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 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.

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.

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,

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.

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 $2R^2/{\delta }^2$.