% FBS (Clonetech #631367), 1mM sodium pyruvate, 1unit/ml penicillin, 1ug/ml streptomycin, 2mM L-glutamine and 1X MEM non-essential amino acids. NIH-3T3 cells were cultured in DMEM supplemented with 10% CCS (Hyclone #SH30087), 1mM sodium pyruvate, 1unit/ml penicillin, 1ug/ml streptomycin and 2mM L-glutamine. ES cells were plated on tissue culture plates pre-coated with 0.1% gelatin and cultured in a standard pluripotency-maintaining conditions (Smith, 2001) using DMEM supplemented with 15% FBS (ES qualified, Gibco #16141), 1mM sodium pyruvate, 1unit/ml penicillin, 1ug/ml streptomycin, 2mM L-glutamine 1X MEM non-essential amino acids 55mM β-mercaptoethanol and 1000 Units/ml leukemia inhibitory factor (LIF).

Sensor cell lines construction

Construction of the reporter cell lines was carried out via random integration of a plasmid harboring the BMP response element (BRE) (Korchynskyi and ten Dijke, 2002) in the enhancer region of a minimal CMV driving the expression of an H2B-Citrine protein fusion. ES cells were transfected using the FugeneHD reagent. NMuMG and 3T3 cells were transfected using Lipofectamine LTX. After transfection, cells were selected with 100 ug/ml hygromycin. All experiments were performed with clonal populations, generated via colony picking (ES) or limiting dilutions (NMuMG, NIH3T3). To ensure results were not dependent on the specific reporter integration site, an independent BRE-reporter cell line was generated using Piggybac integration (SBI) (see Fig. S2C).

METHOD DETAILS

BMP response and flow cytometry

Sensor cell lines were plated at 40% confluency in 96 well plates and cultured under standard conditions (above) for 12h. Media was then replaced and ligand(s) were added at specified concentrations. 24h after ligand addition cells were prepared for flow cytometry in the following way: Cells were washed with PBS and lifted from the plate using either 0.05 ml Accutase (ES cells) or trypsin (NMuMG and 3T3 cells) for 5 minutes at 37°C. Protease activity was quenched by re-suspending the cells in HBSS with 2.5mg/ml Bovine Serum Albumin (BSA). Cells were then filtered with a 40μm mesh and analyzed by flow cytometry (MACSQuant VYB, Miltenyi). All recombinant BMP ligands were acquired from R&D Systems (Table S1), with the exception of Figure S2D where BMP4, BMP10 and GDF5 were acquired from Peprotech.

Ligand integration survey

In order to identify non-additive ligand integration modes, cells were exposed to a matrix of ligands at predetermined concentrations. We selected concentrations that were sufficient to induce responses in cells already known to respond to those ligands, but not so high as to induce potential non-specific responses. For this reason, we based ligand concentrations on supplier data, and selected a concentration at the high end of the input dynamic range for a cell based system susceptible to each ligand (see Table S1). All BMP ligands used in the survey were acquired from R&D Systems (see Table S1 for more information).

SDS-PAGE and immunoblotting

Phoshpo-Smad 1/5/8

For assessment of phoshpo-Smad1/5/8 cells were plated at 40% confluency under standard conditions in 24 well plates. To reduce phospho-Smad1/5/8 background activity, cells were transferred to reduced serum media containing 1.0% FBS for 12 hours. This media was then exchanged for DMEM and cells were incubated at 37°C for another 6 hours. DMEM was then replaced and ligands were added in DMEM at the specified concentrations and incubated at 37°C for 20 minutes. Cells were then treated with 50 μl lysis buffer (Cell Signaling 9803) with the following additions, 0.1M DTT, 50mM NaF, 1mM PMSF and additional protease inhibitors (Thermo 87785). Samples were immediately stored at −80°C until processed for SDS-PAGE. SDS-PAGE was conducted using NuPAGE Bis-Tris Mini Gels 4–12% (Thermo). Approximately 10–20 μg of total protein, denatured by heat, was loaded per well. Samples were run at 50mA for approximately 60 minutes. Protein was transferred from gels to nitrocellulose using the iBlot apparatus and iBlot reagents (Thermo) and program 2 for 8 minutes. Membranes were trimmed and blocked with 5% milk in Tris buffered saline with 0.1% Tween 20 (TBST) for at least 60 minutes at room temperature. Blocking buffer was removed and membranes were briefly washed with TBST. Antibodies against phospho-Smad 1/5/8 (13820 Cell Signaling), phospho-p44/42 MAPK (4370 Cell Signaling), Smad1 (6944 Cell Signaling), GAPDH (2118 Cell Signaling) were than applied at a dilution of 1:1000, 1:2000 for GAPDH, in 1.0% BSA TBST and incubated at 4 ° C for 12 to 16 hours. After incubation with primary antibody, immunoblots were washed with TBST three times for 5 minutes at room temperature and a secondary antibody conjugated with horse radish peroxidase (7074 Cell Signaling) was applied to the blots at 1:1000 in 1.0% BSA TBST for 60 minutes at room temperature. After incubation with the secondary antibody, the immunoblots were washed with TBST three times for 5 minutes and developed using a luminol based substrate (7003 Cell Signaling). The immunoblots were imaged using a BioRad and exposure times that produced signal below saturation. Densitometry was performed using ImageJ (http://imagej.nih.gov).

Bmpr2

For assessment of Bmpr2 protein expression after addition of select BMP ligands, cells were plated at 40% confluency under standard conditions in 24 well plates. Media was replaced, with addition of BMP9 (400ng/ml), and cells were then incubated at 37°C for the specified times. Cells were then treated with 50μl lysis buffer (see above). Samples were immediately stored at −80°C until processed for SDS-PAGE. After electrophoresis, gels were incubated with 20% Ethanol in TBS for 5 minutes. Transfer of protein to nitrocellulose was performed with the iBlot apparatus using program 3 for 8 minutes. Antibodies against BMPR2 (6979 Cell Signaling) and GAPDH (2118 Cell Signaling) were than applied at 1:1000 and 1:2000, respectively, in 1.0% BSA TBST and incubated at 4°C for 12 to 16 hours. Immunoblots were processed, developed and analyzed as described above.

BMP response with heparinase I/III

Cells were plated at 40% confluency in 96 well plates and cultured under standard conditions for 12 hours. Media was exchanged with media containing 2 units of Heparinase I/III (H3917 SIGMA) and cells were incubated at 37° C for 3 hours. Media was then replaced with media containing ligands at the specified concentrations. The cells were then incubated with ligands at 37° C for 20 minutes. After incubation for 20 minutes the cells were processed for phoshpo-Smad 1/5/8 staining and flow cytometry as described above.

BMP response with NaClO3

Sensor cells were plated at 40% confluency under standard conditions including 20 mM NaClO3 (Sigma) and passaged 36 hours later at 40% confluency in 96 well plates under the same conditions and cultured for another 12 hours. Media was then replaced and ligands were added at the specified concentrations. The cells were then incubated with ligands at 37° C for 24 hours and were processed for flow cytometry as previously described.

Receptor over-expression

Overexpression plasmids were constructed for each of the BMP receptors (Bmpr1a, Bmpr1b, Bmpr2, Acvr1, Acvr2a, Acvr2b and Alk1) using the Gibson cloning method (Gibson et al., 2009). Bmpr1b and Alk1 cDNA was purchased from Dharmacon (MMM1013-202858407 and MMM1013-202763719). All other receptor cDNAs were generated by RT-PCR from total RNA extracted from NMuMG cells. The receptor cDNA was concatenated with mTurquoise with an intervening T2A cleavage site (Szymczak and Vignali, 2005), and was expressed under the control of a constitutive PGK promoter integrated in the Pb510b plasmid backbone to enable PiggyBac integration (Ding et al., 2005; Wu et al., 2006). Stable integrations were then generated using the PiggyBac method. Cells were co-transfected with these overexpression plasmids and PB200A to express transposase, and selected with Geneticin. Experiments were performed with polyclonal populations resulting from PiggyBac integrations.

siRNA induced knock-down

Cells were plated at 40% confluency with 30μM total siRNA (ThermoFisher silencer select #4390771) and 3μl RNAiMAX (Life technologies). For every gene, a pool of two distinct siRNA were used, listed in Table S3. Cells were passaged after 24h and were used for the relevant experiments.

Quantitative PCR (qPCR)

Total RNA was harvested from cell lysate using the RNAeasy mini kit (Qiagen) and cDNA was generated from one microgram of RNA using the iScript cDNA synthesis kit (BioRad) following the manufacturer’s instructions. Primers and probes for specific genes (Table S4) were purchased from IDT. Reactions were performed using 1:40 dilution of the cDNA synthesis product with either IQ SYBR Green Supermix or SsoAdvanced Universal probes Supermix (BioRad). Cycling was carried out on a BioRad CFX96 thermocycler using an initial denaturing incubation of 95° for 3 minutes followed by 39 cycles of (95°C for 15 seconds, followed by 60°C for 30 seconds). Each condition was assessed with two biological repeats and each reaction was run at least in triplicate.

Antibody detection for phospho-Smad1/5/8

Cells exposed to specified concentrations of BMP4 for 24 hours were harvested from single wells of a 24 well plate using either 0.05 ml Accutase (ES cells) or trypsin (NMuMG and 3T3 cells). Protease activity was quenched by re-suspending the cells in 0.45 ml HBSS with 1.0% Bovine Serum Albumin (BSA). The cells were then pelleted, washed with 0.5 ml PBS and fixed by re-suspension in 0.5 ml of 4.0% formaldehyde for 5 minutes at room temperature. Following fixation, the cells were washed in 0.5 ml PBS and re-suspended in 0.5 ml PBS with 1.0% Triton X-100 for permeabilization. The cells were then washed with 0.5 ml PBS and re-suspended in blocking solution (PBS with 1.0% BSA and 0.1% Tween 20). Blocking was carried out for 30 minutes at room temperature. The cells were then pelleted and re-suspended in binding solution (PBS with 1.0% BSA) containing a 1:100 dilution of a primary antibody against the phosphorylated form of Smad1/5/8 complex (Cell Signaling Technologies Cat# 13820). The staining proceeded for 12–16 hours at 4°C with constant rocking. Afterwards, cells were washed with 0.5 ml PBS and re-suspended in binding solution containing a 1:500 dilution of a secondary antibody labeled with Alexa 594 (#A21207, ThermoFisher). Secondary detection proceeded for 60 minutes at room temperature with constant rocking. Finally, cells were then pelleted, washed with 0.5 ml PBS filtered with a 40 μm mesh and analyzed by flow cytometry.

Time lapse imaging

Fluorescent reporter cells were first mixed with an excess of non-fluorescent parental cells at a 1:9 ratio to simplify image segmentation and data extraction. Cells were then plated at 1.6·10⁴ cells/well in a 96 well plate equivalent roughly to 15–20% confluency. Cells were grown for 12 hours prior to ligand addition. Each position was imaged every hour starting from the addition of ligands until cells became confluent after about 60h. Images were then analyzed for the number of fluorescent cells and fluorescent signal level.

Mathematical Model for promiscuous interactions

Many signaling pathways comprise multiple ligand and receptor variants that interact promiscuously with one another, with varying affinities, to form many distinct signaling complexes. BMP provides a canonical example of this architecture. However, other pathways, including TGF-β (Smad2/3) signaling, FGF, Wnt, and JAK/STAT, also exhibit similar features. Here we develop a general mathematical model that captures essential aspects of receptor-ligand promiscuity in signaling pathways, and analyze it to understand the functional capabilities this architectural feature provides for cellular signal processing. This model focuses on several features of the natural BMP pathway: promiscuous ligand-receptor interactions, heterodimeric receptors (a simplified version of the natural Type I-Type II receptor tetramers), and variation in the activities of different signaling complexes. To focus on the signaling processing capabilities at the level of receptor-ligand interactions, we neglect other known features of the pathway including preliminary enzymatic processing of ligands, non-canonical signaling, downstream feedback loops (e.g. through Smad6/7), and crosstalk with other signaling pathways. We specifically point out that while this model focuses on mixture of ligand species, each ligand type is composed of two subunits. Thus the model can be used equally well for mixtures of homodimers, heterodimers, or combinations thereof. Finally, we note that while the model applies most directly to the BMP pathway, variants of it could also describe other systems that similarly form multi-part signaling complexes, including receptor aggregates, such as those listed above.

We consider a system with n_L ligands, L_j, each of which can bind to one of n_A type A receptor sub-units, A_i, to form n_L·n_A intermediate dimeric ligand-type A receptor complexes, D_ij. These complexes can in turn bind to one of n_B type B receptor sub-units, B_k, to form n_L·n_A·n_B different trimeric signaling complexes, T_ijk. We assume that the reactions are reversible and follow first-order reaction kinetics with forward reaction rates given by ${k_f}_{ij}^D$ and ${k_f}_{\mathit{ijk}}^T$ for the formation of dimeric and trimeric complexes, respectively, and with reverse reaction rates similarly given by ${k_r}_{ij}^D$ and ${k_r}_{\mathit{ijk}}^T$. These reactions can be summarized as follows

$$A_i+L_j\underset{{k_r}_{ij}^D}{\overset{{k_f}_{ij}^D}{\rightleftarrows }}{D}_{ij}$$ (1)

$$D_{ij}+B_k\underset{{k_r}_{\mathit{ijk}}^T}{\overset{{k_f}_{\mathit{ijk}}^T}{\rightleftarrows }}{T}_{\mathit{ijk}}.$$ (2)

Next, we can write the dynamical equations that describe these reactions:

$$\frac{d{L}_j}{dt}=\frac{1}{V}\sum _{i=1}^{n_A}\left(-{k_f}_{ij}^D{A}_i{L}_j+{k_r}_{ij}^D{D}_{ij}\right)$$ (3)

$$\frac{d{A}_i}{dt}=\sum _{j=1}^{n_L}\left(-{k_f}_{ij}^D{A}_i{L}_j+{k_r}_{ij}^D{D}_{ij}\right)$$ (4)

$$\frac{d{D}_{ij}}{dt}={k_f}_{ij}^D{A}_i{L}_j-{k_r}_{ij}^D{D}_{ij}+\sum _{k=1}^{n_B}\left(-{k_f}_{\mathit{ijk}}^T{D}_{ij}{B}_k+{k_r}_{\mathit{ijk}}^T{T}_{\mathit{ijk}}\right)$$ (5)

$$\frac{d{B}_k}{dt}=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\left(-{k_f}_{\mathit{ijk}}^T{D}_{ij}{B}_k+{k_r}_{\mathit{ijk}}^T{T}_{\mathit{ijk}}\right)$$ (6)

$$\frac{d{T}_{\mathit{ijk}}}{dt}={k_f}_{\mathit{ijk}}^T{D}_{ij}{B}_k-{k_r}_{\mathit{ijk}}^T{T}_{\mathit{ijk}}.$$ (7)

Here, L_j denotes the concentration of the ligand in a volume V, and A_i, B_k, D_ij and T_ijk are the absolute number of receptors and complexes on the cell surface. We assume here that production and consumption are in steady-state, enabling us to neglect the consumption of receptors and ligands by endocytosis. Subunits combine to form various complexes, however the principle of conservation of mass requires that the total number of each type of molecule remain constant:

$$L_j^0=L_j+\frac{1}{V}\left(\sum _{i=1}^{n_A}{D}_{ij}+\sum _{i=1}^{n_A}\sum _{k=1}^{n_B}{T}_{\mathit{ijk}}\right)$$ (8)

$$A_i^0={\mathrm{A}}_{\mathrm{i}}+\sum _{j=1}^{n_L}{\mathrm{D}}_{\text{ij}}+\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\mathrm{T}}_{\text{ijk}}$$ (9)

$$B_k^0=B_k+\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}{T}_{\mathit{ijk}},$$ (10)

where $L_j^0$ is the total ligand concentration and $A_i^0$ and $B_i^0$ are the total receptor levels. Finally, each complex T_ijk induces phosphorylation of the intracellular signal, S, at some rate ε_ijk so that the rate of change of the total signal is given by

$$\frac{dS}{dt}=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}-\gamma S.$$ (11)

We consider the case where the volume for the ligand is large such that there are significantly more ligand molecules than receptors, which can be expressed by V→∞. This reflects our experimental conditions where the ligands are dissolved within a large excess of cell culture media. With this assumption equations (3) and (8) decouple and become

$$L_j=L_j^0.$$ (12)

Additionally, since binding and unbinding occur on fast timescales (minutes (Heinecke et al., 2009)) compared to the timescales of reporter expression, we focused on the behavior of this system at steady state. In this regime, all time derivatives in equations (4–7) vanish and the system can be solved to give

$$D_{ij}=K_{ij}^D{A}_i{L}_j$$ (13)

$$T_{\mathit{ijk}}=K_{\mathit{ijk}}^T{D}_{ij}{B}_k$$ (14)

where we define $K_{\mathit{ijk}}^T\equiv {k_f}_{\mathit{ijk}}^T/{k_r}_{\mathit{ijk}}^T$, and $K_{ij}^D\equiv {k_f}_{ij}^D/{k_r}_{ij}^D$. Stronger affinity thus corresponds to larger values of the K′s. Similarly, setting equation (11) to zero we get

$$S=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}},$$ (15)

with ε_ijk≡ε_ijk/γ. Therefore, the final system of equations describing our model is given as follows

$$A_i^0=A_i+\sum _{j=1}^{n_L}{D}_{ij}+\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{T}_{\mathit{ijk}}$$ (16)

$$B_k^0=B_k+\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}{T}_{\mathit{ijk}}$$ (17)

$$D_{ij}=K_{ij}^D{A}_i{L}_j$$ (18)

$$T_{\mathit{ijk}}=K_{\mathit{ijk}}^T{D}_{ij}{B}_k$$ (19)

$$S=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}.$$ (20)

This system comprises a set of N_v = n_A + n_B + n_A(1 + n_B)n_L + 1 variables and N_p = n_A + n_B + n_A(l + 2 * n_B)n_L parameters.

Solving the steady state equations

In order to find the total signal, S, we first need to solve the system of equations to find T_ijk. Plugging (18) into (16) we find

$$\begin{gathered} A_i\left(1+\sum _{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}\right)=A_i^0-\sum _{j^{\prime }=1}^{n_L}\sum _{k^{\prime }=1}^{n_B}{T}_{i{j}^{\prime }{k}^{\prime }} \\ {A}_i=\frac{A_i^0-{\sum }_{j^{\prime }=1}^{n_L}{\sum }_{k^{\prime }=1}^{n_B}{T}_{i{j}^{\prime }{k}^{\prime }}}{1+{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}}. \end{gathered}$$ (21)

This can be used to solve for D_ij

$$D_{ij}=K_{ij}^D\frac{A_i^0-{\sum }_{j^{\prime }=1}^{n_L}{\sum }_{k^{\prime }=1}^{n_B}{T}_{i{j}^{\prime }{k}^{\prime }}}{1+{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}}{L}_j$$ (22)

which we can plug into (19) to get a coupled set of N = n_A · n_L · n_B quadratic equations for T_ijk

$$T_{\mathit{ijk}}=K_{\mathit{ijk}}^T{K}_{ij}^D\left(\frac{A_i^0-{\sum }_{j^{\prime }=1}^{n_L}{\sum }_{k^{\prime }=1}^{n_B}{T}_{i{j}^{\prime }{k}^{\prime }}}{1+{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}}\right)L_j\left(B_k^0-\sum _{i^{\prime }=1}^{n_A}\sum _{j^{\prime }=1}^{n_L}{T}_{i^{\prime }{j}^{\prime }k}\right).$$ (23)

From (23) we can obtain the signal S, using equation (20).

The error function and least square minimization

In order to solve Eq. 23, we minimize an error function, defined as follows:

$$E(T_{\mathit{ijk}})\equiv \sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\left[K_{\mathit{ijk}}^T{K}_{ij}^D\left(\frac{A_i^0-{\sum }_{j^{\prime }=1}^{n_L}{\sum }_{k^{\prime }=1}^{n_B}{T}_{i{j}^{\prime }{k}^{\prime }}}{1-{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}}\right)L_j\left(B_k^0-\sum _{i^{\prime }=1}^{n_A}\sum _{j^{\prime }=1}^{n_L}{T}_{i^{\prime }{j}^{\prime }k}\right)-T_{\mathit{ijk}}\right]}^2.$$ (24)

Here, E is a function of the complete set of T_ijk’s. It is always positive, being a sum of squares, and vanishes if and only if T_ijk is a solution to equation (23), which can now be written as

$$E(T_{\mathit{ijk}})=0.$$ (25)

This equation is now in a form that can be solved numerically for any given set of parameters via standard optimization methods such as MATLAB’s fmincon and lsqnonlin functions.

Dimensional reduction

The system of equations describing our model can be simplified by dimensional reduction, in which we redefine the variables to reduce the number of parameters and make the remaining parameters dimensionless.

First, we change the units of signal strength using a scaling factor, α.

$$\begin{gathered} S\to \alpha ·S \\ {\epsilon }_{\mathit{ijk}}\to \alpha ·{\epsilon }_{\mathit{ijk}}. \end{gathered}$$ (26)

By choosing a value of α = (Σ_i,j,kε_ijk)⁻¹, we can obtain units such that the phosphorylation rate constants for all complexes sum to 1:

$$\sum _{i,j,k}{\epsilon }_{\mathit{ijk}}=1.$$ (27)

Similarly, changing the receptor units by rescaling with a factor β gives rise to the following transformation:

$$\begin{gathered} A_i\to \beta ·A_i \\ {B}_k\to \beta ·B_k \\ {D}_{ij}\to \beta ·D_{ij} \\ {T}_{\mathit{ijk}}\to \beta ·T_{\mathit{ijk}} \\ S\to \beta ·S \\ {K}_{\mathit{ijk}}^T\to {\beta }^{-1}·K_{\mathit{ijk}}^T. \end{gathered}$$ (28)

By choosing $\beta ={\sum }_{i,j,k}{K}_{\mathit{ijk}}^T$ we effectively obtain units for the receptors and receptor complexes in which the $K_{\mathit{ijk}}^T$ sum to 1:

$$\sum _{i,j,k}{K}_{\mathit{ijk}}^T=1.$$ (29)

Finally, we can also independently choose new units for each individual ligand species:

$$\begin{gathered} L_j\to {\gamma }_j·L_j \\ {K}_{ij}^D\to {\gamma }_j^{-1}{K}_{ij}^D \end{gathered}$$ (30)

We can make these dimensionless by choosing ${\gamma }_j={\sum }_i{K}_{ij}^D$, such that for every j,

$$\sum _i{K}_{ij}^D=1.$$ (31)

Using these re-scaled variables and parameters, we can explore the complete parameter space by examining only parameter values satisfying equations (27), (29), and (31). These constraints reduce the number of independent parameters, N_p, by 2 + n_L.

The (2,2,2) model and parameter selection

In order to see what behaviors arise from the model of promiscuous interactions, we focused on a specific instantiation of the model with N_L = 2 ligands, N_A = 2 A-type receptors and N_B = 2 B-type receptors, which we describe as the (2,2,2) model. In this case there are 20 independent biochemical parameters, $K_{ij}^D{K}_{\mathit{ijk}}^T$ and ε_ijk, restricted by equations (27, 29, 31), and 4 receptor expression level parameters $A_i^0$ and $B_k^0$. In order to study all possible behaviors, random sets of parameters were chosen. We chose random biochemical parameters distributed uniformly over the bounded domains defined by equations (27, 29, 31), while the receptor parameters were chosen from a log-uniform distribution in the range [10⁻³,10³]. Simulations were performed for 100,000 random parameter sets, and an entire 2D input-output function, across 15×15 log-uniform ligand concentrations, was numerically computed for each set. Results are plotted in Figure S4A.

Phenotypical parameters

A useful representation of the modeling results can be achieved by extracting parameters that measure phenotypic characteristic of the computation performed for each of the parameter sets. In this study we focused on two such parameters, the relative ligand strength (RLS) and the ligand interaction coefficient (LIC), as diagrammed in Fig. S4B–D. More specifically, we define RLS as follows:

$$\text{RLS}=\frac{S_{\text{weak}}}{S_{\text{strong}}},$$ (32)

where S_strong and S_weak are the activation strengths of the pathway when induced by the stronger or weaker ligand individually, at saturating levels. In the case where one ligand is a potent activator while the second ligand is weak this index drops to 0. However, when both ligands individually activate the pathway to a similar extent, this index approaches 1.

LIC measures the effective interaction between the ligands at high ligand concentration (L_j ≫ 1). We define the satmax and satmin functions to be the maximal or minimal response, respectively, over varying ligand ratios, at saturating total ligand concentrations (Figure S4B–D). Using these functions, the ligand interference index can be defined as

$$\text{LIC}=\frac{\text{satmin}(S)}{S_{\text{weak}}}-\frac{S_{\text{strong}}}{\text{satmax}(S)}.$$ (33)

For non-interacting ligands, we expect that mixed levels of ligands produce responses that lie within the range defined by the effects of the individual ligands, in which case the coefficient vanishes. However, if a combination of ligands give rise to a stronger response than that of the stronger ligand individually, the first term generates a positive value for this index. On the other hand, if the response to a combination of ligands is smaller than the weaker ligand, the second term dominates, and the index becomes negative. The range for this index is therefore [−1, 1].

The four archetypes and the structure of parameter space

Plotting the values of LIC and RLS for each simulation, we find that the response profiles form a continuous distribution that interpolates between 4 archetypal computations (Figure 4C, S4E). These archetypal computations generally map to different regions in parameter space. Here we describe in more detail the parameter regimes that give rise to each of these archetypes.

Additive (Fig. S6A)

The additive integration can be thought of as the “default” computation. It occurs when both ligands have equivalent receptor affinities ( $K_{i1}^D~K_{i2}^D,K_{i1k}^T~K_{i2k}^T$) and they produce equivalently active complexes (ε_i1k ~ ε_i2k). In such a regime, similarly active complexes form regardless of whether one ligand, the other or both are present, and thus, the response does not depend on the composition of the ligands but only on the total ligand concentration in the environment. This simple computation can occur even when there is only a single receptor variant (n_A = n_B = 1).

Ratiometric (Fig. S6B)

Ratiometric computation occurs when the ligands produce signaling complexes with significantly different activity levels (ε_i1k ≫ ε_i2k). When L₁ is present, high activity complexes (T_i1k) form and the pathway is strongly activated. In contrast when only L₂ is present, only low activity complexes (T_i2k) form and the pathway is weakly activated. In a mixed environment L₂ will compete with L₁ for receptor binding, competitively inhibiting formation of the more active complexes, and thus reducing pathway activation below its maximal level in a ratiometric manner. Note that this computation can also occur with only a single receptor variant (n_A = n_B = 1), and that similar ratiometric sensing behaviors have been observed in other systems (Atkinson, 1968; Berg et al., 2009; Escalante-Chong et al., 2015; Madl and Herman, 1979).

Imbalance detection (Fig. S6C)

Imbalance detection can be thought as a combination of two opposing ratiometric computations, and thus requires the existence of at least two receptor variants, A₁ with ε_11k ≫ ε_12k and A₂ with ε_21k ≪ ε_22k. Moreover, receptor-ligand affinities should be such that each receptor preferentially binds to the ligand with which it forms the weaker complex, i.e. $K_{11}^D<K_{12}^D$ and $K_{22}^D<K_{21}^D$. In this regime, when only a single ligand is present, it can bind both type A receptors, leading to formation of both the more active and less active signaling complexes. However, when both ligands are present simultaneously, they compete for type A receptor, producing primarily the high affinity signaling complexes, which, in this regime, are precisely those with weaker activity. Thus, in this case, the ligands effectively reduce each other’s ability to activate the pathway.

Balance detection (Fig. S6D)

Balance detection occurs in a similar way as the imbalance modes, except that the affinities favor the formation of the more active signaling complexes. We still have ε_11k ≫ ε_12k and ε_21k ≪ ε_22k, as with imbalance detection. However, for this computation, $K_{11}^D>K_{12}^D$ and $K_{22}^D>K_{21}^D$, such that higher affinity receptor-ligand pairs now correspond to the higher activity complexes. Consequently, as before, a mixture of ligands will produce mostly the higher affinities complexes (T_11k, and T_22k), but now these complexes have higher, rather than lower, activity.

Furthermore, the balance detection effect can be enhanced by further reducing the activating effects of individual ligands. Consider the case where only L₁ is present. Lack of competition enables binding of L₁ to both type A receptors giving rise to D₁₁ and D₂₁. If affinities of these dimeric complexes for B-type receptors obey $K_{11k}^T<K_{21k}^T$ then there will be more trimeric complexes of the form T_21k than T_11k. Assuming the high activity complexes are similarly active to each other, ε_11k ~ ε_22k, then ε_21k ≪ ε_11k, and therefore, the enrichment for T_21k will tend to decrease the total signal, further enhancing the balance detection effect.

Computations depend only on ligand ratios at high ligand concentrations

A striking feature of the computations performed by the ligand-receptor interactions (Fig. S4A) is the appearance of diagonal contours in the log-log ligand space. These reflect a general dependence of output on ratios of the two ligands. In fact, this behavior is more general, occurring for any number of ligands and receptors, and can be understood from the model. This can be seen by examining equation (23), where the ligand dependence is entirely through the factor

$$\frac{L_j}{1+{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{L}_{j^{\prime }}}=\frac{1}{\frac{1}{L_j}+{\sum }_{j^{\prime }=1}^{n_L}{K}_{i{j}^{\prime }}^D{R}_{j{j}^{\prime }}}.$$ (34)

Here, on the right-hand side, we have introduced R_jj′ ≡ L_j′/L_j to represent the ratios between each pair of ligands. When ligand concentrations are large, L_j ≫ 1, the 1/L_j vanishes and the solution depends only on ratios between ligands and not on their absolute levels.

Archetypal computations differ from Boolean logic gates

It is also interesting to note that the previous observation suggests a significant distinction between the observed computations, e.g. the archetypes in Fig. 4, and Boolean logic functions. Superficially, the additive, imbalance and balance computations resemble Boolean OR, XOR, and AND gates, respectively. However, a key feature of Boolean logic is the existence of threshold levels that can be used to binarize inputs. For example, to behave like a Boolean AND gate, one would expect that when both inputs are each above some threshold, the output should always be “on.” By contrast, in the computations analyzed here, no such thresholds exist. For example, in an imbalance detection mode, consider two ligand concentrations L₁ and L₂ that individually activate, but produce a weaker response together, i.e. appear to represent two “high” input levels that together produce a “low” output. Because output depends only on ligand ratios, for any value of L₁ chosen here, we can find a concentration of the second ligand, $L_2^{\prime }>L_2$ that will decrease the ligand ratio enough to generate a “high” output. Similar considerations apply for the balance detection computation.

Biological parameter range

In our analysis of model response profiles, we selected random values for dimensionless parameters, distributed across the entire theoretical possible range. This allowed us to understand the full repertoire of theoretically possible responses. We then restricted the analysis to biologically plausible values for each of the parameters to constrain the analysis to biologically relevant regimes. For receptor expression, based on previous measurements of the number of TGF-β receptors in different cell lines (Wakefield et al., 1987), we chose the total receptor counts $A_i^0$ and $B_k^0$ in a log-uniform distribution in the range [0,9 · 10⁴]. For the ligand levels $L_j^0$, we considered the range of experimentally utilized ligand concentrations: [10⁻¹, 10³] ng/mL. To make it comparable with the units for the receptor we converted this range to units of molecules, assuming a typical molecular weight for BMP ligands of about 30 kDa. This produced a concentration range of [10⁻¹², 10⁻⁸] M. Multiplying by a typical eukaryotic cell volume of 2 · 10³μm³, we estimate ligand numbers per cell volume in the range of [1, 10⁴] molecules.

When receptors and ligands are measured in number of molecules, the affinity parameters K_ij and K_ijk have units of molecule⁻¹ (i.e. per molecule). Using estimates in the literature based on surface plasmon resonance measurements (Aykul Martinez-Hackert, 2016), as well as theoretical models (Nicklas and Saiz, 2013), we conservatively selected affinities from a log-uniform distribution over the range [10⁻³, 10⁻¹]. Finally, the efficiency parameters ε_ijk have arbitrary units that define the scale of the response and thus were chosen uniformly in the range [0, 1] without loss of generality.

Using these ranges, we selected 100,000 random parameter sets and performed simulations to compute the input-output functions for each parameter set across a 9×9 matrix of ligand concentrations. The results show that a full range of response profile could occur (Fig S5C–D). Further quantification of the relative frequencies of each response profile revealed a decreased frequency of ratiometric responses and increased frequencies of other functions, compared with the unrestricted parameter screen (Figure S5E).

Versatility search

A key aspect of the model is that it permits the cell to change the computation performed on a pair of ligands by modulating receptor expression. In order to study the effect of receptor expression levels on the computation performed, we selected 1000 random biochemical parameter sets. For each, we simulated the model with varying expression levels of each of the 4 receptor subunits (A₁, A₂, B₁, B₂) systematically chosen from a log-uniform distribution over the range [10⁻³, 10³]. The distribution of LIC and RLS values over the receptor expression levels was then calculated. These distributions were plotted for two specific biochemical parameter sets (one versatile and one non-versatile) in Fig. 6A. The biochemical parameters are provided in Table S5.

Robustness to perturbations in receptor expression

Both theoretical and experimental analysis revealed that changes in receptor expression levels can reprogram the computation performed by cells. This observation provokes the question of how the computation could vary in response to fluctuations, or ‘noise’, in receptor expression levels (Elowitz et al., 2002).

We consider two types of fluctuations in receptor levels. First, there might be an overall, correlated fluctuations across all receptors. Such extrinsic noise might reflect global changes in expression machinery, as well as fluctuations due to cell growth and division. Second, each receptor could vary stochastically in its expression, leading to independent fluctuations, or intrinsic noise. More quantitatively, we defined the receptor noise v as the coefficient of variation of receptor level. In addition, we defined α_E and α_I as the relative proportions of extrinsic and intrinsic noise, with α_E + α_I = 1.

To simulate the extrinsic noise, we generated a scale factor, s, drawn from a gamma distribution with shape parameter α_E/v², and scale parameter v² (giving a mean of α_E and variance α_Ev²). Similarly, intrinsic noise was simulated by a choosing a receptor dependent scale factor s_i, drawn from a gamma distribution with shape parameter α_I/v² and scale parameter v² (mean α_I and variance α_Iv²). To add noise to the receptor expression levels we multiplied the mean receptor level, $R_i^0$, by a combined scale factor (s + s_i). This combined scale factor has mean 1 and variance v², giving rise to the desired coefficient of variation.

We then set out to analyze cellular sensitivity to receptor perturbation across the spectrum of ligand computations. We randomly chose 100 parameter sets from each of 5 regions in the phenotypic space defined by RLS and LIC. These regions were chosen to represent the additive (−0.05 < LIC < 0.05, RLS > 0.8), ratiometric (−0.05 < LIC < 0.05, RLS < 0.2), imbalance (LIC < −0.1, RLS > 0.8), and balance (LIC > 0.1, RLS > 0.8) archetypes, as well as an intermediate region (−0.05 < LIC < 0.05, 0.4 < RLS < 0.6) that represented part of the spectrum of input-output functions but were not specifically associated with a single archetype. For each parameter set, we performed perturbations using 3 types of noise: purely extrinsic noise (α_E = 1, α_I = 0), purely intrinsic noise (α_E = 0, α_I = 1), or a combination of extrinsic and intrinsic noise (α_E = 0.5, α_I = 0.5). To enable comparison of the effects of each type of perturbation, we used a fixed coefficient of variation, v = 0.25.

Representative examples for each region (Fig. S7D–F) show the spread of the resulting functions in phenotypic space. We find that extrinsic noise generates comparatively small quantitative changes in response profiles relative to intrinsic noise, with the combination of extrinsic and intrinsic noise showing an intermediate effect. To systematically characterize the variation in phenotypic space, we analyzed the distribution of the standard deviations of the change in RLS and LIC across all perturbations for each parameter set (Fig. S7G,H). While these differences are generally insufficient to change the class of input-output function observed, we find that intrinsic noise produces greater variability in the phenotypic parameters. These results suggest that the observed input-output functions in the model of the BMP signaling pathway depend primarily on relative ratios of receptor levels rather than absolute numbers.

Finite volume of extracellular space

In the experimental context, cells were exposed to a large volume of media such that the total amount of ligand was much greater than the number of receptors. Using this high volume assumption results in constant ligand levels, even as they bind receptors to form active complexes (Eq. 12). However, this regime might not necessarily be applicable to all in vivo contexts where the exact ratio of ligands to receptors could vary, and titration of ligands by receptors can affect their concentration. Therefore, it is noteworthy to consider also the case of a finite volume. Under these conditions, the steady state equations, (Eq 16–20) become

$$L_j^0=L_j+\sum _{i=1}^{n_A}{D}_{ij}+\sum _{i=1}^{n_A}\sum _{k=1}^{n_B}{T}_{\mathit{ijk}}$$ (35)

$$A_i^0=A_i+\sum _{j=1}^{n_L}{D}_{ij}+\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{T}_{\mathit{ijk}}$$ (36)

$$B_k^0=B_k+\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}{T}_{\mathit{ijk}}$$ (37)

$$D_{ij}=K_{ij}^D{A}_i{L}_j$$ (38)

$$T_{\mathit{ijk}}=K_{\mathit{ijk}}^T{D}_{ij}{B}_k$$ (39)

$$S=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}.$$ (40)

Plugging (38) and (39) in (35–37) we get:

$$L_j^0=L_j\left(1+\sum _{i=1}^{n_A}{K}_{ij}^D{A}_i+\sum _{i=1}^{n_A}\sum _{k=1}^{n_B}{K}_{\mathit{ijk}}^T{K}_{ij}^D{A}_i{B}_k\right)$$ (41)

$$A_i^0=A_i\left(1+\sum _{j=1}^{n_L}{K}_{ij}^D{L}_j+\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{K}_{\mathit{ijk}}^T{K}_{ij}^D{L}_j{B}_k\right)$$ (42)

$$B_k^0=B_k\left(1+\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}{K}_{\mathit{ijk}}^T{K}_{ij}^D{A}_i{L}_j\right),$$ (43)

which can be numerically solved as before. We see that the distribution of behaviors remains similar with all four behavior arising even in the finite volume regime (Fig S5A).

Additionally, one can also ask how the finite volume assumption impacts the computations. To test this, we considered both models for every set of parameters and compared the resulting behavior. While the two models give rise to qualitatively similar responses to increasing ligand levels, they differ specifically at intermediate ligand levels (Fig S5B), where the finite model produces a sharper dependence on ligand concentration, or, equivalently, a reduced input dynamic range.

Signaling modifiers

In addition to receptors and ligands, other secreted and cell bound proteins could in principle reshape the activity of pathway elements. In the BMP pathway many such modifiers are known, including secreted ligand-binding molecules (e.g. Twsg, Chordin, Noggin) and pseudo-receptors (e.g. BAMBI). While these factors were not explicitly modelled, they can be incorporated within the same formalism. Ligand binding molecules can generally form complexes with ligands, just as the receptors do. Therefore, mathematically they are equivalent to type A receptors, with affinities that can be specified in the corresponding elements of $K_{ij}^D$. Since these modifiers are secreted, the resulting complexes will not form complexes with type B receptors, and will not produce active signaling complexes. In the model, this can be represented by 0 values for the corresponding elements of both $K_{\mathit{ijk}}^T$ and ε_ijk. We thus see that the secreted ligand-binding modifiers are mathematically equivalent to orphan type A receptors. Pseudo-receptors similarly have 0 values of ε_ijk, but could have non-zero values for $K_{\mathit{ijk}}^T$.

We therefore see that the modeling framework can naturally extend to include many biologically relevant components of the BMP pathway. While these elements do not play critical roles in the in vitro setting explored in this manuscript, they could play more important roles in vivo.

Nonlinear signal accumulation

Another assumption in the model is that the receptors all contribute linearly to the overall total phosphorylation rate (Eq. 11). This assumption holds when the dephosphorylation rate is large compared with the phosphorylation rate, such that there is a large pool of dephosphorylated SMAD protein. A similar assumption was utilized in various models of the BMP and TGF-β pathway such as in (Vilar et al., 2006), and is motivated by experimental results of the TGF-β dependent phosphorylation and dephosphorylation rates (Inman et al., 2002). Since distinct cell types can differ in their kinase level, amongst others, it is biologically interesting to consider what happens in a regime where this assumption does not hold. The full equation describing the phosphorylated SMAD signal (S_p) should be written as

$$\frac{d{S}_p}{dt}=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}{S}_u-\gamma S_p,$$ (44)

where S_u is the amount of unphosphorylated SMAD and γ is the dephosphorylation rate. Using S_tot = S_u + S_p, we get

$$\frac{d{S}_p}{dt}=\sum _{i=1}^{n_A}\sum _{j=1}^{n_L}\sum _{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}\left(S_{\mathit{tot}}-S_p\right)-\gamma S_p.$$ (45)

Solving for the steady state solution one finds

$$S_p=\frac{{\sum }_{i=1}^{n_A}{\sum }_{j=1}^{n_L}{\sum }_{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}}{1+{\sum }_{i=1}^{n_A}{\sum }_{j=1}^{n_L}{\sum }_{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}}{S}_{\mathit{tot}}$$ (46)

where we define ε_ijk = ε_ijk / γ as before. From this we see that for slow total phosphorylation rates, ${\sum }_{i=1}^{n_A}{\sum }_{j=1}^{n_L}{\sum }_{k=1}^{n_B}{\epsilon }_{\mathit{ijk}}{T}_{\mathit{ijk}}\ll 1$, we recover the linear behavior (Eq. 20). However, when the phosphorylation rates become faster, compared to the characteristic dephosphorylation rate, the response reaches saturation. It is important to note that overall the signal is still determined by a monotonically increasing function of the weighted sum of all trimeric complexes. Therefore, while the quantitative nature of the computations can depend on the exact ratio between phosphorylation and dephosphorylation rates, the qualitative behaviors remain similar.

Additional features of the BMP pathway

The model above neglects a number of known features of the natural BMP pathway in order to focus on the specific effects of promiscuity, and to demonstrate its sufficiency for explaining observed computations. These other features are likely to play additional functional roles in the signaling pathway. One example is the dynamical nature of the receptor-ligand interactions, arising from receptor internalization, trafficking, and degradation. While the steady state response studied in this paper is consistent with the BMP data, the parallel TGF-β branch of the pathway appears to respond transiently (adaptively) in some contexts (Warmflash et al., 2012). Alternative models (Vilar et al., 2006; Zi et al., 2012) were previously developed that focused on these dynamical aspects of the TGF-β response, and showed how these dynamics can give rise to either transient or sustained responses, as well as absolute or relative ligand response profiles, although not the balance and imbalance detection modes described here.

Spatial heterogeneity

The plasma membrane has been shown to contain microdomains differing in lipid and protein composition, or interactions with the cytoskeleton, which could in principle affect spatial and temporal receptor distribution (Delos Santos et al., 2015). This provokes the question of how receptor localization in microdomains could affect the computational behavior of the system. For example, a given receptor could be partially or completely segregated into certain domains which may or may not overlap with the localization of other receptors. Such effects can be modeled by replacing the existing affinity parameters with effective affinity parameters. Specifically, if two receptors are localized to completely different microdomains their effective affinity would be zero. By contrast, if they have a slight preference for distinct microdomains, their effective affinities would only be reduced. Representing domain preference this way provides two potential advantages for the system: First, it enables more flexibility in reaching diverse effective affinities among components. Second, if microdomain localization can be regulated by the cell, it can allow for dynamical tuning of effective affinity parameters, an additional mode of control.

QUANTIFICATION AND STATISTICAL ANALYSIS

Average and variability analysis

All single cell flow cytometry data were averaged by taking the population median. Repeats were averaged by taking the mean of at least 3 repeats. Variability was assessed either using standard deviation or standard error of the mean, as indicated in the legend. To remove bias due to day-to-day variability we normalized each repeat by an overall scale factor. This was determined using total least square fit between each two experiments.

Assignment of integration modes in survey

In the survey (Fig. 2A), for each ligand pair, (L₁, L₂), five different combinations (Fig. S1G) were measured: L₁, L₂, L₁+L₂, L₁+L₁, L₂+L₂, where the latter two indicate double the base concentration of a single ligand. Of these 5 combinations, the first three were assayed four times each, while the latter 2 were measured in duplicate. To estimate the relative likelihood of each ligand integration mode (Types I–IV in Fig. S1G), we examined all 256 possible measurement combinations. For each one, we determined the corresponding integration mode based on the classification scheme in Fig. S1G. The relative likelihoods were then estimated by calculating the frequency of each integration mode. This is plotted in Fig. S1H, as shown in the inset.

RNAseq Analyses

Total RNA was collected from cells using the RNeasy Mini Kit (Qiagen) following the manufacturer’s instructions. Sequencing libraries were constructed using NEBNext Ultra RNA-seq kit (NEB #E7530) and sequenced on Illumnia HiSeq2500. Results were then analyzed using the web-based galaxy platform (https://usegalaxy.org/). Alignment was performed using the TopHat algorithm followed by transcript assembly and FPKM estimates using the Cufflinks algorithm.

DATA AND SOFTWARE AVAILABILITY

Flow cytometry data was analyzed in MATLAB using a custom software (EasyFlow). Mathematical model simulations were performed and analyzed in MATLAB. All the analysis code is available upon request (see CONTACT FOR REAGENT AND RESOURCE SHARING).

The RNAseq data reported in this paper have been deposited in GEO under ID code GSE98674.

Supplementary Material

1. Figure S1 Ligand pairs can show complex integration modes. Related to Figure 2.

(A) NMuMG BMP reporter cells were stimulated with different concentrations of BMP4 (colored dots) and analyzed by flow cytometry for both reporter H2B-Citrine expression (x-axis), and immunostaining of phosphorylated SMAD1/5/8 (y-axis). Note strong correlation both within (scatter) and between (larger circles) ligand concentrations. (B) qRT-PCR measurements show endogenous BMP-responsive gene expression levels correlate with H2B-Citrine reporter expression. Plots show relationships between H2B-Citrine and specific indicated target genes. (C) Correlation coefficients for each pair of target genes shown in (B). (D) Flow cytometry of reporter H2B-Citrine expression showed unimodal distributions 24h after stimulation across the indicated range of BMP4 concentrations (colors). (E) Dynamics of phosphorylated Smad1/5/8 were measured using immunoblotting at time-points up to 48 hours after BMP addition. After a short transient of a few hours, phosphorylated Smad1/5/8 levels remained constant. The plot shows mean and standard deviation (error bars) of 3 independent repeats. (F) Reporter cells were exposed to different concentrations of BMP4 (left) or BMP10 (right). Fluorescence was monitored using time-lapse microscopy over more than 48 hours. Continuous increases in mean fluorescence per cell occurred in most conditions. This result contrasts with the adaptive dynamics observed in response to stimulation by TGF-β ligands (Warmflash et al., 2012). Vertical dashed line indicates the 24h timepoint used for most experiments in the paper. (G) 4 modes of ligand integration are shown schematically. Type I is characterized by a strong response to mixed ligands (green), with weaker responses to the individual ligands (gray). Type II is characterized by a weak response to mixed ligands (red), in comparison to individual ligands. In cases where the mixed response is intermediate (dark blue), two additional integration modes can be realized: The type III integration mode is characterized by decreased activity in response to removal of one ligand (dark blue to light blue). Finally, a type IV integration mode occurs when removal of one of the ligands causes an increase in the response (dark blue to purple). (H) Using the low resolution ligand survey (Fig. 2A), all pairs of ligands were classified across these 4 integration modes. For every pair, the likelihood of each mode was calculated (see Methods) and the corresponding square was colored by bands with widths proportional to the relative likelihood of each mode. The appearance of multiple colors in the same square thus indicates uncertainty about the integration mode.

2. Figure S2 Cellular response to specific relations between ligands can be seen theoretically and experimentally with different reporter lines, ligand sources and readouts. Related to Figures 2 and 3.

(A) For each ligand pair, experimentally measured pathway activity is plotted across all points in the ligand matrix, as a function of either the adjusted ratio of the two ligand concentrations (upper plot) or the sum of the two ligand concentrations (lower plot). Most of the variation in activity in BMP4-BMP9 can be explained by the sum of the two ligands (left plots). For BMP4-GDF5, the data are better explained as a function of an adjusted ratio, where the GDF5 concentration was offset by a constant, representing the threshold above which the response becomes approximately ratiometric. For BMP4-BMP10, the response approximately follows a non-monotonic function of the ratio. (B) Similar plots for the archetypes were generated in the model. (C) An independent BRE-based sensor cell line was generated from NMuMG using a different integration technique (PiggyBAC, see Methods). It was exposed to the same BMP ligand pairs, giving rise to similar combined responses (cf. Fig. 2C–E). (D) Ligands acquired from a different source (Peprotech, see Methods), show similar responses to those acquired from R&D systems (cf. Fig. 2D, E). (E–F) Phosphorylated Smad1/5/8 was analyzed using immunoblotting in cells exposed to single ligand, ligand combinations, or no ligand. BMP4 and BMP10 exhibited imbalance detection (E), while BMP4 and GDF5 exhibited a ratiometric response (F). (G) Erk phosphorylation in response to BMP ligands was analyzed using immunoblotting. While both Erk1 and Erk2 respond dose-dependently to EGF1, they show no response to BMP4, BMP10 and GDF5. In C–E, results are normalized to the un-activated condition, and represent the mean and standard deviation of at least 3 independent repeats.

3. Figure S3 Ligand response profiles are stable over time and in response to perturbations of feedback loops and HSPG components. Related to Figure 3.

(A) Cells were stimulated with combinations of BMP4 and BMP10. Pathway responses were analyzed at different time points after ligand addition. Absolute fluorescence levels increased over time. However, the imbalance response is visible at all timepoints, from 6–96 hours after stimulations. (B) BMP4-BMP10 antagonism from (A) was quantified as the ratio of the least active individual ligand to activation by both ligands. This quantity was stable over the duration of the experiment. (C–E) Smad6 knockdown does not disrupt ligand integration. (C) The input-output response is plotted for cells treated with siRNA against Smad6 (siSmad6) or with a random sequence (siRND). (D) qPCR analysis shows that siRNA treatment reduced Smad6 transcript by ~90%. (E) The results with siRNA for Smad6 were plotted against those with a random siRNA sequence. Each dot represents a single ligand combination. Different colors represent different ligand pairs and the black line represent the line y=x for reference. (F) Cells were stimulated with BMP9 and BmpR2 protein levels were measured at several time points after stimulation using immunoblotting. BmpR2 protein levels were normalized by Gapdh protein levels and the fold change from t=0 was plotted. Error bars represent standard deviation between 3 independent repeats. (G–J) The role of HSPGs was analyzed by inhibiting its biosynthesis using NaClO₃ (G) or by enzymatically removing HSPG using heparinase (H). Results with and without treatment show a high level of correlation around the line y=x plotted in black (I, J).

4. Figure S4 Analysis of ligand integration functions in the mathematical model. Related to Figure 4.

(A) Different biochemical parameter sets generate a range of 2-ligand integration functions. Here, we plotted the steady-state response for 50 randomly selected parameter sets (grid of heat maps). These responses are not organized spatially. Note the broad range of behaviors and the general dependence on ratiometric features at high total ligand concentrations, reflected by the diagonal contours. (B) 2-ligand response profiles can be parameterized by Relative Ligand Strength (RLS) and Ligand Interference Coefficient (LIC). These coefficients are determined by four activity levels that can be extracted from the high total ligand regime: the activity generated by the weaker (a) and stronger (b) ligands individually, as well as the maximal (c) and minimal (d) activity over the entire high ligand region, denoted by satmax and satmin, respectively. (C, D) Determination of RLS and LIC for balance (C) and imbalance (D) detection. (E) For each (LIC, RLS) coordinate pair, we computed the mean response functions for 5 biochemical parameter sets generating phenotypic parameters close to the indicated (LIC, RLS) point (location of heatmap). Inset zooms in one specific (RLS, LIC) point. (F–G) Activity parameters can produce distinct response profiles from the same set of affinity parameters. (F) For a specific set of K_ij, K_ijk values, indicated, the level of each trimeric signaling complex, T_ijk, is plotted as a function of the concentrations of two ligands. (G) The total pathway response depends in general on the levels of all trimeric complexes, each multiplied by a corresponding activity parameter. Here we plot 4 specific sets of activities (ε_ijk), each of which generates a distinct response profile, despite using the same affinity parameters.

5. Figure S5. Changing model assumptions has mild effects on ligand computations. Related to Figure 4.

(A) 100,000 simulations were performed on randomly chosen parameter sets with (bottom) and without (top) allowing for consumption of ligands by cells. The calculated ligand interference coefficient and relative ligand strength (cf. Fig. S4B–D) show similar distributions and produce all computations in both cases. (B) Parameter sets corresponding to the 4 archetypes were selected and the full 2D input-output matrices are ploted for models with (center) and without (left) ligand cosumption. The difference (right column) between the two models (constant ligand, left, vs. consumed ligand, middle) demonstrate that the effects of ligand consumption are most significant at intermediate ligand levels, giving rise to a sharper signal response. (C, D) 100,000 parameter sets were randomly selected either from the complete theoretically available parameter space, assuming a uniform distribution for the dimensional reduced parameters (C) or restricted to a biologically relevant range, based on previously measured values for BMP affinities (Supplementary Text) (D). Resulting response profiles are plotted in the RLS-LIC space (see Fig. S4). Specific regions in the neighborhood of each archetype are shown (colored boxes). (E) The percent of parameter sets giving rise to each response type is shown for the unrestricted parameter selection (black) and for parameters restricted to the biologically relevant range (grey).

7. Figure S7 Receptor expression levels reprogram ligand response profiles. Related to Figure 6.

(A) The relative ligand strength (RLS) and ligand interference coefficient (LIC) are plotted for each ligand pair (shape), for different cell lines (fill style). (B) RLS and LIC are plotted for each ligand pair (shape), for wild-type (filled), and each indicated receptor perturbation (hollow). (C) Expression levels of all 7 BMP receptors in NMuMG cells were measured using RT-qPCR, for each receptor perturbation (indicated) to quantify the effect size and specificity of knockdown or overexpression. Values represent fold expression relative to Sdha expression, relative to control cells, either wildtype (for receptor overexpression) or a non-speicifc siRNA (for receptor knock down). Error bars represent s.e.m. from 4 independent measurements. (D–F) Effects of noise in receptor expression on ligand integration mode. We analyzed the effects of noise on 5 specific parameter sets representing different response profiles (colors). For each parameter set, we analyzed 25 randomly perturbed receptor expression profiles chosen from a gamma distribution with a coefficient of variation (CV) of 0.25. Each resulting interation profile is plotted in the LIC-RLS phenotypic space. When the noise is extrinsic (correlated between all receptors) its effect in the phenotypic space is minimal, as shown by relatively small scatter of colored dots (D). Intrinsic noise (uncorrelated fluctuations in each receptor) increases the scatter (E, F). (G, H) To generalize these results, we repeated the procedure in D–F for 100 parameter sets from each of the 5 regions (balance, imbalance, additive, ratiometric and intermediate regions). For each choice of receptor level, LIC and RLS parameters were calculated and the standard deviation for the 25 choices was calculated. The cumulative distribution function of the standard deviations in either the RLS (G) or LIC (H) is shown to indicate the distribution of sensitivities of ligand integration behavior to each category of noise.

Supplemental Figure 6. Figure S6 Archetypal computations emerge from the interplay between receptor preference and signaling complex activity in the model. Related to Figure 5.

For each archetypal computation (rows), the left-hand schematic represents a parameter regime sufficient for the computation (re-plotted from Fig. 5A–D). Arrow thicknesses represent the relative affinities or activities of indicated complexes. Arrow color represents the identity of the ligand in a given complex. To the right, the response profile across ligand compositions is shown (plot). The behavior of the system is also indicated schematically above the plot for three ligand composition regimes: only one ligand present (left and right) or an equal mixture of ligands (center). Hollow ligands represent those not present in each case. In each regime, some reactions don’t occur (because a particular ligand is not present) or are disfavored (because of competition). Arrows for these reactions are omitted in the corresponding regimes. The total activity of the system in each of these three regimes is indicated by the number of copies of the phosphorylated second messenger.

• Cells perform complex computations on combinations of BMP ligands

• A mathematical model shows how computations arise from receptor-ligand promiscuity

• A single cell type can perform different computations on different ligand pairs

• Changes in receptor profiles can reprogram the computations