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Register a new userBeyond icosahedral symmetry in packings of proteins in spherical shells
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The formation of quasi-spherical cages from protein building blocks is a remarkable self-assembly process in many natural systems, where a small number of elementary building blocks are assembled to build a highly symmetric icosahedral cage. In turn, this has inspired synthetic biologists to design de novo protein cages. We use simple models, on multiple scales, to investigate the self-assembly of a spherical cage, focusing on the regularity of the packing of protein-like objects on the surface. Using building blocks, which are able to pack with icosahedral symmetry, we examine how stable these highly symmetric structures are to perturbations that may arise from the interplay between flexibility of the interacting blocks and entropic effects. We find that, in the presence of those perturbations, icosahedral packing is not the most stable arrangement for a wide range of parameters; rather disordered structures are found to be the most stable. Our results suggest that (i) many designed, or even natural, protein cages may not be regular in the presence of those perturbations, and (ii) that optimizing those flexibilities can be a possible design strategy to obtain regular synthetic cages with full control over their surface properties.
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We report on the buckling and subsequent collapse of orthotropic elastic spherical shells under volume and pressure control. Going far beyond what is known for isotropic shells, a rich morphological phase space with three distinct regimes emerges upon variation of shell slenderness and degree of orthotropy. Our extensive numerical simulations are in agreement with experiments using fabricated polymer shells. The shell buckling pathways and corresponding strain energy evolution are shown to depend strongly on material orthotropy. We find surprisingly robust orthotropic structures with strong similarities to stomatocytes and tricolpate pollen grains, suggesting that the shape of several of Natures collapsed shells could be understood from the viewpoint of material orthotropy.
The dense packing of interacting particles on spheres has proved to be a useful model for virus capsids and colloidosomes. Indeed, icosahedral symmetry observed in virus capsids corresponds to potential energy minima that occur for magic numbers of, e.g., 12, 32 and 72 identical Lennard-Jones particles, for which the packing has exactly the minimum number of twelve five-fold defects. It is unclear, however, how stable these structures are against thermal agitation. We investigate this property by means of basin-hopping global optimisation and Langevin dynamics for particle numbers between ten and one hundred. An important measure is the number and type of point defects, that is, particles that do not have six nearest neighbours. We find that small icosahedral structures are the most robust against thermal fluctuations, exhibiting fewer excess defects and rearrangements for a wide temperature range. Furthermore, we provide evidence that excess defects appearing at low non-zero temperatures lower the potential energy at the expense of entropy. At higher temperatures defects are, as expected, thermally excited and thus entropically stabilised. If we replace the Lennard-Jones potential by a very short-ranged (Morse) potential, which is arguably more appropriate for colloids and virus capsid proteins, we find that the same particle numbers give a minimum in the potential energy, although for larger particle numbers these minima correspond to different packings. Furthermore, defects are more difficult to excite thermally for the short-ranged potential, suggesting that the short-ranged interaction further stabilises equilibrium structures.
We investigate the mechanical response of jammed packings of circulo-lines, interacting via purely repulsive, linear spring forces, as a function of pressure $P$ during athermal, quasistatic isotropic compression. Prior work has shown that the ensemble-averaged shear modulus for jammed disk packings scales as a power-law, $langle G(P) rangle sim P^{beta}$, with $beta sim 0.5$, over a wide range of pressure. For packings of circulo-lines, we also find robust power-law scaling of $langle G(P)rangle$ over the same range of pressure for aspect ratios ${cal R} gtrsim 1.2$. However, the power-law scaling exponent $beta sim 0.8$-$0.9$ is much larger than that for jammed disk packings. To understand the origin of this behavior, we decompose $langle Grangle$ into separate contributions from geometrical families, $G_f$, and from changes in the interparticle contact network, $G_r$, such that $langle G rangle = langle G_frangle + langle G_r rangle$. We show that the shear modulus for low-pressure geometrical families for jammed packings of circulo-lines can both increase {it and} decrease with pressure, whereas the shear modulus for low-pressure geometrical families for jammed disk packings only decreases with pressure. For this reason, the geometrical family contribution $langle G_f rangle$ is much larger for jammed packings of circulo-lines than for jammed disk packings at finite pressure, causing the increase in the power-law scaling exponent.
Chiral heteropolymers such as larger globular proteins can simultaneously support multiple length scales. The interplay between different scales brings about conformational diversity, and governs the structure of the energy landscape. Multiple scales produces also complex dynamics, which in the case of proteins sustains live matter. However, thus far no clear understanding exist, how to distinguish the various scales that determine the structure and dynamics of a complex protein. Here we propose a systematic method to identify the scales in chiral heteropolymers such as a protein. For this we introduce a novel order parameter, that not only reveals the scales but also probes the phase structure. In particular, we argue that a chiral heteropolymer can simultaneously display traits of several different phases, contingent on the length scale at which it is scrutinized. Our approach builds on a variant of Kadanoffs block-spin transformation that we employ to coarse grain piecewise linear chains such as the C$alpha$ backbone of a protein. We derive analytically and then verify numerically a number of properties that the order parameter can display. We demonstrate how, in the case of crystallographic protein structures in Protein Data Bank, the order parameter reveals the presence of different length scales, and we propose that a relation must exist between the scales, phases, and the complexity of folding pathways.
Within the framework of continuum theory, we draw a parallel between ferromagnetic materials and nematic liquid crystals confined on curved surfaces, which are both characterized by local interaction and anchoring potentials. We show that the extrinsic curvature of the shell combined with the out-of-plane component of the director field gives rise to chirality effects. This interplay produces an effective energy term reminiscent of the chiral term in cholesteric liquid crystals, with the curvature tensor acting as a sort of anisotropic helicity. We discuss also how the different nature of the order parameter, a vector in ferromagnets and a tensor in nematics, yields different textures on surfaces with the same topology as the sphere. In particular, we show that the extrinsic curvature governs the ground state configuration on a nematic spherical shell, favouring two antipodal disclinations of charge +1 on small particles and four $+1/2$ disclinations of charge located at the vertices of a square inscribed in a great circle on larger particles.
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