Brian Seguin

Kinematics, material symmetry, and energy densities for lipid bilayers with spontaneous curvature

Abstract Continuum mechanical tools are used to describe the deformation, energy density, and material symmetry of a lipid bilayer with spontaneous curvature. In contrast to conventional approaches in which lipid bilayers are modeled by material surfaces, here we rely on a three-dimensional approach in which a lipid bilayer is modeling by a shell-like body with finite thickness. In this setting, the interface between the leaflets of a lipid bilayer is assumed to coincide with the mid-surface of the corresponding shell-like body. The three-dimensional deformation gradient is found to involve the curvature tensors of the mid-surface in the spontaneous and the deformed states, the deformation gradient of the mid-surface, and the transverse deformation. Attention is also given to the coherency of the leaflets and to the area compatibility of the closed lipid bilayers (i.e., vesicles). A hyperelastic constitutive theory for lipid bilayers in the liquid phase is developed. In combination, the requirements of frame indifference and material symmetry yield a representation for the energy density of a lipid bilayer. This representation shows that three scalar invariants suffice to describe the constitutive response of a lipid bilayer exhibiting in-plane fluidity and transverse isotropy. In addition to exploring the geometrical and physical properties of these invariants, fundamental constitutively associated kinematical quantities are emphasized. On this basis, the effect on the energy density of assuming that the lipid bilayer is incompressible is considered. Lastly, a dimension reduction argument is used to extract an areal energy density per unit area from the three-dimensional energy density. This step explains the origin of spontaneous curvature in the areal energy density. Importantly, along with a standard contribution associated with the natural curvature of the lipid bilayer, our analysis indicates that constitutive asymmetry between the leaflets of the lipid bilayer gives rise to a secondary contribution to the spontaneous curvature.

Microphysical derivation of the Canham-Helfrich free-energy density.

The Canham–Helfrich free-energy density for a lipid bilayer has drawn considerable attention. Aside from the mean and Gaussian curvatures, this free-energy density involves a spontaneous mean-curvature that encompasses information regarding the preferred, natural shape of the lipid bilayer. We use a straightforward microphysical argument to derive the Canham–Helfrich free-energy density. Our derivation (1) provides a justification for the common assertion that spontaneous curvature originates primarily from asymmetry between the leaflets comprising a bilayer and (2) furnishes expressions for the splay and saddle-splay moduli in terms of derivatives of the underlying potential.

Roughening it — evolving irregular domains and transport theorems

Well-known examples of transport theorems include the Leibniz integral rule and a result established by Reynolds for three-dimensional regions that convect with the motion of a continuum. Here, we prove a generalized transport theorem by relaxing the regularity assumptions on the domain of integration to include domains that may, among other things, develop holes, split into pieces, or whose fractal dimension may evolve with time. Our results are of potential value in continuum physics and the calculus of variations.

Extending the Transport Theorem to Rough Domains of Integration

Transport theorems, such as that named after Reynolds, are an important tool in the field of continuum physics. Recently, Seguin and Fried used Harrisons theory of differential chains to establish a transport theorem valid for evolving domains that may become irregular. Evolving irregular domains occur in many different physical settings, such as phase transitions or fracture. Here, emphasizing concepts over technicalities, we present Harrisons theory of differential chains and the results of Seguin and Fried in a way meant to be accessible to researchers in continuum physics. We also show how the transport theorem applies to three concrete examples and approximate the resulting terms numerically. Furthermore, we discuss how the transport theorem might be used to weaken certain basic assumptions underlying the description of continua and the challenges associated with doing so.

Calculating the Bending Moduli of the Canham–Helfrich Free-Energy Density

The Canham–Helfrich free-energy density for a lipid bilayer involves the mean and Gaussian curvatures of the midsurface of the bilayer. The splay and saddle-splay moduli \(\kappa \) and \(\bar{\kappa }\) regulate the sensitivity of the free-energy density to changes in the values of these curvatures. Seguin and Fried derived the Canham–Helfrich energy by taking into account the interactions between the molecules comprising the bilayer, giving rise to integral representations for the moduli in terms of the interaction potential. In the present work, two potentials are chosen and the integrals are evaluated to yield expressions for the moduli, which are found to depend on parameters associated with each potential. These results are compared with values of the moduli found in the current literature.