Effects of local mutations in quadratic iterations

We introduce mutations in replication systems, in which the intact copying mechanism is performed by discrete iterations of a complex quadratic map. More specifically, we consider a "correct" function acting on the complex plane (representing the space of genes to be copied). A "mutation" is a different ("erroneous") map acting on a complex locus of given radius r around a mutation focal point. The effect of the mutation is interpolated radially to eventually recover the original map when reaching an outer radius R. We call the resulting map a "mutated" map. In the theoretical framework of mutated iterations, we study how a mutation (replication error) affects the temporal evolution of the system, on both a local and global scale (from cell diffetentiation to tumor formation). We use the Julia set of the system to quantify simultaneously the long-term behavior of the entire space under mutated maps. We analyze how the position, timing and size of the mutation can alter the topology of the Julia set, hence the system's long-term evolution, its progression into disease, but also its ability to recover or heal. In the context of genetics, mutated iterations may help shed some light on aspects such as the importance of location, size and type of mutation when evaluating a system's prognosis, and of customizing intervention.Comment: 15 pages, 15 figures, 7 reference

A Thermal Gradient Approach for the Quasi-Harmonic Approximation and its Application to Improved Treatment of Anisotropic Expansion

We present a novel approach to efficiently implement thermal expansion in the quasi-harmonic approximation (QHA) for both isotropic and more importantly, anisotropic expansion. In this approach, we rapidly determine a crystal's equilibrium volume and shape at a given temperature by integrating along the gradient of expansion from zero Kelvin up to the desired temperature. We compare our approach to previous isotropic methods that rely on a brute-force grid search to determine the free energy minimum, which is infeasible to carry out for anisotropic expansion, as well as quasi-anisotropic approaches that take into account the contributions to anisotropic expansion from the lattice energy. We compare these methods for experimentally known polymorphs of piracetam and resorcinol and show that both isotropic methods agree to within error up to 300 K. Using the Gr\"{u}neisen parameter causes up to 0.04 kcal/mol deviation in the Gibbs free energy, but for polymorph free energy differences there is a cancellation in error with all isotropic methods within 0.025 kcal/mol at 300 K. Anisotropic expansion allows the crystals to relax into lattice geometries 0.01-0.23 kcal/mol lower in energy at 300 K relative to isotropic expansion. For polymorph free energy differences all QHA methods produced results within 0.02 kcal/mol of each other for resorcinol and 0.12 kcal/mol for piracetam, the two molecules tested here, demonstrating a cancellation of error for isotropic methods. We also find that when expanding in more than a single volume variable, there is a non-negligible rate of failure of the basic approximations of QHA. Specifically, while expanding into new harmonic modes as the box vectors are increased, the system often falls into alternate, structurally distinct harmonic modes unrelated by continuous deformation from the original harmonic mode.Comment: 38 pages, including 9 pages supporting informatio

Noncommutative theories and general coordinate transformations

We study the class of noncommutative theories in $d$ dimensions whose spatial coordinates $(x_i)_{i=1}^d$ can be obtained by performing a smooth change of variables on $(y_i)_{i=1}^d$, the coordinates of a standard noncommutative theory, which satisfy the relation $[y_i, y_j] = i \theta_{ij}$, with a constant $\theta_{ij}$ tensor. The $x_i$ variables verify a commutation relation which is, in general, space-dependent. We study the main properties of this special kind of noncommutative theory and show explicitly that, in two dimensions, any theory with a space-dependent commutation relation can be mapped to another where that $\theta_{ij}$ is constant.Comment: 21 pages, no figures, LaTeX. v2: section 5 added, typos corrected. Version to appear in Physical Review