# On The Critical Casimir Interaction Between Anisotropic Inclusions On A Membrane

## TL;DR

This study investigates the critical Casimir interactions between anisotropic inclusions in membrane-like binary mixtures, revealing long-range, affinity-dependent forces that influence clustering and patterning of membrane proteins.

## Contribution

The paper introduces a lattice model and thermodynamic integration scheme to analyze how anisotropic inclusions interact at criticality, highlighting the effects of domain composition and orientation.

## Key findings

- Interactions are long-range and depend on inclusion affinity.
- Opposing affinities lead to repulsive interactions, same affinities to attraction.
- Multiple domains cause anisotropic interactions that influence clustering.

## Abstract

Using a lattice model and a versatile thermodynamic integration scheme, we study the critical Casimir interactions between inclusions embedded in a two-dimensional critical binary mixtures. For single-domain inclusions we demonstrate that the interactions are very long range, and their magnitudes strongly depend on the affinity of the inclusions with the species in the binary mixtures, ranging from repulsive when two inclusions have opposing affinities to attractive when they have the same affinities. When one of the inclusions has no preference for either of the species, we find negligible critical Casimir interactions. For multiple-domain inclusions, mimicking the observations that membrane proteins often have several domains with varying affinities to the surrounding lipid species, the presence of domains with opposing affinities does not cancel the interactions altogether. Instead we can observe both attractive and repulsive interactions depending on their relative orientations. With increasing number of domains per inclusion, the range and magnitude of the effective interactions decrease in a similar fashion to those of electrostatic multipoles. Finally, clusters formed by multiple-domain inclusions can result in an effective affinity patterning due to the anisotropic character of the Casimir interactions between the building blocks.