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Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments

  • Regular Article - Soft Matter
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Abstract

We consider the dynamic structure factor (DSF) of quasi-spherical vesicles and present a generalization of an expression that was originally formulated by Zilman and Granek (ZG) for scattering from isotropically oriented quasi-flat membrane plaquettes. The expression is obtained in the form of a multi-dimensional integral over the undulating membrane surface. The new expression reduces to the original stretched exponential form in the limit of sufficiently large vesicles, i.e., in the micron range or larger. For much smaller unilamellar vesicles, deviations from the asymptotic, stretched exponential equation are noticeable even if one assumes that the Seifert-Langer leaflet density mode is completely relaxed and membrane viscosity is neglected. To avoid the need for an exhaustive numerical integration while fitting to neutron spin echo (NSE) data, we provide a useful approximation for polydisperse systems that tests well against the numerical integration of the complete expression. To validate the new expression, we performed NSE experiments on variable-size vesicles made of a POPC/POPS lipid mixture and demonstrate an advantage over the original stretched exponential form or other manipulations of the original ZG expression that have been deployed over the years to fit the NSE data. In particular, values of the membrane bending rigidity extracted from the NSE data using the new approximations were insensitive to the vesicle radii and scattering wavenumber and compared very well with expected values of the effective bending modulus (\(\tilde{\kappa }\)) calculated from results in the literature. Moreover, the generalized scattering theory presented here for an undulating quasi-spherical shell can be easily extended to other models for the membrane undulation dynamics beyond the Helfrich Hamiltonian and thereby provides the foundation for the study of the nanoscale dynamics in more complex and biologically relevant model membrane systems.

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Data availability

Raw data of the experiment are available at doi.ill.fr/10.5291/ILL-DATA.DIR-277.

Notes

  1. Following the theoretical soft-matter literature [8, 10,11,12, 14,15,16], the term “dynamic structure factor” is used here instead of the common equivalent experimental term “intermediate scattering function”, even though in the experimental community, DSF typically refers to the energy domain.

  2. The term vesicle “static structure factor” used here is equivalent to the term vesicle “form factor” commonly used in the small angle scattering community.

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Acknowledgements

The authors gratefully acknowledge Ryan Murphy and Lionel Porcar for assistance with the SANS data collection, and John Nagle for useful discussions. The authors also acknowledge the Partnership for Soft Condensed Matter (PSCM) for providing access to the DLS instrument and laboratory infrastructure used for sample preparation. EGK and MN acknowledge support from the Center for High Resolution Neutron Scattering, a partnership between the National Institute of Standards and Technology (NIST) and the National Science Foundation under Agreement No. DMR-2010792. The identification of any commercial products does not imply endorsement or recommendation by NIST. This work benefited from the use of the SasView application, originally developed under NSF award DMR-0520547. SasView contains code developed with funding from the European Union’s Horizon 2002 research and innovation programme under the SINE2020 project, grant agreement No 654000. This research was also supported in part by the National Science Foundation under Grant No. NSF PHY-1748958

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Authors and Affiliations

  1. Avram and Stella Goldstein-Goren Department of Biotechnology Engineering, and Ilse Katz Institute for Nanoscale Science and Technology, Ben-Gurion University of the Negev, 84105, Beer Sheva, Israel

    Rony Granek

  2. Institut Laue-Langevin (ILL), 71 Avenue des Martys, 38042, Grenoble, CEDEX 9, France

    Ingo Hoffmann

  3. Center for Neutron Research, National Institute of Standards and Technology, 100 Bureau Drive, Gaithersburg, MD, 20899, USA

    Elizabeth G. Kelley & Michihiro Nagao

  4. Department of Materials Science and Engineering, University of Maryland, College Park, MD, 20742, USA

    Michihiro Nagao

  5. Department of Physics and Astronomy, University of Delaware, Newark, DE, 19716, USA

    Michihiro Nagao

  6. Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, IL, 60208, USA

    Petia M. Vlahovska

  7. Department of Physics, University of Toronto, 60 St George St, Toronto, ON, M5S 1A7, Canada

    Anton Zilman

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Contributions

RG, AZ, and PMV developed the theory. IH, EGK, and MN performed the experiment. RG, IH, EGK, and MN wrote the article.

Corresponding authors

Correspondence to Rony Granek, Ingo Hoffmann or Elizabeth G. Kelley.

Additional information

This article is dedicated to Fyl Pincus whose pioneering achievements, inspiring approach, and revolutionary ideas in soft condensed matter and biological physics have had a great impact on the entire community in general, and especially on the present authors, including on RG, EGK, MN, and AZ.

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Granek, R., Hoffmann, I., Kelley, E.G. et al. Dynamic structure factor of undulating vesicles: finite-size and spherical geometry effects with application to neutron spin echo experiments. Eur. Phys. J. E 47, 12 (2024). https://doi.org/10.1140/epje/s10189-023-00400-9

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