A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes
Cell membranes contain a variety of biomolecules, especially various kinds of lipids and proteins, which constantly change with fluidity and environmental stimuli. Though Helfrich curvature elastic energy has successfully explained many phenomena for single-component membranes, a new theoretical fra...
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MDPI AG
2025-06-01
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| Series: | Membranes |
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| Online Access: | https://www.mdpi.com/2077-0375/15/6/182 |
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| author | Hao Wu Zhong-Can Ou-Yang |
| author_facet | Hao Wu Zhong-Can Ou-Yang |
| author_sort | Hao Wu |
| collection | DOAJ |
| description | Cell membranes contain a variety of biomolecules, especially various kinds of lipids and proteins, which constantly change with fluidity and environmental stimuli. Though Helfrich curvature elastic energy has successfully explained many phenomena for single-component membranes, a new theoretical framework for multicomponent membranes is still a challenge. In this work, we propose a generalized Helfrich free-energy functional describe equilibrium shapes and phase behaviors related to membrane heterogeneity with via curvature-component coupling in a unified framework. For multicomponent membranes, a new but important Laplace–Beltrami operator is derived from the variational calculation on the integral of Gaussian curvature and applied to explain the spontaneous nanotube formation of an asymmetric glycolipid vesicle. Therefore, our general mathematical framework shows a predictive capabilities beyond the existing multicomponent membrane models. The set of new curvature-component coupling EL equations have been derived for global vesicle shapes associated with the composition redistribution of multicomponent membranes for the first time and specified into several typical geometric shape equations. The equilibrium radii of isotonic vesicles for both spherical and cylindrical geometries are calculated. The analytical solution for isotonic vesicles reveals that membrane stability requires distinct elastic moduli among components (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi mathvariant="normal">A</mi></msub><mo>≠</mo><msub><mi>k</mi><mi mathvariant="normal">B</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>k</mi><mo stretchy="false">¯</mo></mover><mi mathvariant="normal">A</mi></msub><mo>≠</mo><msub><mover accent="true"><mi>k</mi><mo stretchy="false">¯</mo></mover><mi mathvariant="normal">B</mi></msub></mrow></semantics></math></inline-formula>), which is consistent with experimental observations of coexisting lipid domains. Furthermore, we elucidate the biophysical implications of the derived shape equations, linking them to experimentally observed membrane remodeling processes. Our new free-energy framework provides a baseline for more detailed microscopic membrane models. |
| format | Article |
| id | doaj-art-e9878cb8d2fd41428033df56f4304c28 |
| institution | Kabale University |
| issn | 2077-0375 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Membranes |
| spelling | doaj-art-e9878cb8d2fd41428033df56f4304c282025-08-20T03:27:22ZengMDPI AGMembranes2077-03752025-06-0115618210.3390/membranes15060182A Generalized Helfrich Free Energy Framework for Multicomponent Fluid MembranesHao Wu0Zhong-Can Ou-Yang1Wenzhou Institute, University of Chinese Academy of Sciences, Wenzhou 325001, ChinaInstitute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, ChinaCell membranes contain a variety of biomolecules, especially various kinds of lipids and proteins, which constantly change with fluidity and environmental stimuli. Though Helfrich curvature elastic energy has successfully explained many phenomena for single-component membranes, a new theoretical framework for multicomponent membranes is still a challenge. In this work, we propose a generalized Helfrich free-energy functional describe equilibrium shapes and phase behaviors related to membrane heterogeneity with via curvature-component coupling in a unified framework. For multicomponent membranes, a new but important Laplace–Beltrami operator is derived from the variational calculation on the integral of Gaussian curvature and applied to explain the spontaneous nanotube formation of an asymmetric glycolipid vesicle. Therefore, our general mathematical framework shows a predictive capabilities beyond the existing multicomponent membrane models. The set of new curvature-component coupling EL equations have been derived for global vesicle shapes associated with the composition redistribution of multicomponent membranes for the first time and specified into several typical geometric shape equations. The equilibrium radii of isotonic vesicles for both spherical and cylindrical geometries are calculated. The analytical solution for isotonic vesicles reveals that membrane stability requires distinct elastic moduli among components (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>k</mi><mi mathvariant="normal">A</mi></msub><mo>≠</mo><msub><mi>k</mi><mi mathvariant="normal">B</mi></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mover accent="true"><mi>k</mi><mo stretchy="false">¯</mo></mover><mi mathvariant="normal">A</mi></msub><mo>≠</mo><msub><mover accent="true"><mi>k</mi><mo stretchy="false">¯</mo></mover><mi mathvariant="normal">B</mi></msub></mrow></semantics></math></inline-formula>), which is consistent with experimental observations of coexisting lipid domains. Furthermore, we elucidate the biophysical implications of the derived shape equations, linking them to experimentally observed membrane remodeling processes. Our new free-energy framework provides a baseline for more detailed microscopic membrane models.https://www.mdpi.com/2077-0375/15/6/182multicomponent membranescurvature-component couplinggeneralized Helfrich free energyEuler-Lagrange equations |
| spellingShingle | Hao Wu Zhong-Can Ou-Yang A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes Membranes multicomponent membranes curvature-component coupling generalized Helfrich free energy Euler-Lagrange equations |
| title | A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes |
| title_full | A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes |
| title_fullStr | A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes |
| title_full_unstemmed | A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes |
| title_short | A Generalized Helfrich Free Energy Framework for Multicomponent Fluid Membranes |
| title_sort | generalized helfrich free energy framework for multicomponent fluid membranes |
| topic | multicomponent membranes curvature-component coupling generalized Helfrich free energy Euler-Lagrange equations |
| url | https://www.mdpi.com/2077-0375/15/6/182 |
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