Free energy profile and kinetics of the formation of membrane invaginations by membrane-bending peripheral proteins

Peripheral proteins, known to induce curvature, have been identified as key agents in the spontaneous remodeling of bilayer membranes, leading to invaginations and the genesis of membrane tubules. For proteins like cholera and Shiga toxin, which impart the membrane with locally isotropic curvatures, the resultant membrane-mediated interactions remain notably subtle. Consequently, the collective action of these proteins, culminating in the formation of dense clusters on the membrane and subsequent invagination, unfolds over an extended timescale, often spanning several minutes. This gradual progression challenges direct simulation of the invagination process, even with coarsegrained models. In this study, we introduce a steered molecular dynamics protocol wherein peripheral proteins are impelled to converge on a membrane patch, instigating an invagination. Utilizing the Jarzynski equality, we derive the free energy profile of this process from a suite of non-equilibrium simulation replicas. Examining two distinct membrane-associated proteins, we elucidate the influence of protein flexibility and the distribution of induced curvatures on both the remodeling process and the corresponding free energy profile. We delve into the role of membrane-mediated effects in shaping protein organization within the invaginated domain. Building on the free energy profile, we model the formation of invaginations as a Markovian process, and offer estimates of the corresponding timescales. Our findings yield minute-long implied timescales that resonate well with empirical observations.