Ashutosh Agrawal

Modeling protein-mediated morphology in biomembranes

The equilibrium theory for lipid membranes is used to describe the structure of nuclear pores and the membrane shapes accompanying endocytosis. The commonly used variant of the theory contains a fixed parameter called the spontaneous curvature which accounts for asymmetry in the bending response of the membrane. This is replaced here by a variable distribution of spontaneous curvature representing the influence of attached proteins. The required adjustments to the standard theory are described and the resulting model is applied to the study of membrane morphology at the cites of protein-assisted nuclear pore formation and endocytosis.

Interaction between surface shape and intra-surface viscous flow on lipid membranes

The theory of intra-surface viscous flow on lipid bilayers is developed by combining the equations for flow on a curved surface with those that describe the elastic resistance of the bilayer to flexure. The model is derived directly from balance laws and augments an alternative formulation based on a variational principle. Conditions holding along an edge of the membrane are emphasized, and the coupling between flow and membrane shape is simulated numerically.

Endocytic proteins drive vesicle growth via instability in high membrane tension environment.

Significance Biological cells are engaged in an incessant uptake of macromolecules for nutrition and inter- and intracellular communication; this entails significant local bending of the plasma membrane and formation of cargo-carrying vesicles executed by a designated set of membrane-deforming proteins. The energetic cost incurred in forming vesicles is directly related to the stressed state of the membrane and, hence, that of the cell. In this study, we reveal a protein-induced “snap-through instability” that offsets tension and drives vesicle growth during clathrin-mediated endocytosis, the main pathway for the transport of macromolecules into cells. Because these proteins (actin and BAR proteins) are involved in other interfacial rearrangements in cells, the predicted instability could be at play in cells at-large. Clathrin-mediated endocytosis (CME) is a key pathway for transporting cargo into cells via membrane vesicles; it plays an integral role in nutrient import, signal transduction, neurotransmission, and cellular entry of pathogens and drug-carrying nanoparticles. Because CME entails substantial local remodeling of the plasma membrane, the presence of membrane tension offers resistance to bending and hence, vesicle formation. Experiments show that in such high-tension conditions, actin dynamics is required to carry out CME successfully. In this study, we build on these pioneering experimental studies to provide fundamental mechanistic insights into the roles of two key endocytic proteins—namely, actin and BAR proteins—in driving vesicle formation in high membrane tension environment. Our study reveals an actin force-induced “snap-through instability” that triggers a rapid shape transition from a shallow invagination to a highly invaginated tubular structure. We show that the association of BAR proteins stabilizes vesicles and induces a milder instability. In addition, we present a rather counterintuitive role of BAR depolymerization in regulating the shape evolution of vesicles. We show that the dissociation of BAR proteins, supported by actin–BAR synergy, leads to considerable elongation and squeezing of vesicles. Going beyond the membrane geometry, we put forth a stress-based perspective for the onset of vesicle scission and predict the shapes and composition of detached vesicles. We present the snap-through transition and the high in-plane stress as possible explanations for the intriguing direct transformation of broad and shallow invaginations into detached vesicles in BAR mutant yeast cells.

Mechanics of membrane-membrane adhesion

Curvature elasticity is used to derive the equilibrium conditions that govern the mechanics of membrane–membrane adhesion. These include the Euler–Lagrange equations and the interface conditions which are derived here for the most general class of strain energies permissible for fluid surfaces. The theory is specialized for homogeneous membranes with quadratic ‘Helfrich’-type energies with non-uniform spontaneous curvatures. The results are employed to solve four-point boundary value problems that simulate the equilibrium shapes of lipid vesicles that adhere to each other. Numerical studies are conducted to investigate the effect of relative sizes, osmotic pressures, and adhesion-induced spontaneous curvature on the morphology of adhered vesicles.

Stability of lipid membranes

Lipid membranes are versatile biological structures that undergo significant structural remodelling, often triggered by instabilities. Since they invariably possess heterogeneous properties, owing to the presence of multiple lipid species and their interactions with proteins, heterogeneity can have a significant impact on their equilibrium state and stability. In this work, we use curvature elasticity to derive the generalized stability criterion for heterogeneous lipid membranes. Our formulation entertains strain energies that go beyond the Helfrich energy and exhibit higher-order dependence on curvature invariants or spatially varying properties.

Ultradonut topology of the nuclear envelope

Significance Lipid membranes exhibit a variety of morphologies tailored to perform specific functions of cells and their organelles. A unique lipid structure is the nuclear envelope which houses the genome and plays a vital role in genome organization and signaling pathways. The nuclear envelope is composed of two fused membranes with thousands of toroid-shaped pores with extremely high curvatures, the origin of which remains an open question in biology. Here, we show that the architecture of this “ultradonut” may be generated by nanoscale buckling instabilities triggered by membrane stresses during nuclei growth. Our findings may help understand the impact of membrane mechanics on the geometry and the functionality of the nucleus and more generally, other double-membrane organelles in cells. The nuclear envelope is a unique topological structure formed by lipid membranes in eukaryotic cells. Unlike other membrane structures, the nuclear envelope comprises two concentric membrane shells fused at numerous sites with toroid-shaped pores that impart a “geometric” genus on the order of thousands. Despite the intriguing architecture and vital biological functions of the nuclear membranes, how they achieve and maintain such a unique arrangement remains unknown. Here, we used the theory of elasticity and differential geometry to analyze the equilibrium shape and stability of this structure. Our results show that modest in- and out-of-plane stresses present in the membranes not only can define the pore geometry, but also provide a mechanism for destabilizing membranes beyond a critical size and set the stage for the formation of new pores. Our results suggest a mechanism wherein nanoscale buckling instabilities can define the global topology of a nuclear envelope-like structure.

Vesicle adhesion reveals novel universal relationships for biophysical characterization

Adhesion plays an integral role in diverse biological functions ranging from cellular transport to tissue development. Estimation of adhesion strength, therefore, becomes important to gain biophysical insight into these phenomena. In this study, we use curvature elasticity to present non-intuitive, yet remarkably simple, universal relationships that capture vesicle–substrate interactions. These relationships not only provide efficient strategies to tease out adhesion energy of biological molecules but can also be used to characterize the physical properties of elastic biomimetic nanoparticles. We validate the modeling predictions with experimental data from two previous studies.

Mechanics of Cellular Membranes

In this chapter we summarize the theory of cellular membranes required to model a diverse range of biological phenomena. A typical lipid bilayer is modeled as a two dimensional fluid shell with flexural resistance. We discuss the notion of fluidity and obtain the governing equilibrium equations for membranes with inhomogeneous properties. The theory is specialized to axisymmetric problems and employed to model protein mediated endocytosis. We obtain the contact conditions required to model the interactions of membranes with curved substrates in the presence of wetting and adhesion. Finally, we discuss the theory of membranes with coexistent phases.