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A Beginner’s Short Guide to Membrane Biophysics

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Modeling Biomaterials

Part of the book series: Nečas Center Series ((NECES))

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Abstract

This chapter provides a basic guide to the essentials of membrane biophysics. The review consists of three parts: The first part focuses on physicochemical aspects of biomembranes. The origin of the hydrophobic attraction that drives the self-assembly of lipids into large structures like membranes is explained, and an overview of the thermodynamic properties of bilayer membranes is given. The second part introduces the Helfrich model for the curvature elasticity of two-dimensional manifolds. This is the most commonly used framework for studying the mechanical properties of lipid membranes. The third part of the chapter is dedicated to molecular simulations of membranes. The principles of molecular dynamics simulations are explained, and the different modeling strategies for membranes, ranging from atomistic to highly coarse-grained lattice-based models, are reviewed. These different classes of models address membrane properties and processes across a wide spectrum of lengths and time scales.

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Notes

  1. 1.

    Water also has three vibrational degrees of freedom, but they are irrelevant at room temperature and become excited only at temperatures of several thousand degrees.

  2. 2.

    In contrast to A p, the physical area A is not a thermodynamic variable that can be easily fixed. In some theoretical studies [27], the term “surface tension” is used to describe a Lagrange multiplier that fixes the average physical area.

  3. 3.

    Equations 7–9 cannot obviously (and do not intend to) summarize the entire field of classical statistical mechanics. They are only meant to demonstrate the basic idea that the free energy of a thermodynamic system can be derived from the statistics of an ensemble of microscopic (molecular) states of that system. This concept will be used in Sect. 3.4 where the statistical–mechanical behavior of thermally fluctuating membranes will be discussed.

  4. 4.

    Carrying the integration over the projected rather than the surface area generates an error of order |∇ h|2 in the calculated Helfrich energy. The approximation in Eq. 16 involves a correction of the same order.

  5. 5.

    The extrapolation of the simulated fluctuation data to the small q limit involves several technical issues that influence the determined values of the bending modulus. It is also worthwhile to mention the existence of alternative computational methods for determining κ. A comprehensive and updated review with many references on these topics can be found in [36]. A computational method to accelerate the slow relaxation of the small q (large wavelength) Fourier modes of fluctuating membranes has been proposed in [37].

  6. 6.

    The pore will not grow indefinitely. The parabolic shape of the free energy Eq. 41 assumes a linear dependence of G on the pore area, Eq. 40, but this form is only valid for holes that are much smaller than the membrane area (but not too small to be considered as a local defect). For larger holes, the quadratic form for the elastic stretch energy Eq. 4 must be taken into account, which limits the size of the pore. See, e.g., [25].

  7. 7.

    The existence of a shadow trajectory also follows from this property.

  8. 8.

    The Langevin dynamics of a single free particle (f = 0) is underdamped at time scales dt ≪ m∕a (a, b → 1) and overdamped for dt ≫ m∕a (a →−1, b → 0).

  9. 9.

    Any numerical integration scheme becomes unstable and fails to follow the continuous-time trajectory when the integration time step becomes \(dt_s\gtrsim \omega _0^{-1}\), where ω 0 is the largest vibrational frequency of the system (the resonance frequency of the strongest interaction term).

  10. 10.

    Many simulations are conducted in the isobaric–isothermal ensemble, which differ from the canonical ensemble in that the pressure rather than the volume of the system is fixed. Algorithms for fixing the pressure in MD simulations are called “barostats,” and they suffer from similar implementation problems to thermostat algorithms. A relatively robust G-JF barostat algorithm, which is based on concepts used for the derivation of the G-JF thermostat, has been presented in [62].

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Authors and Affiliations

  1. Department of Biomedical Engineering, Ben-Gurion University of the Negev, Be’er Sheva, Israel

    Oded Farago

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Correspondence to Oded Farago .

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  1. Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic

    Josef Málek

  2. Mathematical Institute, University of Oxford, Oxford, UK

    Endre Süli

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Farago, O. (2021). A Beginner’s Short Guide to Membrane Biophysics. In: Málek, J., Süli, E. (eds) Modeling Biomaterials. Nečas Center Series. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-88084-2_1

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