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Understanding the mechanisms that dictate the localization of cytoskeletal filaments is crucial for elucidating cell shape regulation in prokaryotes. The actin homolog MreB plays a pivotal role in maintaining the shape of many rod-shaped bacteria such asEscherichia coliby directing cell-wall synthesis according to local curvature cues. However, the basis of MreB’s curvature-dependent localization has remained elusive. Here, we develop a biophysical model for the energetics of a filament binding to a surface that integrates the complex interplay between filament twist and bending and the two-dimensional surface geometry. Our model predicts that the spatial localization of a filament like MreB with substantial intrinsic twist is governed by both the mean and Gaussian curvatures of the cell envelope, which strongly covary in rod-shaped cells. Using molecular dynamics simulations to estimate the mechanical properties of MreB filaments, we show that their thermodynamic preference for regions with lower mean and Gaussian curvatures matches experimental observations for physiologically relevant filament lengths of ~50 nm. We find that the experimentally measured statistical curvature preference is maintained in the absence of filament motion and after a cycle of depolymerization, repolymerization, and membrane rebinding, indicating that equilibrium energetics can explain MreB localization. These findings provide critical insights into the physical principles underlying cytoskeletal filament localization and suggest design principles for synthetic shape-sensing nanomaterials. 

Geometric frustration is recognized to generate complex morphologies in self-assembling particulate and molecular systems. In bulk states, frustration drives structured arrays of topological defects. In the dilute limit, these systems have been shown to form a novel state of self-limiting assembly, in which the equilibrium size of multiparticle domains are finite and well defined. In this article we employ Monte Carlo simulations of a recently developed 2D lattice model of geometrically frustrated assembly [Hackney et al., Phys. Rev. X 13, 041010 (2023)] to study the phase transitions between the self-limiting and defect bulk phase driven by two distinct mechanisms: (1) increasing concentration and (2) decreasing temperature or frustration. The first transition is mediated by a concentration-driven percolation transition of self-limiting, wormlike domains into an intermediate heterogeneous network mesophase, which gradually fills in at high concentration to form a quasiuniform defect bulk state. We find that the percolation threshold is weakly dependent on frustration and shifts to higher concentration as frustration is increased, but depends strongly on the ratio of cohesion to elastic stiffness in the model. The second transition takes place between self-limiting assembly at high-temperature or frustration and phase separation into a condensed bulk state at low temperature or frustration. We consider the competing influences that translational and conformational entropy have on the critical temperature or frustration and show that the self-limiting phase is stabilized at higher frustrations and temperatures than previously expected. Taken together, this understanding of the transition pathways from self-limiting to bulk defect phases of frustrated assembly allows us to map the phase behavior of this 2D minimal model over the full range of concentration. 

Bottom-up self-assembly is a powerful approach to engineering at small scales. Special strategies are needed to formulate components that assemble into predetermined shapes with precise sizes. 

We study the ground state thermodynamics of a model class of geometrically frustrated assemblies, known as warped-jigsaw particles. While it is known that frustration in soft matter assemblies has the ability to propagate up to mesoscopic, multi-particle size scales, notably through the selection of the self-limiting domain, little is understood about how the symmetry of shape-misfit at the particle scale influences emergent morphologies at the mesoscale. Here we show that polarity in the shape-misfit of warped-jigsaw puzzles manifests at a larger scale in the morphology and thermodynamics of the ground-state assembly of self-limiting domains. We use a combination of continuum theory and discrete particle simulations to show that the polar misfit gives rise to two mesoscopically distinct polar, self-limiting ribbon domains. Thermodynamic selection between the two ribbon morphologies is controlled by a combination of the binding anisotropy along distinct neighbor directions and the orientation of polar shape-misfit. These predictions are valuable as design features for ongoing efforts to program self-limiting assemblies through the synthesis of intentionally frustrated particles, further suggesting a generic classification of frustrated assembly behavior in terms of the relative symmetries of shape-misfit and the underlying long-range inter-particle order it frustrates.