Function through form in soft matter: The influence of bounded geometries in heated gels and fluctuating proteins
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The ability to control the behavior of a material through the prescription of spatial inhomogeneity is of increasing interest in condensed matter physics. Examples of materials such as auxetic metamaterials, which widen when stretched, and hydrogels that change shape when swollen involve patterned inhomogeneity to achieve these responses. However, as the boundary of a material breaks translational symmetry, it may also be considered a spatial inhomogeneity. We consider materials in which the boundary is the operative inhomogeneity and plays a key role in determining response to changes in environmental conditions. We first consider a swollen hydrogel rod whose equilibration to a deswollen phase is frustrated due to an impermeable skin that forms at its boundary, trapping solvent within the gel. This constrained gel undergoes internal phase separation, which results in the formation of a solvent-poor shell, enclosing a solvent-rich core. The coaxial arrangement of these regions is unstable to symmetry breaking, which leads to polarization of the solvent distribution across the rod's cross-section, and an internal elastic stress distribution that causes the rod to bend. If the rod is curved when originally fabricated, the stress generated as a result of phase separation leads this polarization to align with the rod's curvature. In the case of a ring formed from a uniformly curved rod, this stress leads to a "Pringling"' instability that buckles the ring out of its plane. If, instead, the rod is originally fabricated straight, its symmetry is spontaneously broken by the gel's polarization, which causes it to bend in the spontaneously selected direction. The Goldstone modes associated with this broken continuous symmetry twist the rod and leads to a reduction of correlations between bending directions along the rod. We also consider allosteric regulation of proteins where the ability of a protein to bind a ligand molecule is modified by the presence of a bound ligand elsewhere on the protein. In particular, we examine a model in which a bound ligand modifies the thermal fluctuations of elastic deformations about a static mean conformation. We model the protein as an elastic continuum and determine the change in fluctuation correlations due to the presence of a bound ligand. By treating the bound ligand as a small adjustment of the protein's boundary, we develop a perturbative approach to calculating the change in fluctuation correlations. We show that, to leading order, the increase in vibrational entropy due to binding is further increased by the presence of an already-bound ligand, which provides for a cooperative effect whose magnitude is governed by the protein shape and binding site locations.