# The Impact of Membrane Protein Diffusion on GPCR Signaling

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## Abstract

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## 1. Introduction

## 2. Diffusion Limitation in GPCR-G Protein Binding

#### 2.1. Reduced Diffusional Model

#### 2.2. Scaling Analysis

#### 2.3. More Complicated Scenarios: Clustering and Geometries Other Than Planar

## 3. Measuring the Second Messenger Concentration: The Reaction Network

#### 3.1. The Signaling Network

#### 3.2. Concentration of Activated G Protein

#### 3.3. cAMP Production

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PBRD | particle-based reaction-diffusion |

GPCR | G protein-coupled receptor |

cAMP | cyclic adenosine monophosphate |

GTP | guanosine triphosphate |

GDP | guanosine diphosphate |

AC | adenolyate cyclase |

ATP | adenosine triphosphate |

LatB | Latrunculin B |

AR | adrenergic receptor |

$\left[\mathrm{X}\right]$ | concentration of X (in cell or membrane, depending on context) |

G | unactivated G protein |

${\mathrm{G}}^{*}$ | activaced G protein |

R | receptor |

S | stimulus, agonistic ligand |

$\mathrm{XY}$ | complex of molecules X and Y |

## Appendix A. Simulation Details

## References

- Hauser, A.S.; Attwood, M.M.; Rask-Andersen, M.; Schiöth, H.B.; Gloriam, D.E. Trends in GPCR drug discovery: New agents, targets and indications. Nat. Rev. Drug Discov.
**2017**, 16, 829–842. [Google Scholar] [CrossRef] [PubMed] - Bers, D.M.; Ziolo, M.T. When is cAMP not cAMP? Circ. Res.
**2001**, 89, 373–375. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Calebiro, D.; Koszegi, Z. The subcellular dynamics of GPCR signaling. Mol. Cell. Endocrinol.
**2019**, 483, 24–30. [Google Scholar] [CrossRef] [Green Version] - Bock, A.; Annibale, P.; Konrad, C.; Hannawacker, A.; Anton, S.E.; Maiellaro, I.; Zabel, U.; Sivaramakrishnan, S.; Falcke, M.; Lohse, M.J. Optical mapping of cAMP signaling at the nanometer scale. Cell
**2020**, 182, 1519–1530. [Google Scholar] [CrossRef] - Maudsley, S.; Martin, B.; Luttrell, L.M. The origins of diversity and specificity in G protein-coupled receptor signaling. J. Pharmacol. Exp. Ther.
**2005**, 314, 485–494. [Google Scholar] [CrossRef] [Green Version] - Wacker, D.; Stevens, R.C.; Roth, B.L. How Ligands Illuminate GPCR Molecular Pharmacology. Cell
**2017**, 170, 414–427. [Google Scholar] [CrossRef] [Green Version] - Zaccolo, M.; Pozzan, T. Discrete microdomains with high concentration of cAMP in stimulated rat neonatal cardiac myocytes. Science
**2002**, 295, 1711–1715. [Google Scholar] [CrossRef] - De Lean, A.; Stadel, J.; Lefkowitz, R. A ternary complex model explains the agonist-specific binding properties of the adenylate cyclase-coupled beta-adrenergic receptor. J. Biol. Chem.
**1980**, 255, 7108–7117. [Google Scholar] [CrossRef] - Sutherland, E.W.; Wosilatt, W.D. Inactivation and activation of liver phosphorylase. Nature
**1955**, 175, 169–170. [Google Scholar] [CrossRef] - Neves, S.R.; Ram, P.T.; Iyengar, R. G protein pathways. Science
**2002**, 296, 1636–1639. [Google Scholar] [CrossRef] - Fay, S.P.; Posner, R.G.; Swann, W.N.; Sklar, L.A. Real-Time Analysis of the Assembly of Ligand, Receptor, and G Protein by Quantitative Fluorescence Flow Cytometry. Biochemistry
**1991**, 30, 5066–5075. [Google Scholar] [CrossRef] [PubMed] - Shea, L.D.; Neubig, R.R.; Linderman, J.J. Timing is everything: The role of kinetics in G protein activation. Life Sci.
**2000**, 68, 647–658. [Google Scholar] [CrossRef] - Sungkaworn, T.; Jobin, M.L.; Burnecki, K.; Weron, A.; Lohse, M.J.; Calebiro, D. Single-molecule imaging reveals receptor–G protein interactions at cell surface hot spots. Nature
**2017**, 550, 543–547. [Google Scholar] [CrossRef] [PubMed] - Alhadeff, R.; Vorobyov, I.; Yoon, H.W.; Warshel, A. Exploring the free-energy landscape of GPCR activation. Proc. Natl. Acad. Sci. USA
**2018**, 115, 10327–10332. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Lamb, T.D. Stochastic simulation of activation in the G-protein cascade of phototransduction. Biophys. J.
**1994**, 67, 1439–1454. [Google Scholar] [CrossRef] [Green Version] - Linderman, J.J.; Mahama, P.A.; Forsten, K.E.; Lauffenburger, D.A. Diffusion and probability in receptor binding and signaling. Adv. Chem. Eng.
**1994**, 19, 51–127. [Google Scholar] [CrossRef] - Melo, E.; Martins, J. Kinetics of bimolecular reactions in model bilayers and biological membranes. A critical review. Biophys. Chem.
**2006**, 123, 77–94. [Google Scholar] [CrossRef] - Goldstein, B.; Levine, H.; Torney, D. Diffusion limited reactions. SIAM J. Appl. Math.
**2007**, 67, 1147–1165. [Google Scholar] [CrossRef] - Kenakin, T. Drug efficacy at G protein–coupled receptors. Annu. Rev. Pharmacol. Toxicol.
**2002**, 42, 349–379. [Google Scholar] [CrossRef] [Green Version] - Bornheimer, S.J.; Maurya, M.R.; Farquhar, M.G.; Subramaniam, S. Computational modeling reveals how interplay between components of a GTPase-cycle module regulates signal transduction. Proc. Natl. Acad. Sci. USA
**2004**, 101, 15899–15904. [Google Scholar] [CrossRef] [Green Version] - Jaeger, J.C.; Carslaw, H.S. XVIII.—Heat Flow in the Region bounded Internally by a Circular Cylinder. Proc. R. Soc. Edinburgh. Sect. Math. Phys. Sci.
**1943**, 61, 223–228. [Google Scholar] [CrossRef] - Waite, T.R. Theoretical treatment of the kinetics of diffusion-limited reactions. Phys. Rev.
**1957**, 107, 463–470. [Google Scholar] [CrossRef] - Szabo, A.; Schulten, K.; Schulten, Z. First passage time approach to diffusion controlled reactions. J. Chem. Phys.
**1980**, 72, 4350–4357. [Google Scholar] [CrossRef] - Torney, D.C.; McConnell, H.M. Diffusion-limited reaction rate theory for two-dimensional systems. Proc. R. Soc. Lond. Ser. Math. Phys. Sci.
**1983**, 387, 147–170. [Google Scholar] [CrossRef] - Molski, A. A Model of Diffusion-Influenced Enzyme Activation. J. Phys. Chem.
**2000**, 104, 4532–4536. [Google Scholar] [CrossRef] - Mahama, P.A.; Linderman, J.J. A Monte Carlo study of the dynamics of G-protein activation. Biophys. J.
**1994**, 67, 1345–1357. [Google Scholar] [CrossRef] [Green Version] - Lauffenburger, D.A.; Linderman, J. Receptors: Models for Binding, Trafficking, and Signaling; Oxford University Press: Oxford, UK, 1996. [Google Scholar]
- Doi, M. Stochastic theory of diffusion-controlled reaction. J. Phys. A Math. Theor.
**1976**, 9, 1479. [Google Scholar] [CrossRef] - Calebiro, D.; Rieken, F.; Wagner, J.; Sungkaworn, T.; Zabel, U.; Borzi, A.; Cocucci, E.; Zurn, A.; Lohse, M.J. Single-molecule analysis of fluorescently labeled G-protein-coupled receptors reveals complexes with distinct dynamics and organization. Proc. Natl. Acad. Sci. USA
**2013**, 110, 743–748. [Google Scholar] [CrossRef] [Green Version] - Scarselli, M.; Annibale, P.; McCormick, P.J.; Kolachalam, S.; Aringhieri, S.; Radenovic, A.; Corsini, G.U.; Maggio, R. Revealing G-protein-coupled receptor oligomerization at the single-molecule level through a nanoscopic lens: Methods, dynamics and biological function. FEBS J.
**2016**, 283, 1197–1217. [Google Scholar] [CrossRef] - Boltz, H.H.; Sirbu, A.; Stelzer, N.; Lohse, M.J.; Schütte, C.; Annibale, P. Quantitative spectroscopy of single molecule interaction times. Opt. Lett.
**2021**, 46, 1538–1541. [Google Scholar] [CrossRef] - Bathe-Peters, M.; Gmach, P.; Boltz, H.-H.; Einsiedel, J.; Gotthardt, M.; Hübner, H.; Gmeiner, P.; Lohse, M.J.; Annibale, P. Visualization of β-adrenergic receptor dynamics and differential localization in cardiomyocytes. Proc. Natl. Acad. Sci. USA
**2021**, 118, e2101119118. [Google Scholar] [CrossRef] [PubMed] - Bakardjieva, A.; Galla, H.J.; Helmreich, E.J. Modulation of the β-Receptor Adenylate Cyclase Interactions in Cultured Chang Liver Cells by Phospholipid Enrichment. Biochemistry
**1979**, 18, 3016–3023. [Google Scholar] [CrossRef] [PubMed] - Heron, D.S.; Shinitzky, M.; Hershkowitz, M.; Samuel, D. Lipid fluidity markedly modulates the binding of serotonin to mouse brain membranes. Proc. Natl. Acad. Sci. USA
**1980**, 77, 7463–7467. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bockaert, J.; Fagni, L.; Dumuis, A.; Marin, P. GPCR interacting proteins (GIP). Pharmacol. Ther.
**2004**, 103, 203–221. [Google Scholar] [CrossRef] [PubMed] - Stelzer, N. Investigating How Cortical Actin Modulates GPCR Dynamics and Signaling. Master’s Thesis, TU Berlin, Berlin, Germany, 2021. [Google Scholar]
- Bockaert, J.; Marin, P.; Dumuis, A.; Fagni, L. The ’magic tail’ of G protein-coupled receptors: An anchorage for functional protein networks. FEBS Lett.
**2003**, 546, 65–72. [Google Scholar] [CrossRef] [Green Version] - Di Rienzo, C.; Gratton, E.; Beltram, F.; Cardarelli, F. Fast spatiotemporal correlation spectroscopy to determine protein lateral diffusion laws in live cell membranes. Proc. Natl. Acad. Sci. USA
**2013**, 110, 12307–12312. [Google Scholar] [CrossRef] [Green Version] - Posern, G.; Sotiropoulos, A.; Treisman, R. Mutant actins demonstrate a role for unpolymerized actin in control of transcription by serum response factor. Mol. Biol. Cell
**2002**, 13, 4167–4178. [Google Scholar] [CrossRef] [Green Version] - Gronewold, T.M.A.; Sasse, F.; Lünsdorf, H.; Reichenbach, H. Effects of rhizopodin and latrunculin B on the morphology and on the actin cytoskeleton of mammalian cells. Cell Tissue Res.
**1999**, 295, 121–129. [Google Scholar] [CrossRef] - Bornancin, F.; Pfister, C.; Chabre, M. The transitory complex between photoexcited rhodopsin and transducin. Eur. J. Biochem.
**1989**, 184, 687–698. [Google Scholar] [CrossRef] - Taylor, C.W. The role of G proteins in transmembrane signalling. Biochem. J.
**1990**, 272, 1–13. [Google Scholar] [CrossRef] [Green Version] - Gregorio, G.G.; Masureel, M.; Hilger, D.; Terry, D.S.; Juette, M.; Zhao, H.; Zhou, Z.; Perez-Aguilar, J.M.; Hauge, M.; Mathiasen, S.; et al. Single-molecule analysis of ligand efficacy in β2AR–G-protein activation. Nature
**2017**, 547, 68–73. [Google Scholar] [CrossRef] [PubMed] - Guo, C.; Levine, H. A thermodynamic model for receptor clustering. Biophys. J.
**1999**, 77, 2358–2365. [Google Scholar] [CrossRef] [Green Version] - Broday, D.M. Diffusion of clusters of transmembrane proteins as a model of focal adhesion remodeling. Bull. Math. Biol.
**2000**, 62, 891–924. [Google Scholar] [CrossRef] [Green Version] - Cairo, C.W. Signaling by Committee: Receptor Clusters Determine Pathways of Cellular Activation. ACS Chem. Biol.
**2007**, 2, 652–655. [Google Scholar] [CrossRef] [Green Version] - Caré, B.R.; Soula, H.A. Impact of receptor clustering on ligand binding. BMC Syst. Biol.
**2011**, 5, 48. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Caré, B.R.; Soula, H.A. Receptor clustering affects signal transduction at the membrane level in the reaction-limited regime. Phys. Rev. -Stat. Nonlinear Soft Matter Phys.
**2013**, 87, 1–7. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Watabe, M.; Yoshimura, H.; Arjunan, S.N.; Kaizu, K.; Takahashi, K. Signaling activations through G-protein-coupled-receptor aggregations. Phys. Rev. E
**2020**, 102, 32413. [Google Scholar] [CrossRef] - Woolf, P.J.; Linderman, J.J. An algebra of dimerization and its implications for G-protein coupled receptor signaling. J. Theor. Biol.
**2004**, 229, 157–168. [Google Scholar] [CrossRef] - Rosholm, K.R.; Leijnse, N.; Mantsiou, A.; Tkach, V.; Pedersen, S.L.; Wirth, V.F.; Oddershede, L.B.; Jensen, K.J.; Martinez, K.L.; Hatzakis, N.S.; et al. Membrane curvature regulates ligand-specific membrane sorting of GPCRs in living cells. Nat. Chem. Biol.
**2017**, 13, 724–729. [Google Scholar] [CrossRef] - Nikolaev, V.O.; Moshkov, A.; Lyon, A.R.; Miragoli, M.; Novak, P.; Paur, H.; Lohse, M.J.; Korchev, Y.E.; Harding, S.E.; Gorelik, J. β 2-adrenergic receptor redistribution in heart failure changes cAMP compartmentation. Science
**2010**, 327, 1653–1657. [Google Scholar] [CrossRef] - Daniels, D.R. Receptor-ligand diffusion-limited reaction rates on curved membranes. Chem. Phys. Lett.
**2022**, 795, 139516. [Google Scholar] [CrossRef] - Dorsaz, N.; De Michele, C.; Piazza, F.; De Los Rios, P.; Foffi, G. Diffusion-limited reactions in crowded environments. Phys. Rev. Lett.
**2010**, 105, 1–4. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Schütz, G.J.; Schindler, H.; Schmidt, T. Single-molecule microscopy on model membranes reveals anomalous diffusion. Biophys. J.
**1997**, 73, 1073–1080. [Google Scholar] [CrossRef] [Green Version] - Wawrezinieck, L.; Rigneault, H.; Marguet, D.; Lenne, P.F. Fluorescence correlation spectroscopy diffusion laws to probe the submicron cell membrane organization. Biophys. J.
**2005**, 89, 4029–4042. [Google Scholar] [CrossRef] [Green Version] - Haugh, J.M. Analysis of reaction-diffusion systems with anomalous subdiffusion. Biophys. J.
**2009**, 97, 435–442. [Google Scholar] [CrossRef] [Green Version] - Debnath, T.; Ghosh, P.K.; Li, Y.; Marchesoni, F.; Nori, F. Active diffusion limited reactions. J. Chem. Phys.
**2019**, 150. [Google Scholar] [CrossRef] - Di Rienzo, C.; Annibale, P. Visualizing the molecular mode of motion from a correlative analysis of localization microscopy datasets. Opt. Lett.
**2016**, 41, 4503. [Google Scholar] [CrossRef] - Sarkar, S. Concentration Dependence of Diffusion-Limited Reaction Rates and Its Consequences. Phys. Rev. X
**2020**, 10, 41032. [Google Scholar] [CrossRef] - Irannejad, R.; von Zastrow, M. GPCR signaling along the endocytic pathway. Curr. Opin. Cell Biol.
**2014**, 27, 109–116. [Google Scholar] [CrossRef] [Green Version] - Yu, J.Z.; Rasenick, M.M. Real-Time Visualization of a Fluorescent Gαs: Dissociation of the Activated G Protein from Plasma Membrane. Mol. Pharmacol.
**2002**, 61, 352–359. [Google Scholar] [CrossRef] [Green Version] - Gillespie, D.T. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys.
**1976**, 22, 403–434. [Google Scholar] [CrossRef] - Gillespie, D.T. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem.
**1977**, 81, 2340–2361. [Google Scholar] [CrossRef]

**Figure 1.**Simplified kinetic model illustrating how the second messenger molecule cAMP is produced downstream. The initial ternary reaction between the ligand (S), receptor (R) and G protein (G) on the cell membrane (gray line) leads to activated G protein (G${}^{*}$). This then is used together with membrane-bound AC and ATP from the cell cytosol (green) for cAMP production. We only depict the steps along the line of the conversion of the extra-cellular signal to production of the second messenger protein cAMP, which we explicitly model in this short perspective.

**Figure 2.**Sketch of the toy model. The particle (red dot) diffuses with a diffusion constant $D>0$ in a circular domain of radius $L>0$, searching for the active zone of radius ℓ ($0<\ell <L$). One example trajectory is depicted by the blue line. Within the active zone, the particle can get absorbed at rate $k>0$. For the target problem of this work, the particle can be thought of as the ligand-bound receptor $\mathrm{RS}$ finding the G protein, but it is moving with the combined diffusion constant.

**Figure 3.**Experimental evidence for how receptor dynamics can be modulated by the subplasmalemmal environment, in particular for the cortical actin mesh, showing the diffusion coefficient of the ${\beta}_{2}$-adrenergic receptor (${\beta}_{2}$-AR) in rat myoblast H9c2 cells for different regions of the cell and conditions. The ${\beta}_{2}$-AR is significantly slower in cells overexpressing actin. When actin is hyperpolimerized, using the mutant S14C, the diffusion is additionally reduced. The receptors are slowest around large actin bundles (fibers). We added a dashed line corresponding to the wild-type findings to highlight the effect of actin overexpression. Actin depolymerization by Latrunculin B (2 $\mathsf{\mu}$M) leads to a statistically significant recovery in the diffusion rate of the receptor. Error bars indicate the standard error of the mean. This figure reproduces data from [36].

**Figure 4.**Effective production rate ${k}_{\mathrm{eff}}$ given by the inverse of the mean first reaction time in the Doi model presented in the main text, with some additional details given in Appendix A. We present the results in dimensionless units (upper ordinate axis) and using exemplary physical values (lower axis).

**Left**: Direct results for ${k}_{\mathrm{eff}}$ as a function of the particle diffusion coefficient for varying ratios of domain size L and interaction range ℓ as coded by line color.

**Right**: The same data, but directly employing the diffusion-limited scaling ${k}_{\mathrm{eff}}\sim {\tau}_{L}^{-1}$. Here, we show only data for larger values of $L/\ell >3$, where the scaling is expected to hold.

**Figure 5.**(

**A**) Sketch of the change in the Doi model geometry on a cylinder for varying cylinder radii (cylinder is also sketched as an inset of (

**B**)). We compare models of the same interaction area $\pi {L}^{2}$. Due to the periodic boundary conditions, this leads to a larger extension in the axial direction.

**Figure 6.**(

**Left**) Visualization of the results from Equation (9). We show artificial dose–response curves for various values of the effective rate ${k}_{\mathrm{eff}}$ which, in the diffusion-limited regime, corresponds to the relative diffusion constant. In the upper x-axis, we provide a possible instance of physical values for a typical value of ${F}_{0}$. (

**Right**) The $EC50$ values (i.e., the stimulus concentrations for which the response is half maximal as a function of the dynamically controlled effective rate (lower x-axis) or the diffusion constant (using typical values; upper x-axis)).

**Figure 7.**Pilot experiment displaying the dose–response curves for cells stably expressing a FRET cAMP biosensor (Epac1-camps(H187)) and exposed to actin depolymerization. Depicted is the steady state response to an external increasing isoprenaline concentration. The control curve corresponds to two measurements with untreated cells, while the LatB curve is the response with additional LatB (25 nM) and thus a lesser degree of polymerization in the actin. Lower connectivity in the mesh should reduce the dynamical inhibition of the receptors and therefore increase their mobility. The normalization and the fits are performed using a Hill–Langmuir function with $n=1$. The numerical values for the constant C (in M) are given in the plot and quantify the obvious left shift upon the addition of LatB that is qualitatively in line with the reasoning for an increase in mobility.

**Figure 8.**In this perspective, we consider GPCR signaling as a reaction-diffusion process (A). Due to the unavailability of G protein following activation, the diffusive travel times are relevant and can be limiting to the observed reaction rates. From this, local modulation of signaling can be phrased as a local modulation of diffusive properties. We focus on a change in the subplasmalemmal environment (B) through a change in actin expression. This aims to directly modulate the diffusion constant D of the relative motion between GPCR and G protein. Less directly, the distribution of diffusive travel time and, thus, the global reaction kinetics can be affected by a change in geometry (i.e., local curvature (C)). The diffusion constant can also be modulated by other means, with possibilities including membrane composition (D) or GIPs binding to the receptor (E). A less direct way for changing diffusion times would be non-trivial spatial statistics (i.e., clustering or oligomerization (F)). The effect of C and F can be partially captured in the simple model discussed here (see Section 2.3). They also are related to the issue of anomalous diffusion (Section 2.3).

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**MDPI and ACS Style**

Boltz, H.-H.; Sirbu, A.; Stelzer, N.; de Lanerolle, P.; Winkelmann, S.; Annibale, P.
The Impact of Membrane Protein Diffusion on GPCR Signaling. *Cells* **2022**, *11*, 1660.
https://doi.org/10.3390/cells11101660

**AMA Style**

Boltz H-H, Sirbu A, Stelzer N, de Lanerolle P, Winkelmann S, Annibale P.
The Impact of Membrane Protein Diffusion on GPCR Signaling. *Cells*. 2022; 11(10):1660.
https://doi.org/10.3390/cells11101660

**Chicago/Turabian Style**

Boltz, Horst-Holger, Alexei Sirbu, Nina Stelzer, Primal de Lanerolle, Stefanie Winkelmann, and Paolo Annibale.
2022. "The Impact of Membrane Protein Diffusion on GPCR Signaling" *Cells* 11, no. 10: 1660.
https://doi.org/10.3390/cells11101660