Lipid bilayer

A lipid bilayer is a self-assembled double layer of amphiphilic molecules. Its bending modulus controls thermal undulations and the resulting interaction between nearby membranes.

A lipid assembly containing a bilayer-like region whose edges are stabilized by other components, often short-chain lipids or detergents. Size and composition affect its use as a membrane mimic.

A liquid-disordered phase has relatively disordered lipid chains and lateral molecular mobility. Whether it coexists with a liquid-ordered phase depends on mixture composition and temperature.

A liquid-ordered phase combines relatively ordered lipid chains with lateral molecular mobility. It is distinct from a solid gel and can coexist with a liquid-disordered phase in suitable mixtures.

spontaneous curvature is the preferred curvature of a membrane or monolayer in a specified elastic description. Headgroup hydration, chain geometry and leaflet composition can change it; the sign depends on the chosen surface-normal convention.

A lipid bilayer nanodisc is a small bilayer patch stabilized at its edge, often by a surrounding scaffold. It permits controlled lipid composition and study of membrane-associated proteins, while lacking the two separately sealed aqueous compartments of a closed lipid vesicle.

In an incompressible, noninterdigitating lipid bilayer with negligible head-group thickness, let the inner radius be $r>0$, molecular tail volume $v>0$ and head-group area $a>0$. Counting heads on both membrane leaflets and conserving tail volume gives $R^3-r^3=3(v/a)(R^2+r^2)$. For $R>r$, the function $q(R)=(R^3-r^3)/(R^2+r^2)$ has derivative $(R^4+3R^2{r}^2+2R{r}^3)/(R^2+r^2{)}^2>0$, starts at zero and tends to infinity. Thus exactly one candidate outer radius exists in this geometric model. A maximum tail length $l$ additionally requires $r_m-r\le l$ and $R-r_m\le l$, where $r_m^3=r^3+3(v/a)r^2$. Uniqueness does not establish compatibility with arbitrary molecular lengths.

A membrane leaflet is one of the two layers forming a lipid bilayer. In a phospholipid-based cell membrane, the hydrophilic regions face the adjacent water, and the hydrophobic regions face the interior shared by the two membrane leaflets.

Line tension is energy per unit length of a boundary, such as the edge between two phases of a lipid bilayer. Positive line tension tends to minimize the perimeter at fixed domain area. Its dimensions and geometry differ from surface tension, which is energy per unit area.

For a circular domain of radius $R_0$ with line tension $\gamma$, the stiffness of a real polar sine or cosine amplitude with integer angular harmonic $m$ is $K_m=\pi \gamma (m^2-1)/R_0$, when the enclosed area is fixed. The equipartition theorem gives $⟨u_m^2⟩=k_{B}T{R}_0/[\pi \gamma (m^2-1)]$ for $m\ge 2$. A positive-frequency complex Fourier mode coefficient has half this variance. The uniform mode is forbidden and $m=1$ is translation.

The first polar sine and cosine harmonics of a circular boundary represent displacements of its centre. A complete translation leaves perimeter and area unchanged, so line tension supplies zero stiffness to these modes. The centre of an unconfined domain has no normalizable positional thermal equilibrium in an infinite plane; recentering the boundary removes this freedom from a shape spectrum.

A lipid vesicle is a closed lipid bilayer enclosing fluid. Its shape and thermal membrane undulations depend on membrane tension, membrane bending modulus, and constraints such as fixed enclosed volume.

A liposome is a closed artificial lipid vesicle enclosing an aqueous compartment. It may contain one lipid bilayer or several concentric bilayers. A single-bilayer spherical model distinguishes the inner aqueous radius, the boundary between opposing tail regions, and the outer head-group radius; finite molecular packing constrains all three.

A proteoliposome is a liposome containing one or more reconstituted membrane proteins in its lipid bilayer. It combines a controlled membrane boundary with a chosen molecular machine. A solution containing detergent-solubilized protein is not automatically a proteoliposome: a closed vesicle and a membrane-incorporated protein must actually be formed.

The artificial compartment is useful because material can be placed on one side of a lipid bilayer and transport measured across it. In an originating experiment on the NhaA sodium/proton antiporter, purified protein was reconstituted into sodium-loaded proteoliposomes. Parallel sodium and proton flux measurements, together with an independent membrane-potential equilibrium method, supported a stoichiometry of two protons per sodium ion over the tested pH range 7.2–8.1.^[ref]

For a bookkeeping example,100 sodium ions moved outward through a perfectly coupled $2{\mathrm{H}}^{+}/\mathrm{N}{\mathrm{a}}^{+}$ exchanger would accompany 200 protons moving inward. The net movement is 100 positive elementary charges inward, which explains the word electrogenic. These invented counts illustrate coupling, not a measured single-vesicle trace.

Interpreting a proteoliposome assay still requires knowing which way the membrane protein faces, whether the vesicles leak, and how much protein is incorporated. An otherwise matched empty liposome helps estimate membrane background. Reconstitution simplifies a biological system; it does not automatically eliminate every alternative source of the measured flux.

For a tension-dominated cylindrical lipid vesicle of fixed length $L$ and volume, a real axisymmetric sine amplitude $u_q$ has stiffness $K_q=\pi \sigma L[(q{R}_0{)}^2-1]/R_0$. Fixed volume requires a mean-radius shift $-u_q^2/(4R_0)$. Stable modes have $⟨u_q^2⟩=k_{B}T/K_q$ by the equipartition theorem. Modes with $q{R}_0<1$ have the Rayleigh–Plateau instability, rather than a negative equilibrium variance. Bending modifies very short wavelengths and near-marginal modes.

An axisymmetric cylindrical membrane deformation is invariant under rotations about the cylinder axis, so its radius depends only on the axial coordinate. Its surface area is $2\pi \int R\sqrt{1+R_z^2}dz$, and its enclosed volume is $\pi \int R^{2}dz$. These scalar shape constraints are the basis of the fixed-volume capillary spectrum of a cylindrical membrane.

Nearby lipid bilayers experience attractive van der Waals forces, screened electrostatic repulsion, and fluctuation-induced steric repulsion.

The Hamaker constant sets the strength of pairwise-summed van der Waals attraction between macroscopic bodies across a medium.

The Debye–Hückel screening length is the distance over which mobile ions suppress an electrostatic potential in the linearized electrolyte theory.

For fully dissociated electrolytes at the same temperature and solvent permittivity, linear screening theory gives ${\lambda }_D^{-2}\propto {\sum }_i{c}_i{z}_i^2=2I$. A divalent symmetric salt of formula concentration $c$ has $I=4c$, not $c$ or $2c$. ion association and nonideal concentrated-solution effects limit this comparison.

Helfrich repulsion is the entropic interaction produced when neighboring membranes suppress each other's thermal undulations. Its free energy per area scales as $(k_{B}T{)}^2/(k_c{d}^2)$.

A membrane unbinding transition sends the equilibrium layer spacing to infinity. In a dilute lamellar stack, competition between a quadratic virial attraction and cubic Helfrich repulsion gives a continuous transition with spacing exponent one.

The Helfrich energy of a fluid membrane combines surface tension with curvature elasticity. Without spontaneous curvature or a topological term,

In a bound-cap model, an adhered sphere of radius $r$ covers area $A=2\pi r^2\xi$ with $\xi =1-\cos \alpha \in [0,2]$. Its excess area over its circular footprint is $\pi r^2{\xi }^2$. The Helfrich energy of that spherical cap is $4\pi \kappa \xi$, adhesion gives $-wA$, and membrane tension gives $\sigma \pi r^2{\xi }^2$. Setting $\bar{w}=w{r}^2/\kappa$ and $\bar{\sigma }=\sigma r^2/\kappa$ yields the displayed expression. Free-surface deformation, neck structure, Gaussian curvature and topology changes are omitted. Full wrapping alone does not prove scission.

For positive membrane tension, the bound-cap energy is convex in indentation. Its constrained minimum is the unconstrained value $(\bar{w}-2)/\bar{\sigma }$ clipped to $[0,2]$. No wrapping occurs at $\bar{w}\le 2$, partial wrapping at $2<\bar{w}<2+2\bar{\sigma }$, and full wrapping at $\bar{w}\ge 2+2\bar{\sigma }$. At zero tension the expression is linear, with all indentations degenerate at $\bar{w}=2$. These phase boundaries belong to the specified cap model and can move when free-membrane and neck energies are included.

A fluid surface has no equilibrium static in-plane shear modulus, because material can rearrange after shear. This does not set its membrane tension or membrane bending modulus to zero. Time-dependent deformation can also produce viscous in-plane shear stress; absence of a static elastic coefficient is not absence of dissipation.

A thermal membrane undulation is a shape fluctuation excited by thermal energy. In a planar membrane, a Fourier mode has stiffness $k_c{q}^4+\sigma q^2$, so tension suppresses long-wavelength excess area while bending suppresses short wavelengths.

In a membrane-mechanics experiment, a narrow glass pipette applies a controlled pressure difference and draws a tongue of membrane into its opening. Imaging the aspirated length and vesicle geometry estimates the change in apparent membrane area. At low tension, much of that change can flatten thermal undulations; at higher tension it can measure actual area dilation. For ideal spherical caps with pipette radius $R_p$ and vesicle radius $R_v>R_p$, the Young–Laplace equation gives $\Delta p=2\sigma (1/R_p-1/R_v)$. Bending at the rim, adhesion and a membrane-associated cytoskeleton can require corrections to this simple geometric inference.

This coefficient multiplies the squared mean-curvature term $\frac{\kappa }{2}\int (2H-C_0{)}^{2}dA$ in the Helfrich membrane energy. It has units of energy, unlike a beam's flexural rigidity. For small height modes of a nearly flat membrane of area $A$, with Fourier normalization $h_q=A^{-1}\int h(\mathbf{x})e^{-i\mathbf{q}\cdot \mathbf{x}}{d}^{2}x$, quadratic bending and tension energies give $⟨|h_q{|}^2⟩=k_{B}T/[A(\kappa q^4+\sigma q^2)]$ by the equipartition theorem. Thermal-flicker measurements can therefore separate short-wavelength bending from longer-wavelength tension effects.

For small-slope height fluctuations of a nearly planar membrane, $A$ is projected area, $\sigma$ is membrane tension, $\kappa$ is membrane bending modulus, and $q$ is a wavenumber. With a Fourier convention in which the mode amplitude $h_{\mathbf{q}}$ has units of length, the quadratic Helfrich membrane energy and the equipartition theorem give the displayed thermal variance. Bending dominates for $q\gg \sqrt{\sigma /\kappa }$, giving a $q^{-4}$ spectrum. Tension dominates at smaller $q$, giving $q^{-2}$. The corresponding membrane elastic length is $\sqrt{\kappa /\sigma }$ when length is measured as $1/q$; a wavelength convention multiplies it by $2\pi$.

The membrane bending modulus sets the energy cost of curvature. A larger $k_c$ suppresses thermal undulations and resists wrapping around small objects.

Membrane tension is the free-energy cost per added projected area. Together with bending elasticity it selects a length scale and controls the radius of a pulled membrane tube.

The membrane elastic length is the distance over which bending-dominated deformation relaxes under tension. In a small-slope one-dimensional membrane it is $\xi =\sqrt{k_c/\sigma }$.

For a nearly flat membrane represented by height $h(x)$, the quadratic bending-and-tension energy per invariant transverse length isIts free equilibrium equation is $k_c{h}_{xxxx}-\sigma h_{xx}=0$.

For a tension-dominated membrane with internal dimension $D$, the massless Gaussian height covariance has spectrum $k_{B}T/(\sigma q^2)$. Removing the uniform translation mode leaves the height difference well-defined. Its variance grows linearly with distance for $D=1$, logarithmically for $D=2$, and approaches a cutoff-dependent constant for $D>2$. A microscopic cutoff is necessary when the short-distance integral diverges. This distinguishes rough positional order from a failure of the local small-slope approximation.

Adding positive quadratic confinement $t\int h^2/2$ to the membrane energy changes its height spectrum to $k_{B}T/(\sigma q^2+t)$. The continuous height-shift symmetry is explicitly broken. Connected correlation function decay on the finite length $\xi$ and the height-difference variance saturates for every internal dimension, although its saturation value can depend on a microscopic cutoff. A negative confinement coefficient gives an unstable quadratic model rather than pinning.

For a membrane of bending modulus $k_c$ adhering with energy per area $\mathcal{U}$ to a cylinder of radius $R$, $U=R^2\mathcal{U}/k_c$ compares adhesion with bending cost.

A membrane-mediated interaction potential is the separation-dependent energy produced when deformation fields around two membrane-bound objects overlap. Its sign and range depend on membrane tension, bending stiffness, inclusion geometry, and which side of the membrane each object occupies.

Minimizing the Helfrich energy per length of an unconstrained cylindrical membrane gives $r_0=\sqrt{k_c/(2\sigma )}$, balancing the area cost against bending curvature.

When a tether draws area from the thermal undulations of a finite vesicle, extension raises membrane tension rather than drawing area from a fixed-tension reservoir. The tether therefore narrows and its required pulling force increases with length.

Thermal membrane undulations generate Helfrich repulsion from a nearby particle. For a sphere of radius $R$ tightly enclosed by a high-tension membrane,

For a sphere of radius $R$ moving through a membrane tube with a uniform narrow gap $\delta \ll R$, lubrication theory givesIt exceeds the unbounded-fluid Stokes drag law coefficient by $2R^2/{\delta }^2$.