Statistical Mechanical Models Of Virus Capsid Assembly

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Viruses have become an increasingly popular subject of physics investigation, particularly in the last decade. Advances in imaging of virus capsids-the protective protein shells-in a wide variety of stages of assembly have encouraged physical assembly models at a similarly wide variety of scales, while the apparent simplicity of the capsid system-typically, many identical units assembling spontaneously into an icosahedrally symmetric (rather than amorphous) shell-makes the problem particularly interesting. We take a look at the existing physical assembly models in light of the question of how a particular assembly target can be consistently achieved in the presence of so many possible incorrect results. This review leads us to pose our own model of fully irreversible virus assembly, which we study in depth using a large ensemble of simulated assembled capsids, generated under a variety of capsid shell elastic parameters. While this irreversible model (predictably) did not yield consistently symmetric results, we do glean some insight into the effect of elasticity on growth, as well as an understanding of common failure modes. In particular, we found that (i) capsid size depends strongly on the spontaneous curvature and weakly on the ratio of bending to stretching elastic stiffnesses, (ii) the probability of successful capsid completion decays exponentially with capsid size, and (iii) the degree of localization of Gaussian curvature depends heavily on the ratio of elastic stiffnesses. We then go on to consider more thoroughly the nature of the ensem- ble of symmetric and almost-symmetric capsids-ultimately computing a phase diagram of minimum-energy capsids as a function of the two above-mentioned elastic parameters-and also look at a number of modifications we can make to our irreversible model, finally putting forth a rather different type of model potentially appropriate for understanding immature HIV assembly, and concluding with a fit of this new model's parameters to recent experimental structures. A common thread between the coarse-grained models we discuss in the first part of the thesis is that they all depend explicitly on elastic parameters that are otherwise completely unmotivated. We thus devote the second part to the question of how (elastic) model parameters can be determined from ab initio methods. Modeling protein interactions as springs with very general quadratic potentials, we run atomistic molecular dynamics simulations and analyze the trajectories to determine stiffness tensors for these generalized springs. After a thorough examination of the mathematical structure of our springs-including transformations of the stiffness tensors into different reference frames and gauges, and an analytical formula for composing generalized springs in series-we go on to apply the technique to measure the elasticity of a mature HIV capsid lattice by simulating isolated pairs of interacting protein domains. We compute the relaxation times for each bond, and for the entire lattice, which both gives the stiffness as a physical, comparable timescale, and also provides a way to invalidate many simulations with too short a run time. Because calculation of the relaxation matrix requires a measurement of the diffusion of the individual proteins, we conclude with a brief study of the effects of finite box sizes and differing thermostat strengths on diffusion measurements from atomistic simulations.
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