Chapter II: Combinatorial, context-dependent logic of BMP signaling
2.3 Results
2.3.8 A mathematical model of competitive ligand-receptor interactions can ex-
As shown above, ligands can interact in different ways depending on receptor context. To understand how such complex, contextual responses could emerge, we analyzed a mathematical model of the pathway developed in a parallel study.68This model assumes that each ligand variant,πΏπ, can bind to any pair of Type I and Type II receptors, π΄π andπ΅π respectively, with affinity πΎπ π π to produce a trimeric, ligand-receptor signaling complex, ππ π π (Figure 2.7A), whose binding is captured by a set of ordinary differential equations describing mass-action kinetics. Each signaling complex can then phosphorylate SMAD proteins with its own specific kinase activity ππ π π. Receptors are
expressed at specific total levels, π΄0
π or π΅0
π. The limited total amount of each receptor generates competition among ligands for available receptors. Other features of the BMP systemβincluding step-wise assembly of ligand-receptor complexes, the heterotetrameric stoichiometry of actual receptor complexes, co-receptors, and other factorsβplay important roles in the natural system but were not required to explain observations in this work (STAR Methods).
Figure 2.7: Mathematical model of receptor competition can explain contextual ligand equiv- alence groups. (caption on following page)
Figure 2.7(previous page):
(A) In a minimal BMP receptor competition model, mass action kinetics govern one-step assembly of ligand (πΏπ), Type I receptor (π΄π), and Type II receptor (π΅π) into trimeric signaling complexes
(ππ π π) that activate pathway output with some activity (ππ π π). The included components are listed
beneath their associated variable.
(B) Observed responses, normalized between 0 and 1, correlate with simulated responses in top parameter fits.
(C) Total complex abundance (in arbitrary units, normalized to total possible number of complexes) does not correlate with overall output from a given ligand. Ligands of intermediate strength (e.g.
BMP10) can bind more receptors than strongly activating ligands.
(D) For individual ligands activating NMuMG, the inferred output of each complex is plotted against its percentage of the total complexes for all top parameter fits. A minority of the complexes that form can produce the majority of the output.
(E) The percent change in output (relative to NMuMG) from BMPR1A- and ACVR1-containing complexes following BMPR1A knockdown is shown for all five ligands. BMPR1A knockdown reduces the number of BMPR1A-containing complexes and reduces their output as expected, but also can reduce output from ACVR1-containing complexes, as BMPR1A loss increases availability of Type II receptors. All top parameter fits are plotted, unless the absolute change is small (i.e.
<0.05 change in response). GDF5, which does not activate these cells and has only small changes in ACVR1 output, is not shown (N/A).
(F) Percent change in output (relative to signaling alone) from complexes containing BMP4 or another ligand are shown for BMP4 paired with four other ligands: BMP7, BMP9, BMP10, and GDF5. Equal sharing between the two ligands would reduce output by 50%, whereas ligands that do not share complexes produce as much as output alone as they do in a pair. All top parameter fits are plotted, unless the absolute change is small (i.e. <0.05 change in response). GDF5, which does not activate these cells and has only small changes in overall output, is not shown (N/A).
(G) A schematic parameter set shows three ligands with different affinities and activities for four possible receptor dimers. In the model, these ligands have different individual strengths and pairwise strengths which change with receptor perturbations.
(H) In a simulation where each receptor subunit is available, the three ligands form both signaling and nonsignaling complexes. Bar height shows complex abundance. Empty bars show nonsignaling complexes, whereas filled bars show signaling complexes.
(I) After simulated knockout of the black Type I receptor, the abundances of many signaling complexes change, even if they did not involve the knocked out receptor.
(J) In a simulation where each receptor subunit is available, competition for shared receptors changes the formation of signaling complexes by each ligand. The pink and gold ligands can respectively shift the blue ligand to preferentially form its signaling and nonsignaling complexes.
See also Figure S2.8.
To identify parameter values that can generate the range of interactions observed experimentally, we focused our analysis on a single ligand from each global equivalence group. Similarly, we considered only the five broadly-expressed receptors (Figures 2.7A, S2.4A, S2.5A): the Type I
receptors ACVR1 and BMPR1A and the Type II receptors ACVR2A, ACVR2B, and BMPR2. We fit the model simultaneously to measurements of individual and pairwise responses in the four NMuMG-derived cell lines whose receptor profiles are restricted to this set (i.e. NMuMG and the knockdowns of ACVR1, BMPR2, and BMPR1A). We then analyzed a set of 22 solutions from nearly 7,000 least-squares fits that produced the lowest fitting error and preserved key pairwise interactions (Figures 2.7B, S2.8A; STAR Methods).
Within these best fit parameter sets, all ligands were promiscuous, with substantial but variable affinities for most or all receptors (Figure S2.8B, top). BMP7 and GDF5 had lower affinities overall across all complexes, while BMP4, BMP9, and BMP10 had higher affinities for select complexes (Figure S2.8B, top). The ligands also differed in the activity of the resulting com- plexes. GDF5 weakly activated most signaling complexes, while other ligands strongly activated multiple receptor complexes (Figure S2.8B, bottom). Many signaling complexes exhibited strong affinity but weak activity or vice versa (Figure S2.8C). A parallel computational study of the BMP pathway showed that this inverse relationship between activity and affinity generates the complex multi-ligand responses necessary for combinatorial addressing, in which ligand combinations can selectively activate cell types.68Taken together, these results suggest that the context-dependent ligand interactions observed experimentally can be explained by affinity-based competition to form complexes with different activities. They further suggest that the effective biochemical parameters for mammalian BMPs exhibit the features associated with combinatorial addressing.
2.3.9 In the model, context-dependence emerges from redistribution of signaling complexes