Physical mechanisms of phase separation in living systems play key physiological roles and have recently been the focus of intensive studies. The strongly heterogeneous nature of such phenomena poses difficult modeling challenges that require going beyond mean-field approaches based on postulating a free energy landscape. The pathway we take here is to calculate the partition function starting from microscopic interactions by means of cavity methods, based on a tree approximation for the interaction graph. We illustrate them on the binary case and then apply them successfully to ternary systems, in which simpler one-factor approximations are proved inadequate. We demonstrate the agreement with lattice simulations and contrast our theory with coacervation experiments of associative de-mixing of nucleotides and poly-lysine. Different types of evidence are provided to support cavity methods as ideal tools for modeling biomolecular condensation, giving an optimal balance between the consideration of spatial aspects and fast computational results.

Biofisika Institute Barrio Sarriena s n 48940 Leioa Bizkaia Spain

Department of Philosophy Avenida de Tolosa 70 20018 Donostia San Sebastian Gipuzkoa Spain

Department of Physics Universitá di Roma la Sapienza Piazzale Aldo Moro 5 00185 Rome Italy

Donostia International Physics Center Paseo Manuel de Lardizabal 4 20018 Donostia San Sebastian Gipuzkoa Spain

Ikerbasque Foundation Alameda Urquijo 36 48011 Bilbao Bizkaia Spain

Institute for Theoretical Chemistry University of Vienna Vienna Austria

Institute of Nanotechnology Soft and Living Matter Laboratory Consiglio Nazionale delle Ricerche Piazzale Aldo Moro 5 00185 Rome Italy

Institute of Organic Chemistry and Biochemistry of the Czech Academy of Sciences Flemingovo náměstí 542 2 160 00 Praha 6 Czech Republic

Max Planck Institute of Molecular Cell Biology and Genetics Pfotenhauerstraße 108 01307 Dresden Germany

Zobrazit více v PubMed

Harold F.M. Oxford University Press; 2003. The Way of the Cell: Molecules, Organisms, and the Order of Life.

M Berg J., Stryer L., Tymoczko J.L. Springer-Verlag; 2015. Biochemistry.

Landau L., Lifshitz E.M., Pitaevskii L.P. Course of theory physics. Statistical Physics. 1980;vol. 9 chap. IX.

Wurtz J.D., Lee C.F. Stress granule formation via ATP depletiontriggered phase separation. New J. Phys. 2018;20:045008.

Shin Y., Brangwynne C.P. Liquid phase condensation in cell physiology and disease. Science. 2017;357:eaaf4382. doi: 10.1126/science.aaf4382. PubMed DOI

Conte M., Irani E., Chiariello A.M., Abraham A., Bianco S., Esposito A., Nicodemi M. Loop-extrusion and polymer phase-separation can co-exist at the single-molecule level to shape chromatin folding. Nat. Commun. 2022;13:4070. doi: 10.1038/s41467-022-31856-6. PubMed DOI PMC

Sartori P., Leibler S. Lessons from equilibrium statistical physics regarding the assembly of protein complexes. Proc. Natl. Acad. Sci. USA. 2020;117:114–120. doi: 10.1073/pnas.1911028117. PubMed DOI PMC

Frayn K.N. John Wiley & Sons; 2009. Metabolic Regulation: A Human Perspective.

Zhang Y., Bertulat B., Tencer A.H., Ren X., Wright G.M., Black J., Cardoso M.C., Kutateladze T.G. MORC3 forms nuclear condensates through phase separation. iScience. 2019;17:182–189. doi: 10.1016/j.isci.2019.06.030. PubMed DOI PMC

Petronilho E.C., Pedrote M.M., Marques M.A., Passos Y.M., Mota M.F., Jakobus B., de Sousa G.D.S., Pereira da Costa F., Felix A.L., Ferretti G.D.S., et al. Phase separation of p53 precedes aggregation and is affected by oncogenic mutations and ligands. Chem. Sci. 2021;12:7334–7349. doi: 10.1039/D1SC01739J. PubMed DOI PMC

Lu J., Qian J., Xu Z., Yin S., Zhou L., Zheng S., Zhang W. Emerging roles of liquid-liquid phase separation in cancer: from protein aggregation to immune-associated signaling. Front. Cell Dev. Biol. 2021;9:631486. doi: 10.3389/fcell.2021.631486. PubMed DOI PMC

Ahn J.H., Davis E.S., Daugird T.A., Zhao S., Quiroga I.Y., Uryu H., Li J., Storey A.J., Tsai Y.H., Keeley D.P., et al. Phase separation drives aberrant chromatin looping and cancer development. Nature. 2021;595:591–595. doi: 10.1038/s41586-021-03662-5. PubMed DOI PMC

Dora Tang T.Y., Rohaida Che Hak C., Thompson A.J., Kuimova M.K., Williams D.S., Perriman A.W., Mann S. Fatty acid membrane assembly on coacervate microdroplets as a step towards a hybrid protocell model. Nat. Chem. 2014;6:527–533. doi: 10.1038/nchem.1921. PubMed DOI

Zwicker D., Seyboldt R., Weber C.A., Hyman A.A., Jülicher F. Growth and division of active droplets provides a model for protocells. Nat. Phys. 2017;13:408–413. doi: 10.1038/nphys3984. DOI

Donau C., Späth F., Sosson M., Kriebisch B.A.K., Schnitter F., Tena-Solsona M., Kang H.S., Salibi E., Sattler M., Mutschler H., Boekhoven J. Active coacervate droplets as a model for membraneless organelles and protocells. Nat. Commun. 2020;11:5167. doi: 10.1038/s41467-020-18815-9. PubMed DOI PMC

Seyboldt R., Jülicher F. Role of hydrodynamic flows in chemically driven droplet division. New J. Phys. 2018;20:105010. doi: 10.1088/1367-2630/aae735. DOI

AI O. Moscowskii, Rabochii (Repr. in transl. Bernal 1967, S 1-214) 1924. Proiskhozhdenie Zhizny, Moscow; Izd.

Lauber N., Flamm C., Ruiz-Mirazo K. “Minimal metabolism”: a key concept to investigate the origins and nature of biological systems. Bioessays. 2021;43:2100103. doi: 10.1002/bies.202100103. PubMed DOI

Floris E., Piras A., Dall'Asta L., Gamba A., Hirsch E., Campa C.C. Physics of compartmentalization: how phase separation and signaling shape membrane and organelle identity. Comput. Struct. Biotechnol. J. 2021;19:3225–3233. doi: 10.1016/j.csbj.2021.05.029. PubMed DOI PMC

Milo R., Phillips R. Garland Science; 2015. Cell Biology by the Numbers.

Sear R.P., Cuesta J.A. Instabilities in complex mixtures with a large number of components. Phys. Rev. Lett. 2003;91:245701. doi: 10.1103/PhysRevLett.91.245701. PubMed DOI

Jacobs W.M., Frenkel D. Phase transitions in biological systems with many components. Biophys. J. 2017;112:683–691. doi: 10.1016/j.bpj.2016.10.043. PubMed DOI PMC

Jacobs W.M. Self-assembly of biomolecular condensates with shared components. Phys. Rev. Lett. 2021;126:258101. doi: 10.1103/PhysRevLett.126.258101. PubMed DOI

Carugno G., Neri I., Vivo P. Physical Biology. 2022. Instabilities of complex fluids with partially structured and partially random interactions. PubMed

Krishna S., Brenner M.P. Phase separation in fluids with many interacting components. Proc. Natl. Acad. Sci. USA. 2021;118 e2108551118. PubMed PMC

Flory P.J. Thermodynamics of high polymer solutions. J. Chem. Phys. 1942;10:51–61. doi: 10.1063/1.1723621. DOI

Huggins M.L. Some properties of solutions of long-chain compounds. J. Phys. Chem. 1942;46:151–158. doi: 10.1021/j150415a018. DOI

Overbeek J.T.G., Voorn M.J. Phase separation in polyelectrolyte solutions Theory of complex coacervation. J. Cell. Comp. Physiol. 1957;49:7–26. doi: 10.1002/jcp.1030490404. PubMed DOI

Lin Y.-H., Forman-Kay J.D., Chan H.S. Sequence-specific polyampholyte phase separation in membraneless organelles. Phys. Rev. Lett. 2016;117:178101. doi: 10.1103/PhysRevLett.117.178101. PubMed DOI

Lin Y.-H., Brady J.P., Forman-Kay J.D., Chan H.S. Charge pattern matching as a ’fuzzy’mode of molecular recognition for the functional phase separations of intrinsically disordered proteins. New J. Phys. 2017;19:115003. doi: 10.1088/1367-2630/aa9369. DOI

Cardy J. Vol. 5. Cambridge university press; 1996. (Scaling and Renormalization in Statistical Physics).

Bethe H.A. Statistical theory of superlattices. Proc. R. Soc. London Series A Math. Phys. Sci. 1935;150:552–575. doi: 10.1098/rspa.1935.0122. DOI

Peierls R. Statistical theory of superlattices with unequal concentrations of the components. Proc. R. Soc. London Series A Math. Phys. Sci. 1936;154:207–222. doi: 10.1098/rspa.1936.0047. DOI

Mézard M., Parisi G. The cavity method at zero temperature. J. Stat. Phys. 2003;111:1–34. doi: 10.1023/A:1022221005097. DOI

Yedidia J.S., Freeman W.T., Weiss Y. Understanding belief propagation and its generalizations. Exploring artificial Intelligence in the New Millennium, 8; 2003. p. 236–239.

Mezard M., Montanari A. Oxford University Press; 2009. Information, Physics, and Computation.

Baxter R.J. Elsevier; 2016. Exactly Solved Models in Statistical Mechanics.

Mertens S. Computational complexity for physicists. Comput. Sci. Eng. 2002;4:31–47. doi: 10.1109/5992.998639. DOI

Duke T., Graham I. Equilibrium mechanisms of receptor clustering. Prog. Biophys. Mol. Biol. 2009;100:18–24. PubMed

Sarker R., Mohammadian M., Yao X. Vol. 48. Springer Science & Business Media; 2002. (Evolutionary Optimization).

Kawasaki K. Diffusion constants near the critical point for time-dependent Ising models. I. Phys. Rev. 1966;145:224–230. doi: 10.1103/PhysRev.145.224. DOI

Schupper N., Shnerb N.M. Inverse melting and inverse freezing: a spin model. Phys. Rev. E. 2005;72:046107. doi: 10.1103/PhysRevE.72.046107. PubMed DOI

Gross D., Kanter I., Sompolinsky H. Mean-field theory of the Potts glass. Phys. Rev. Lett. 1985;55:304–307. doi: 10.1103/PhysRevLett.55.304. PubMed DOI

Krząkała F., Zdeborová L. Potts glass on random graphs. Europhys. Lett. 2008;81:57005. doi: 10.1209/0295-5075/81/57005. DOI

McCarty J., Delaney K.T., Danielsen S.P.O., Fredrickson G.H., Shea J.E. Complete phase diagram for liquid-liquid phase separation of intrinsically disordered proteins. J. Phys. Chem. Lett. 2019;10:1644–1652. doi: 10.1021/acs.jpclett.9b00099. PubMed DOI PMC

Pal T., Wessén J., Das S., Chan H.S. Subcompartmentalization of polyampholyte species in organelle-like condensates is promoted by charge-pattern mismatch and strong excluded-volume interaction. Phys. Rev. E. 2021;103 doi: 10.1103/PhysRevE.103.042406. PubMed DOI

Wessén J., Das S., Pal T., Chan H.S. Analytical formulation and field-theoretic simulation of sequence- specific phase separation of protein-like heteropolymers with short-and long-spatial-range interactions. J. Phys. Chem. B. 2022;126:9222–9245. doi: 10.1021/acs.jpcb.2c06181. PubMed DOI

Lin Y.-H., Wessén J., Pal T., Das S., Chan H.S. Phase-Separated Biomolecular Condensates. Springer; 2023. Numerical techniques for applications of analytical theories to sequence-dependent phase separations of intrinsically disordered proteins; pp. 51–94. PubMed

Binder K., Landau D.P. Finite-size scaling at first-order phase transitions. Phys. Rev. B. 1984;30:1477–1485.