Curved cellular membranes are both abundant and functionally relevant. While novel tomography approaches reveal the structural details of curved membranes, their dynamics pose an experimental challenge. Curvature especially affects the diffusion of lipids and macromolecules, yet neither experiments nor continuum models distinguish geometric effects from those caused by curvature-induced changes in membrane properties. Molecular simulations could excel here, yet despite community interest toward curved membranes, tools for their analysis are still lacking. Here, we satisfy this demand by introducing CurD, our novel and openly available implementation of the Vertex-oriented Triangle Propagation algorithm to the study of lipid diffusion along membranes with mean and/or Gaussian curvature. This approach, aided by our highly optimized implementation, computes geodetic distances significantly faster than conventional implementations of path-finding algorithms. Our tool, applied to coarse-grained simulations, allows for the first time the analysis of curvature effects on diffusion at size scales relevant to physiological processes such as endocytosis. Our analyses with different membrane geometries reveal that Gaussian curvature plays a surprisingly small role on lipid motion, whereas mean curvature; i.e., the packing of lipid headgroups largely dictates their mobility.

Institute of Biotechnology University of Helsinki FI 00790 Helsinki Finland

Institute of Organic Chemistry and Biochemistry of the Czech Academy of Sciences Flemingovo nám 542 2 CZ 16000 Prague 6 Czech Republic

Zobrazit více v PubMed

Singharoy A.; Maffeo C.; Delgado-Magnero K. H.; Swainsbury D. J.; Sener M.; Kleinekathöfer U.; Vant J. W.; Nguyen J.; Hitchcock A.; Isralewitz B.; et al. Atoms to Phenotypes: Molecular Design Principles of Cellular Energy Metabolism. Cell 2019, 179, 1098–1111. 10.1016/j.cell.2019.10.021. PubMed DOI PMC

Pezeshkian W.; König M.; Wassenaar T. A.; Marrink S. J. Backmapping Triangulated Surfaces to Coarse-Grained Membrane Models. Nat. Commun. 2020, 11, 2296.10.1038/s41467-020-16094-y. PubMed DOI PMC

Enkavi G.; Javanainen M.; Kulig W.; Róg T.; Vattulainen I. Multiscale Simulations of Biological Membranes: The Challenge to Understand Biological Phenomena in a Living Substance. Chem. Rev. 2019, 119, 5607–5774. 10.1021/acs.chemrev.8b00538. PubMed DOI PMC

Vögele M.; Hummer G. Divergent Diffusion Coefficients in Simulations of Fluids and Lipid Membranes. J. Phys. Chem. B 2016, 120, 8722–8732. 10.1021/acs.jpcb.6b05102. PubMed DOI

Ingólfsson H. I.; Neale C.; Carpenter T. S.; Shrestha R.; López C. A.; Tran T. H.; Oppelstrup T.; Bhatia H.; Stanton L. G.; Zhang X.; et al. Machine Learning–Driven Multiscale Modeling Reveals Lipid-Dependent Dynamics of RAS Signaling Proteins. Proc. Natl. Acad. Sci. U.S.A. 2022, 119, e2113297119.10.1073/pnas.2113297119. PubMed DOI PMC

Larsen A. H. Molecular Dynamics Simulations of Curved Lipid Membranes. Int. J. Mol. Sci. 2022, 23, 8098.10.3390/ijms23158098. PubMed DOI PMC

McMahon H. T.; Boucrot E. Membrane Curvature at a Glance. J. Cell Sci. 2015, 128, 1065–1070. 10.1242/jcs.114454. PubMed DOI PMC

Anderson R. G. The Caveolae Membrane System. Annu. Rev. Biochem. 1998, 67, 199–225. 10.1146/annurev.biochem.67.1.199. PubMed DOI

Colina-Tenorio L.; Horten P.; Pfanner N.; Rampelt H. Shaping the Mitochondrial Inner Membrane in Health and Disease. J. Int. Med. 2020, 287, 645–664. 10.1111/joim.13031. PubMed DOI

Olzmann J. A.; Carvalho P. Dynamics and Functions of Lipid Droplets. Nat. Rev. Mol. Cell Biol. 2019, 20, 137–155. 10.1038/s41580-018-0085-z. PubMed DOI PMC

Dasgupta R.; Miettinen M. S.; Fricke N.; Lipowsky R.; Dimova R. The Glycolipid GM1 Reshapes Asymmetric Biomembranes and Giant Vesicles by Curvature Generation. Proc. Natl. Acad. Sci. U.S.A. 2018, 115, 5756–5761. 10.1073/pnas.1722320115. PubMed DOI PMC

Mim C.; Unger V. M. Membrane Curvature and Its Generation by BAR Proteins. Trends Biochem. Sci. 2012, 37, 526–533. 10.1016/j.tibs.2012.09.001. PubMed DOI PMC

Woodward X.; Javanainen M.; Fábián B.; Kelly C. V. Nanoscale Membrane Curvature Sorts Lipid Phases and Alters Lipid Diffusion. Biophys. J. 2023, 122, 2203–2215. 10.1016/j.bpj.2023.01.001. PubMed DOI PMC

Baoukina S.; Ingólfsson H. I.; Marrink S. J.; Tieleman D. P. Curvature-Induced Sorting of Lipids in Plasma Membrane Tethers. Adv. Theory Simul. 2018, 1, 1800034.10.1002/adts.201800034. DOI

Boyd K. J.; May E. R. BUMPy: AModel-Independent Tool for Constructing Lipid Bilayers of Varying Curvature and Composition. J. Chem. Theory Comput. 2018, 14, 6642–6652. 10.1021/acs.jctc.8b00765. PubMed DOI PMC

Yesylevskyy S.; Khandelia H. EnCurv: Simple Technique of Maintaining Global Membrane Curvature in Molecular Dynamics Simulations. J. Chem. Theory. Comput. 2021, 17, 1181–1193. 10.1021/acs.jctc.0c00800. PubMed DOI

Yesylevskyy S. O.; Rivel T.; Ramseyer C. The Influence of Curvature on the Properties of the Plasma Membrane. Insights From Atomistic Molecular Dynamics Simulations. Sci. Rep. 2017, 7, 16078.10.1038/s41598-017-16450-x. PubMed DOI PMC

Bhatia H.; Ingólfsson H. I.; Carpenter T. S.; Lightstone F. C.; Bremer P.-T. MemSurfer: ATool for Robust Computation and Characterization of Curved Membranes. J. Chem. Theory Comput. 2019, 15, 6411–6421. 10.1021/acs.jctc.9b00453. PubMed DOI

Davoudi S.; Ghysels A. Defining Permeability of Curved Membranes in Molecular Dynamics Simulations. Biophys. J. 2023, 122, 2082–2091. 10.1016/j.bpj.2022.11.028. PubMed DOI PMC

Bhaskara R. M.; Grumati P.; Garcia-Pardo J.; Kalayil S.; Covarrubias-Pinto A.; Chen W.; Kudryashev M.; Dikic I.; Hummer G. Curvature Induction and Membrane Remodeling by FAM134B Reticulon Homology Domain Assist Selective ER-Phagy. Nat. Commun. 2019, 10, 2370.10.1038/s41467-019-10345-3. PubMed DOI PMC

Kabbani A. M.; Woodward X.; Kelly C. V. Revealing the Effects of Nanoscale Membrane Curvature on Lipid Mobility. Membranes 2017, 7, 60.10.3390/membranes7040060. PubMed DOI PMC

Adler J.; Sintorn I.-M.; Strand R.; Parmryd I. Conventional Analysis of Movement on Non-flat Surfaces Like the Plasma Membrane Makes Brownian Motion Appear Anomalous. Commun. Biol. 2019, 2, 12.10.1038/s42003-018-0240-2. PubMed DOI PMC

Gesper A.; Wennmalm S.; Hagemann P.; Eriksson S.-G.; Happel P.; Parmryd I. Variations in Plasma Membrane Topography Can Explain Heterogenous Diffusion Coefficients Obtained by Fluorescence Correlation Spectroscopy. Front. Cell Dev. Biol. 2020, 8, 767.10.3389/fcell.2020.00767. PubMed DOI PMC

Metzler R.; Jeon J.-H.; Cherstvy A. G.; Barkai E. Anomalous Diffusion Models and Their Properties: Non-stationarity, Non-ergodicity, and Ageing at the Centenary of Single Particle Tracking. Phys. Chem. Chem. Phys. 2014, 16, 24128–24164. 10.1039/C4CP03465A. PubMed DOI

Adler J.; Shevchuk A. I.; Novak P.; Korchev Y. E.; Parmryd I. Plasma Membrane Topography and Interpretation of Single-Particle Tracks. Nat. Methods 2010, 7, 170–171. 10.1038/nmeth0310-170. PubMed DOI

Reister E.; Seifert U. Lateral Diffusion of a Protein on a Fluctuating Membrane. Europhys. Lett. 2005, 71, 859.10.1209/epl/i2005-10139-6. DOI

Ohta T. Brownian Motion on a Fluctuating Random Geometry. J. Phys. Soc. Jpn. 2020, 89, 074001.10.7566/JPSJ.89.074001. DOI

Naji A.; Brown F. L. Diffusion on Ruffled Membrane Surfaces. J. Chem. Phys. 2007, 126, 06B611.10.1063/1.2739526. PubMed DOI

Boal D.Mechanics of the Cell; Cambridge University Press, 2012.

Slepecky N.; Ulfendahl M.; Flock Å. Effects of Caffeine and Tetracaine on Outer Hair Cell Shortening Suggest Intracellular Calcium Involvement. Hear. Res. 1988, 32, 11–21. 10.1016/0378-5955(88)90143-8. PubMed DOI

Meyer H. W.; Westermann M.; Stumpf M.; Richter W.; Ulrich A. S.; Hoischen C. Minimal Radius of Curvature of Lipid Bilayers in the Gel Phase State Corresponds to the Dimension of Biomembrane Structures “Caveolae. J. Struct. Biol. 1998, 124, 77–87. 10.1006/jsbi.1998.4042. PubMed DOI

Ölveczky B. P.; Verkman A. Monte Carlo Analysis of Obstructed Diffusion in Three Dimensions: Application to Molecular Diffusion in Organelles. Biophys. J. 1998, 74, 2722–2730. 10.1016/S0006-3495(98)77978-0. PubMed DOI PMC

Almeida P. F.; Vaz W. L.. Handbook of Biological Physics; Elsevier, 1995; Vol. 1; pp 305–357.

Sbalzarini I. F.; Hayer A.; Helenius A.; Koumoutsakos P. Simulations of (An)isotropic Diffusion on Curved Biological Surfaces. Biophys. J. 2006, 90, 878–885. 10.1529/biophysj.105.073809. PubMed DOI PMC

Nieto V.; Crowley J.; Santos D. E.; Monticelli L. Birth of an Organelle: Molecular Mechanism of Lipid Droplet Biogenesis. bioRxiv 2023, 10.1101/2023.07.28.550987. DOI

Faraudo J. Diffusion Equation on Curved Surfaces. I. Theory and Application to Biological Membranes. J. Chem. Phys. 2002, 116, 5831–5841. 10.1063/1.1456024. DOI

Yoshigaki T. Theoretically Predicted Effects of Gaussian Curvature on Lateral Diffusion of Membrane Molecules. Phys. Rev. E 2007, 75, 041901.10.1103/PhysRevE.75.041901. PubMed DOI

Gov N. S. Diffusion in Curved Fluid Membranes. Phys. Rev. E 2006, 73, 041918.10.1103/PhysRevE.73.041918. PubMed DOI

Yesylevskyy S.; Ramseyer C. Determination of Mean and Gaussian Curvatures of Highly Curved Asymmetric Lipid Bilayers: The Case Study of the Influence of Cholesterol on the Membrane Shape. Phys. Chem. Chem. Phys. 2014, 16, 17052–17061. 10.1039/C4CP01544D. PubMed DOI

Marrink S. J.; De Vries A. H.; Mark A. E. Coarse Grained Model for Semiquantitative Lipid Simulations. J. Phys. Chem. B 2004, 108, 750–760. 10.1021/jp036508g. DOI

Marrink S. J.; Risselada H. J.; Yefimov S.; Tieleman D. P.; De Vries A. H. The MARTINI Force Field: Coarse Grained Model for Biomolecular Simulations. J. Phys. Chem. B 2007, 111, 7812–7824. 10.1021/jp071097f. PubMed DOI

Crane K.; Livesu M.; Puppo E.; Qin Y. A Survey of Algorithms for Geodesic Paths and Distances. arXiv:2007.10430 2020, 10.48550/arXiv.2007.10430. DOI

Qin Y.; Han X.; Yu H.; Yu Y.; Zhang J. Fast and Exact Discrete Geodesic Computation Based on Triangle-Oriented Wavefront Propagation. ACM Trans. Graph. 2016, 35, 1–13. 10.1145/2897824.2925930. DOI

Hummer G. Position-Dependent Diffusion Coefficients and Free Energies From Bayesian Analysis of Equilibrium and Replica Molecular Dynamics Simulations. New J. Phys. 2005, 7, 34.10.1088/1367-2630/7/1/034. DOI

Christensen M. How to Simulate Anisotropic Diffusion Processes on Curved Surfaces. J. Comput. Phys. 2004, 201, 421–438. 10.1016/j.jcp.2004.06.005. DOI

Pezeshkian W.; Marrink S. J. Simulating Realistic Membrane Shapes. Curr. Opin. Cell Biol. 2021, 71, 103–111. 10.1016/j.ceb.2021.02.009. PubMed DOI

Siggel M.; Kehl S.; Reuter K.; Köfinger J.; Hummer G. TriMem: A Parallelized Hybrid Monte Carlo Software for Efficient Simulations of Lipid Membranes. J. Chem. Phys. 2022, 157, 174801.10.1063/5.0101118. PubMed DOI

Pezeshkian W.; Ipsen J. H. Mesoscale Simulation of Biomembranes With FreeDTS. Nat. Commun. 2024, 15, 548.10.1038/s41467-024-44819-w. PubMed DOI PMC

Nanoscale membrane curvature sorts lipid phases and alters lipid diffusion