Multiary complex formations in GPCR signaling activations

Masaki Watabe,1, ∗ Hideaki Yoshimura,2 Satya N. V. Arjunan,1, 3 Kazunari Kaizu,1 and Koichi Takahashi1, 4, †

1Laboratory for Biologically Inspired Computing, RIKEN Centerfor Biosystems Dynamics Research, Suita, Osaka 565-0874, Japan

2School of Science, The University of Tokyo, 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan3Lowy Cancer Research Centre, The University of New South Wales, Sydney, Australia

4Institute for Advanced Biosciences, Keio University, Fujisawa, Kanagawa 252-8520, Japan(Dated: April 17, 2020)

Eukaryotic cells transmit extracellular signal information to cellular interiors through the for-mation of a ternary complex made up of a ligand (or agonist), G-protein, and G-protein coupledreceptor (GPCR). Previously formalized theories of ternary complex formation have mainly assumedthat receptors can only take the form of monomers. Here, we propose a multiary complex modelof GPCR signaling activations via the formations of various aggregated receptor states. Our re-sults from model simulations imply that the receptor aggregation processes can govern the signalingactivity in a regime inaccessible by previous theories. In particular, we show how the affinity ofligand-receptor binding can be largely varied by various oligomer formations in the low concentra-tion range of G-protein stimulus. More broadly, our work provides concrete modeling principles toexplore general relations between receptor aggregation and various signaling properties.

Introduction.— G-protein coupled receptors (GPCR)in eukaryotic cells form a remarkable modular systemover the cell membrane, its main function being to pro-vide cells with a wide means of signal communicationsbetween extracellular molecules (e.g., hormones and neu-rotransmitters) and intracellular signaling G-proteins.GPCR signal communication can be achieved throughconformational changes in receptors, as well as the com-plex formation with three different components: a ligand(or agonist), G-protein, and GPCR. This complex forma-tion serves as an activated signaling component, accord-ingly changing the affinity of ligand-receptor binding asa function of G-protein stimulus. These ideas have beenformalized into mechanistic theories of GPCR signalingactivation, termed ternary complex models [1, 2], follow-ing in particular, the assumption that receptors can onlytake the form of monomers (see Figure 1).

Thousand of GPCRs diffuses on the cell membrane,randomly interacting with each other and spontaneouslyforming oligomers such as dimers and trimers. Such re-ceptor aggregation may, for example, extend the colocal-ization period of the ternary complex, amplifying signal-ing activity [3]. Likewise, in a wide concentration rangeof ligand stimulus, group behavior such as cooperativityinduced by receptor dimerization may be constrained bythe affinity of higher-order oligomer formations [4, 5]. Re-cent studies analyzing receptor aggregation have shownvarious novel aspects of receptor systems, implying mod-ifications to ternary complex models [3–15]. This lies incontrast to previous theories which have mostly focusedon scenarios where receptor-receptor couplings (i.e., di-rect interaction of two receptors) are weakly linked withGPCR signaling activations.

In this letter, we propose a multiary complex modelof GPCR signaling activations via dimer formations ofligand-bound receptors, represented by a multivalent

FIG. 1. Schematic illustration of the simplest ternary com-plex model [1, 2]. A ligand (red bullet) and G-protein (orangeobject) can bind to a monomeric GPCR (green Y-shaped ob-ject) with equilibrium constants K`0 and Kg0, respectively.A ternary complex composed of a ligand, G-protein and re-ceptor can be formed in two ways: (i) ligands can interactwith the receptors binding to G-proteins with equilibriumconstant αK`0, and (ii) G-proteins can bind to the ligand-bound receptors with equilibrium constant αKg0, where α isa cooperativity factor that denotes the mutual effect of thereceptor-binding affinity to the ligand and G-protein.

form of physical observables under basis vectors of var-ious aggregated receptor states (see Figure 2). We per-form model simulations to explore the functional roleof receptor aggregation in GPCR signaling activations.Crucially, we show how a mixture of various aggre-gated receptor states can lead to the transition of ligand-receptor binding affinity in a regime which cannot bepredicted by ternary complex models. We finally discussthat such receptor aggregation is a phenomenon com-mon in various biological cells, and also of relevance morebroadly beyond GPCR signaling activation. Our worksuggests new avenues for extracting general relations be-tween receptor aggregation and various signaling proper-ties.

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Model framework.— In the receptor state vector repre-sentation of physical observables, the multiary complexmodel is described by a function containing the probabil-ities of biochemical interactions that form various aggre-gated receptor states. All possible aggregated receptorstates via dimer formation of ligand-bound receptors canbe treated mathematically as basis vectors in a multidi-mensional real vector space.

First, we assume that the observed state vectors ofreceptors are represented in terms of ligand-bound andunbound receptors. Figure 2 shows network diagrams of

(a)

(b)

(c)

(d)

FIG. 2. Network diagrams of multiary complex formation vialigand-bound receptor dimerization. (a) Multivalent form ofthe simplest ternary complex model. (b) Dimer formations ofligand-bound receptors. (c) First-order interactions of ligandand G-protein to receptor states. (d) Second-order interac-tions of receptor states to G-protein-bound receptor states.Here ki and di are the association and dissociation rates ofthe i-th index, respectively, and L and G represent ligand andG-protein stimuli, respectively.

the multiary complex model. Null (Φ, Φ′), monomeric(M, M′) and dimeric (D′) observed state vectors of re-ceptors are given by

Φ =

rrrrrr

...

rN

, M =

RRrRrr

...

RrN−1

, Φ′ =

r · rr · rrr · rrr

...

rN · rN

,

M′ =

R · rR · rrR · rrr

...

RrN−1 · rN

, D′ =

R ·RR ·RrR ·Rrr

...

RrN−1 ·RrN−1

(1)

where r, R and G represent the receptors, the ligand-bound receptors and G-proteins, correspondingly. Nrefers to the number of receptors that can be aggregatedin the Φ and M observed states. There are N2 elementsin the Φ′, M′, and D′ observed states.

Null (G ·Φ, G ·Φ′), monomeric (G ·M, G ·M′) anddimeric (G ·D′) observed state vectors of the G-protein-bound receptors are also given by

G ·Φ =

G · rG · rrG · rrr

...

G · rN

, G ·M =

G ·RG ·RrG ·Rrr

...

G ·RrN−1

G ·Φ′ =

G · r · rG · r · rrG · r · rrr

...

G · rN · rN

, G ·M′ =

G ·R · rG ·R · rrG ·R · rrr

...

G ·RrN−1 · rN

,

G ·D′ =

G ·R ·RG ·R ·RrG ·R ·Rrr

...

G ·RrN−1 ·RrN−1

(2)

where G·r and G·R represent G-protein-bound receptors.N refers to the number of receptors that can be aggre-gated in the G ·Φ, and G ·M observed states. There areN2 elements in the G ·Φ′, G ·M′, and G ·D′observedstates.

In first-order interactions of a ligand, G-protein, andreceptor, the rates of association (k`i, kai, kgi, kbi) anddissociation (d`i, dai, dgi, dbi) of the i-th index are rep-resented by N × N diagonal matrices acting upon the

basis vectors, transforming a single aggregated state toan observed state.

The dissociation rates (dxi, dyj) of the i-th and j-thindices for second-order interactions in Figures 2b andd are represented by N2 × N2 diagonal matrices. Non-diagonal matrices of the association rates (kxi, kyj) of thei-th and j-th indices can, however, transform a mixtureof various aggregated states into an observed state. Thenon-diagonal matrices can be written in the form of

kxi = kxiFxi, kyj = kyjFyj (3)

where i = 0, 1, 2 and j = 0, 1, 2, 3; kxi,yj and Fxi,yj arethe second-order association rates and N × N scalingmatrices, respectively.

For convenience, we define a dimensionless lumped pa-rameter that constrains the fraction of oligomer forma-tions in the absence of ligands. The lumped parametercan be written in the matrix form of

kx = kxFx0 (4)

where kx = T/Kx0.Multivalent cell-models.— To begin, we constructed

multivalent (N = 1 ∼ 5) cell-models of multiary com-plex formations. We then used the E-cell system ver-sion 4 [16] to simulate cell-models of biological fluctua-tion that arise from stochastic changes in the cell surfacegeometry, number of receptors, ligand binding, molecularstates, and diffusion constants. These cell-models assumethat non-diffusive receptors are uniformly distributed onthe cell membrane. The source code of the multivalentcell-models is provided in the Supplemental Material [17].

In a concentration range of ligand stimulus from10−3K`0 to 103K`0, we ran model simulations for a pe-riod of 100, 000 sec to verify the complete convergence ofreceptor response to full equilibrium. To quantify coop-erative characteristics in the multivalent cell-models, theHill function can be fitted to the ligand-receptor bindingcurves of the monomeric and dimeric observed state vec-tors of receptors: M, G ·M, M′, G ·M′, D′ and G ·D′.The Hill function can generally be written in the form of

B(L) =B0L

n

Ln +K ′n(5)

where L, B0, and n represent ligand concentration, max-imum area-density of the ligand-bound receptor, and theHill-coefficient, respectively; K ′ denotes overall affinity ofligand-receptor binding as a function of G-protein stim-ulus from 10−3Kg0 to 103Kg0.

The model parameter values are given as follows: to-tal receptor concentration, T = 4.977 receptors/µm2;cooperativity factor, α = Ka0/K`0 = Kb0/Kg0; bind-ing affinity and dissociation rates for each first-orderinteractions, K`1 = K`2 = 100K`0, Kg1 = 100Kg0,Ka1 = Ka2 = 100Ka0, Kb1 = Kb2 = 100Kb0, d`0 =d`1 = d`2 = dg0 = dg1 = da0 = da1 = da2 =

db0 = db1 = db2 = db3 = 1.00 s−1; and binding affin-ity and dissociation rates for each second-order interac-tions, Ky0 = Kx0, Kx1 = Ky1 = Ky2 = K`1Kx0/K`0,Kx2 = Ky3 = K`2K`1Kx0/K

2`0, dx0 = dx1 = dx2 = dy0 =

dy1 = dy2 = dy3 = 1.00 s−1; . The local equilibriumconstants of the association and dissociation rates of thei-th index satisfy the relation Ki = di/ki.

We assume that the scaling factors of diagonal ele-ments in the first-order association (k`i, kai, kgi, kbi)

(a) (d)

(b) (e)

(c) (f)

FIG. 3. Model comparison between the simplest ternarycomplex model (left column) and the monovalent cell-model(right column). (a) Ligand-receptor binding curve as a func-tion of ligand stimulus L/K`0, is given by Eq. (7) whereG/Kg0 = 0.1, α = 0.03 (blue), 0.3 (light-blue), 1.0 (blackdots), 3.0 (light-violet), and 30 (violet). (b) Overall affinityof ligand-receptor binding (K′/K`0) as a function of G-proteinstimulus G/Kg0, is given by the Eq. (8), for G/Kg0 = 0.1 andα = 0.03, 0.3, 1.0, 3.0, 30. (c) Colors represent the bind-ing affinity of the ternary complex model in the range from0 (blue) to ∼ 1 (green) and ≥ 2 (red); the black dashed lineshows K′/K`0 = 1. (d) For α = 1 and kx = 0 (black dots),10−4 (light-orange), 10−3 (orange), and 10−2 (dark-orange),ligand-receptor binding curves are shown as a function of lig-and stimulus L/K`0. (e) For α = 1 and kx = 0, 10−4,10−3, and 10−2, overall affinities of ligand-receptor bindingare shown as a function of G-protein stimulus G/Kg0. (f)Color represents the binding affinity of the monovalent cell-model as a function of α andG/Kg0, assuming kx = 0.001; theblack dashed and solid lines represent K′/K`0 = 1 in the sim-plest ternary complex model and the monovalent cell-model,respectively.

and dissociation rates (d`i, dai, dgi, dbi, dxi, dyj) of thei-th and j-th indices are all unity. The scaling matricesof the second-order association rates (kxi, kyj) are alsogiven by

Fxi = Fyj =J NN

N(6)

where i = 0, 1, 2 and j = 0, 1, 2, 3; J NN represents all-ones matrix where every element is equal to one. In ourmodel framework, the second-order interactions of recep-tors exhibit various oligomers (see Figure 2b): Φ + Φ→Φ′, M + Φ →M′ and M + M → D′. G-protein-boundoligomers are also formed through the second-order in-teractions between G-proteins and receptors (see Fig-ure 2d): G ·Φ + Φ → G ·Φ′, G ·M + Φ → G ·M′,G ·Φ + M → G ·M′ and G ·M + M → G ·D′. Inthe monovalent cell-model (N = 1), there are variousdimers via the second-order interactions: r + r → rr,R+r → rR, R+R→ RR, Gr+r → Grr, GR+r → GrR,Gr + R → GrR and GR + R → GRR. Moreover,the higher-order multivalent cell-models (N = 2 ∼ 5)also exhibit dimers, as well as higher-order oligomers;for example, trimers (rrr, rrR, rRR, Grrr, GrrR andGrRR), tetramers (rrrr, rrrR, rrRR, Grrrr, GrrrRand GrrRR), and pentamers (rrrrr, rrrrR, rrrRR,Grrrrr, GrrrrR and GrrrRR).The simplest ternary complex model [1, 2].— The net-

work diagram of the multiary complex model (see Fig-ure 2) converges to that of the simplest ternary complexmodel (see Figure 1) as N = 1 and kx → 0. Overallligand-receptor binding states (R, GR) in the ternarycomplex model can be written in the form of

B(L) =B0L

L+K ′(7)

where L and B0 represent ligand concentration and max-imum area-density of the ligand-bound receptor, respec-tively. Overall affinity of ligand-receptor binding as afunction of G-protein stimulus is given by

K ′ = K`0

(1 +G/Kg0

1 +G/ (αKg0)

)(8)

where α is the cooperativity factor that satisfies the re-lation α = Ka0/K`0 = Kb0/Kg0. If α = 1, there is noaffinity transition.

Model comparison.— To clearly see the effects arisingfrom the second-order interactions of receptors, we com-pare the ligand-receptor binding curves between the sim-plest ternary complex model (kx = 0) and the monova-lent cell-model (kx > 0 and N = 1). Figure 3 clearlyshows differences in the binding curves. In the simplestternary complex model, the cooperativity factor α inEq. (8) plays a key role in largely varying the overallaffinities of ligand-receptor binding in the high concen-tration range of G-protein stimulus (see Figures 3a,b, and

FIG. 4. Transition of the overall affinity in the multivalentcell-models is shown as a function of the lumped parame-ter kx, assuming α = 1 and G/Kg0 = 10−3. Each coloredline represents the monovalent (red), bivalent (blue), trivalent(green), tetravalent (pink), and pentavalent (violet) models;the dashed black line denotes no receptor-receptor interac-tions of receptors.

c). The affinities of the binding curves shown in Fig-ures 3a and b, can be decreased (K ′ < K`0) or increased(K ′ > K`0) as a function of the cooperativity factor.Such affinity transitions are also shown in Figure 3c, vary-ing from blue to red regions in the high G/Kg0 range.

While there is no direct interaction of receptors in thesimplest ternary complex model, the second-order associ-ation rate (kx0) in the monovalent cell-model plays a keyrole to largely change the overall affinity in the low con-centration range of G-protein stimulus (see Figure 3d,e and f). The affinity of the binding curves shown inFigure 3d and e, can be increased (K ′ > K`0) as a func-tion of the lumped parameter kx. Such affinity shiftsare also shown in Figure 3f, represented by red coloredregion in the low G/Kg0 range. In the absence of G-protein stimulus (G → 0), the overall network diagramof the multiary complex model converges to the dimerformations of ligand-bound receptors (see Figure 2b). Inparticular, the dimer formations are modeled under aspecific parameter condition that can exhibit positive co-operativity (K`1 = K`2), increasing the affinity of ligand-receptor binding [4, 5]. Because of these model parameterrelations, the monovalent cell-model displays an affinitytransition in the low G/Kg0 range.

The affinity transitions can be also seen in the higher-order multivalent cell-models: bivalent (N = 2), triva-lent (N = 3), tetravalent (N = 4), and pentavalent(N = 5). For α = 1 and G/Kg0 = 10−3, Figure 4shows the affinity transitions in multivalent cell-modelsas a function of the lumped parameter kx. While theaffinity in the multivalent cell-models is always unity ifkx = 0 (K ′ = K`0; black dashed line), the affinity canbe increased through the increase of kx (K ′ > K`0; col-ored lines). Also, affinity-splitting between the mono-valent cell-model (red line) and higher-order multivalent

cell-models (blue, green, pink and violet lines) becomesapparent in the high kx range.

Conclusion.— Many diffusive GPCRs in the cell mem-branes randomly collide with each other, spontaneouslytaking the form of various oligomers such as dimers,trimers and tetramers. The functional role of receptoroligomerization (or aggregation) in GPCR signaling acti-vations, however, has been elusive to date. In this letter,we constructed a multiary complex model to investigatebiophysical effects arising from various aggregated recep-tor states in GPCR signaling activations. Our resultsfrom model simulations revealed that receptor oligomer-ization functions to largely vary the overall affinity ofligand-receptor binding in a regime which cannot be ruledby cooperativity factor in the simplest ternary complexmodel.

Furthermore, such receptor aggregation in the cellmembranes is ubiquitous across various biological cellsand of relevance more generally beyond GPCR signal-ing activations presented here. The function of oligomerformation through receptor-receptor interactions leads tothe general questions: what is the relation between recep-tor aggregation and various signaling properties, e.g., theamplification and propagation of noisy signals [18, 19],and the physical limit and sensitivity to chemical con-centration sensing in ligand-receptor binding [20–24]?Our work sheds light on these interesting questions fromthe perspective of theoretical biophysics, and suggestsconcrete modeling principles to explore general rules ofreceptor aggregation governing signaling activities andproperties in various signal transduction systems.

We would like to thank Yasushi Okada, TomonobuM. Watanabe, Jun Kozuka, Michio Hiroshima, KozoNishida, Wei Xiang Chew, Suguru Kato, Toru Niina,Koji Ochiai, Keiko Itano, Kotone Itaya, Takuya Miuraand Kaoru Ikegami for their guidance and supportthroughout this research work.

∗ Corresponding author; masaki@riken.jp† Corresponding author; ktakahashi@riken.jp

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