Abstract
Protein aggregation on the plasma membrane (PM) is of critical importance to many cellular processes such as cell adhesion, endocytosis, fibrillar conformation, and vesicle transport. Lateral diffusion of protein aggregates or clusters on the surface of the PM plays an important role in governing their heterogeneous surface distribution. However, the stability behavior of the surface distribution of protein aggregates remains poorly understood. Therefore, understanding the spatial patterns that can emerge on the PM solely through protein–protein interaction, lateral diffusion, and feedback is an important step toward a complete description of the mechanisms behind protein clustering on the cell surface. In this work, we investigate the pattern formation of a reaction–diffusion model that describes the dynamics of a system of ligand–receptor complexes. The purely diffusive ligand in the cytosol can bind receptors in the PM and the resultant ligand–receptor complexes not only diffuse laterally but can also form clusters resulting in different oligomers. Finally, the largest oligomers recruit ligands from the cytosol using positive feedback. From a methodological viewpoint, we provide theoretical estimates for diffusion-driven instabilities of the protein aggregates based on the Turing mechanism. Our main result is a threshold phenomenon, in which a sufficiently high recruitment of ligands promotes the input of new monomeric components and consequently drives the formation of a single-patch spatially heterogeneous steady state.
Similar content being viewed by others
References
Abel SM, Roose JP, Groves JT, Weiss A, Chakraborty AK (2012) The membrane environment can promote or suppress bistability in cell signaling networks. J Phys Chem B 116(11):3630–3640
Achdou Y, Franchi B, Marcello N, Tesi MC (2013) A qualitative model for aggregation and diffusion of \(\beta \)-amyloid in Alzheimer’s disease. J Math Biol 67(6–7):1369–1392
Adam G, Delbrück M (1968) Reduction of dimensionality in biological diffusion processes. Struct Chem Mol Biol 198:198–215
Albersheim P, Anderson-Prouty AJ (1975) Carbohydrates, proteins, cell surfaces, and the biochemistry of pathogenesis. Annu Rev Plant Physiol 26(1):31–52
Alonso S, Baer M (2010) Phase separation and bistability in a three-dimensional model for protein domain formation at biomembranes. Phys Biol 7(4):046012
Amestoy P, Buttari A, Duff I, Guermouche A, LExcellent JY, Uçar B (2011) Mumps. Encyclopedia of parallel computing, pp 1232–1238
Andreasen M, Lorenzen N, Otzen D (2015) Interactions between misfolded protein oligomers and membranes: a central topic in neurodegenerative diseases? Biochim et Biophys Acta-Biomembr 1848(9):1897–1907
Anoop R, Ralf L (2018) Membranes as modulators of amyloid protein misfolding and target of toxicity. Biochim Biophys Acta Biomembr 1860(9):1863–1875
Arosio P, Rima S, Lattuada M, Morbidelli M (2012) Population balance modeling of antibodies aggregation kinetics. J Phys Chem B 116(24):7066–7075
Arosio P, Knowles TPJ, Linse S (2015) On the lag phase in amyloid fibril formation. Phys Chem Chem Phys 17(12):7606–7618
Baisamy L, Jurisch N, Diviani D (2005) Leucine zipper-mediated homo-oligomerization regulates the Rho–GEF Activity of AKAP-Lbc. J Biol Chem 280(15):15405–15412
Bentz J, Nir S (1981) Mass action kinetics and equilibria of reversible aggregation. J Chem Soc Faraday Trans 1 Phys Chem Condens Ph 77(6):1249–1275
Berg HC (1977) E.M. Purcell physics of chemoreception. Biophys J 20:193–219
Bertsch M, Franchi B, Marcello N, Tesi MC, Tosin A (2016) Alzheimer’s disease: a mathematical model for onset and progression. Math Med Biol J IMA 34(2):193–214
Beta C, Amselem G, Bodenschatz E (2008) A bistable mechanism for directional sensing. N J Phys 10(8):083015
Burke SP, Schumann TEW (1928) Diffusion flames. Ind Eng Chem 20(10):998–1004
Changeux J-P, Thiéry J, Tung Y, Kittel C (1967) On the cooperativity of biological membranes. Proc Natl Acad Sci U S A 57(2):335
Chatani E, Yamamoto N (2018) Recent progress on understanding the mechanisms of amyloid nucleation. Biophys Rev 10(2):527–534
Chen CP, Posy S, Ben-Shaul A, Shapiro L, Honig BH (2005) Specificity of cell–cell adhesion by classical cadherins: critical role for low-affinity dimerization through-strand swapping. Proc Natl Acad Sci 102(24):8531–8536
Choquet D (2010) Fast AMPAR trafficking for a high-frequency synaptic transmission. Eur J Neurosci 32(2):250–260
Christensen SM, Tu H-L, Jun JE, Alvarez S, Triplet MG, Iwig JS, Yadav KK, Bar-Sagi D, Roose JP, Groves JT (2016) One-way membrane trafficking of sos in receptor-triggered ras activation. Nat Struct Mol Biol 23(9):838
Cohen SIA, Vendruscolo M, Dobson CM, Knowles TPJ (2012) From macroscopic measurements to microscopic mechanisms of protein aggregation. J Mol Biol 421(2–3):160–171
Darnell JE, Lodish HF, Baltimore D et al (1990) Molecular cell biology, vol 2. Scientific American Books, New York
Davide C, Leah E-K, Mackenzie John A, Stéphanie P, Anotida M (2018) A coupled bulk–surface model for cell polarisation. J Theor Biol 481:119–135
Denk J, Kretschmer S, Halatek J, Hartl C, Schwille P, Frey E (2018) MinE conformational switching confers robustness on self-organized Min protein patterns. Proc Natl Acad Sci 115(18):4553–4558
Diegmiller R, Montanelli H, Muratov CB, Shvartsman SY (2018) Spherical caps in cell polarization. Biophys J 115(1):26–30
Drake RL (1972) A general mathematical survey of the coagulation equation. Top Curr Aerosol Res (Part 2) 3(Part 2):201–376
Erwin F, Jacob H, Simon K, Petra S (2018) Protein pattern formation. Physics of biological membranes. Springer, Berlin, pp 229–260
Franchi B, Lorenzani S (2016) From a microscopic to a macroscopic model for Alzheimer disease: two-scale homogenization of the Smoluchowski equation in perforated domains. J Nonlinear Sci 26(3):717–753
Gallier Jean (2009) Notes on spherical harmonics and linear representations of lie groups. preprint
Gan Q, Salussolia CL, Wollmuth LP (2015) Assembly of AMPA receptors: mechanisms and regulation. The Journal of Physiology 593(Pt 1):39–48
Getz MC, Nirody JA, Rangamani P (2018) Stability analysis in spatial modeling of cell signaling. Wiley Interdiscip Rev Syst Biol Med 10(1):e1395
Gierer A, Meinhardt H (1972) A theory of biological pattern formation. Kybernetik 12(1):30–39
Giese W, Eigel M, Westerheide S, Engwer C, Klipp E (2015) Influence of cell shape, inhomogeneities and diffusion barriers in cell polarization models. Phys Biol 12(6):066014
Goehring NW, Chowdhury D, Hyman AA, Grill SW (2010) FRAP analysis of membrane-associated proteins: lateral diffusion and membrane-cytoplasmic exchange. Biophys J 99(8):2443–2452
Goryachev AB, Pokhilko AV (2008) Dynamics of Cdc42 network embodies a Turing-type mechanism of yeast cell polarity. FEBS Lett 582(10):1437–1443
Guidotti G (1972) The composition of biological membranes. Arch Intern Med 129(2):194–201
Habchi J, Chia S, Galvagnion C, Michaels TCT, Bellaiche MMJ, Ruggeri FS, Sanguanini M, Idini I, Kumita JR, Sparr E, Linse S, Dobson CM, Knowles TPJ, Vendruscolo M (2018) Cholesterol catalyses \(\beta \)42 aggregation through a heterogeneous nucleation pathway in the presence of lipid membranes. Nat Chem 10(6):673–683
Han S, Kollmer M, Markx D, Claus S, Walther P, Fändrich M (2017) Amyloid plaque structure and cell surface interactions of \(\beta \)-amyloid fibrils revealed by electron tomography. Sci Rep 7:43577
Hashimoto K, Panchenko AR (2010) Mechanisms of protein oligomerization, the critical role of insertions and deletions in maintaining different oligomeric states. Proc Natl Acad Sci 107(47):20352–20357
Haugh JM, Lauffenburger DA (1997) Physical modulation of intracellular signaling processes by locational regulation. Biophys J 72(5):2014–2031
Holmes BB, Diamond MI (2012) Cellular mechanisms of protein aggregate propagation. Curr Opin Neurol 25(6):721–726
Ispolatov I (2005) Binding properties and evolution of homodimers in protein-protein interaction networks. Nucleic Acids Res 33(11):3629–3635
Jain MK, Wagner RC (1988) Introduction to biological membranes. Wiley, New York
Jarrett JT, Lansbury PT Jr (1993) Seeding one-dimensional crystallization of amyloid: a pathogenic mechanism in alzheimer’s disease and scrapie? Cell 73(6):1055–1058
Johannes L, Pezeshkian W, Ipsen JH, Shillcock JC (2018) Clustering on Membranes: fluctuations and More. Trends Cell Biol 28(5):405–415
Johnson SM, Connelly S, Fearns C, Powers ET, Kelly JW (2012) The transthyretin amyloidoses: from delineating the molecular mechanism of aggregation linked to pathology to a regulatory-agency-approved drug. J Mol Biol 421(2–3):185–203
Kholodenko Boris N (2006) Cell-signalling dynamics in time and space. Nat Rev Mol cell Biol 7(3):165–176
Khuc TP, Nicola Ernesto M, Goehring Nathan W, Vijay KK, Grill Stephan W (2014) Parameter-space topology of models for cell polarity. J Phys 16(6):065009
Lao QZ, Kobrinsky E, Liu Z, Soldatov NM (2010) Oligomerization of \(\text{ Ca }_v\beta \) subunits is an essential correlate of Ca2+ channel activity. J Fed Am Soc Exp Biol 24(12):5013–5023
Lemmon Mark A, Joseph S (2010) Cell signaling by receptor tyrosine kinases. Cell 141(7):211–225
Lorent JH, Diaz-Rohrer B, Lin X, Spring K, Gorfe AA, Levental KR, Levental I (2017) Structural determinants and functional consequences of protein affinity for membrane rafts. Nat Commun 8(1):1219
Madzvamuse A, Chung AHW, Venkataraman C (2015) Stability analysis and simulations of coupled bulk–surface reaction–diffusion systems. Proc R Soc A Math Phys Eng Sci 471(2175):20140546–20140546
Manor A, Shnerb NM (2006) Dynamical failure of Turing patterns. Europhys Lett 74(5):837
Marianayagam NJ, Sunde M, Matthews JM (2004) The power of two: protein dimerization in biology. Trends Biochem Sci 29(11):618–625
McCloskey MA, Poo MM (1986) Rates of membrane-associated reactions: reduction of dimensionality revisited. J Cell Biol 102(1):88–96
Meisl G, Kirkegaard JB, Arosio P, Michaels TCT, Vendruscolo M, Dobson CM, Linse S, Knowles TPJ (2016) Molecular mechanisms of protein aggregation from global fitting of kinetic models. Nat Protoc 11(2):252
Mori Y, Jilkine A, Edelstein-Keshet L (2008) Wave-pinning and cell polarity from a bistable reaction–diffusion system. Biophys J 94(9):3684–3697
Mori Y, Jilkine A, Edelstein-Keshet L (2011) Asymptotic and bifurcation analysis of wave-pinning in a reaction–diffusion model for cell polarization. SIAM J Appl Math 71(4):1401–1427
Muratcioglu S, Chavan TS, Freed BC, Jang H, Lyuba Khavrutskii R, Freed N, Dyba MA, Stefanisko K, Tarasov SG, Gursoy A et al (2015) GTP-dependent K-Ras dimerization. Structure 23(7):1325–1335
Murray JD (1993) Mathematical biology, 2nd edn. Springer, New York
Padmanabhan P, Martinez-Mairmol R, Xia D, Gotz J, Meunier FA (2019) Frontotemporal dementia mutant Tau promotes aberrant Fyn nanoclustering in hippocampal dendritic spines. eLife 8:e45040
Porat-Shliom N, Milberg O, Masedunskas A, Weigert R (2013) Multiple roles for the actin cytoskeleton during regulated exocytosis. Cell Mol Life Sci 70(12):2099–2121
Postma M, Bosgraaf L, Loovers HM, Van Haastert PJM (2004) Chemotaxis: signalling modules join hands at front and tail. EMBO Rep 5(1):35–40
Rangamani P, Lipshtat A, Azeloglu EU, Calizo RC, Hu M, Ghassemi S, Hone J, Scarlata S, Neves SR, Iyengar R (2013) Decoding information in cell shape. Cell 154(6):1356–1369
Rappel W-J, Edelstein-Keshet L (2017) Mechanisms of cell polarization. Curr Opin Syst Biol 3:43–53
Rätz A (2015) Turing-type instabilities in bulk–surface reaction–diffusion systems. J Comput Appl Math 289:142–152
Rätz A, Röger M (2012) Turing instabilities in a mathematical model for signaling networks. J Math Biol 65(6–7):1215–1244
Rätz A, Röger M (2014) Symmetry breaking in a bulk-surface reaction–diffusion model for signalling networks. Nonlinearity 27(8):1805
Rosa C, Rocha Fernando A, Damas Ana M, Martins Pedro M (2012) A generic crystallization-like model that describes the kinetics of amyloid fibril formation. J Biol Chem 287(36):30585–30594
Sarabipour S, Hristova K (2016) Mechanism of FGF receptor dimerization and activation. Nat Commun 7:10262
Sarkar B, Das A, Maiti S (2013) Thermodynamically stable amyloid-\(\beta \) monomers have much lower membrane affinity than the small oligomers. Front Physiol 4:84
Scott FH (2005) Elements of chemical reaction engineering, 4th edn. Prentice Hall, New Jersey
Semplice M, Veglio A, Naldi G, Serini G, Gamba A (2012) A bistable model of cell polarity. PLoS ONE 7(2):e30977
Sholpan A, Xiaoguang Y, Lee James C-M (2011) Impacts of membrane biophysics in Alzheimer’s disease: from amyloid precursor protein processing to \(\beta \) peptide-induced membrane changes. Int J Alzheimer’s Dis 17(2011):134971
Silverman RA et al (1972) Special functions and their applications. Courier Corporation, North Chelmsford
Sleno R, Hbert TE (2018) Chapter five—the dynamics of GPCR oligomerization and their functional consequences. In: Arun KS, (ed) International review of cell and molecular biology, vol 338 of G protein-coupled receptors: emerging paradigms in activation, signaling and regulation Part A, pp 141–171. Academic Press
Smoluchowski M (1918) Versuch einer mathematischen Theorie der Koagulationskinetik kolloider Lösungen. Zeitschrift für Physikalische Chemie 92(1):129–168
Stephen S, Neil D (2018) Model reduction permits Turing instability analysis of arbitrary reaction–diffusion models. J R Soc Interface 15:213298
Stillwell W (2013) An introduction to biological membranes: from bilayers to rafts. Newnes, Amsterdam
Strogatz SH (1994) Nonlinear dynamics and chaos with applications to physics. Biology, chemistry and engineering, 1st edn. Westview Press, Routledge
Trong PK, Nicola EM, Goehring NW, Kumar KV, Grill SW (2014) Parameter-space topology of models for cell polarity. N J Phys 16(6):065009
Turing AM (1952) The chemical basis of morphogenesis. Philos Trans R Soc Lond Ser B Biol Sci 237(641):37–72
van Oosterom A (2006) The surface Laplacian operator of the potentials on a bounded volume conductor has a unique inverse. IEEE Trans Biomed Eng 53(7):1449–1450
Yanyan C, Javier B (2019) A non-linear analysis of Turing pattern formation. PLoS ONE 14(8):1–9
Yeagle L (2011) The structure of biological membranes. CRC Press, Boca Raton
Zhang Y-J, Shi J-M, Bai C-J, Wang H, Li H-Y, Yi W, Ji S-R (2012) Intra-membrane oligomerization and extra-membrane oligomerization of amyloid-\(\beta \) peptide are competing processes as a result of distinct patterns of motif interplay. J Biol Chem 287(1):748–756
Zidar M, Kuzman D, Ravnik M (2018) Characterisation of protein aggregation with the smoluchowski coagulation approach for use in biopharmaceuticals. Soft Matter 14(29):6001–6012
Acknowledgements
This work was supported by Air Force Office of Scientific Research (AFOSR) Multidisciplinary University Research Initiative (MURI) Grant FA9550-18-1-0051 to P. Rangamani. M. Holst was supported in part by NSF Awards DMS 1620366 and DMS 1345013. We also thank the Ph.D students Jennifer Fromm, Allen Leung, and Kiersten Scott from the Rangamani Lab for the comments and feedback.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Electronic supplementary material
Below is the link to the electronic supplementary material.
Supplementary material 2 (mp4 7775 KB)
Rights and permissions
About this article
Cite this article
Stolerman, L.M., Getz, M., Smith, S.G.L. et al. Stability Analysis of a Bulk–Surface Reaction Model for Membrane Protein Clustering. Bull Math Biol 82, 30 (2020). https://doi.org/10.1007/s11538-020-00703-4
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s11538-020-00703-4