We model membrane proteins as anisotropic objects characterized by symmetric-traceless tensors and determine the coupling between these orderparameters and membrane curvature. We consider the interactions 1) between transmembrane proteins that respect up-down (reflection) symmetry of bilayer membranes and that have circular or non-circular cross-sectional areas in the tangent-plane of membranes, 2) between transmembrane proteins that break reflection symmetry and have circular or non-circular crosssectional areas, and 3) between non-transmembrane proteins. Using a field theoretic approach, we find non-entropic 1/R 4 interactions between reflection-symmetry-breaking transmembrane proteins with circular crosssectional area and entropic 1/R 4 interactions between transmembrane proteins with circular cross-section that do not break up-down symmetry in agreement with previous calculations. We also find anisotropic 1/R 4 interactions between reflection-symmetry-conserving transmembrane proteins with non-circular cross-section, anisotropic 1/R 2 interactions between reflectionsymmetry-breaking transmembrane proteins with non-circular cross-section, and non-entropic 1/R 4 many-particle interactions among non-transmembrane proteins. For large R, these interactions are considerably larger than Van der Waals interactions or screened electrostatic interactions and might provide 1 the dominant force inducing aggregation of the membrane proteins.

We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-GordonHamiltonian to describe it. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising from the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization group recursion relations that predict a transition with decreasing bending rigidity κ.

We consider in mean-field theory the continuous development below a secondorder phase transition of n-atic tangent plane order on a deformable surface of genus zero with order parameter ψ = e inθ . Tangent plane order expels Gaussian curvature. In addition, the total vorticity of orientational order on a surface of genus zero is two. Thus, the ordered phase of an n-atic on such a surface will have 2n vortices of strength 1/n, 2n zeros in its order parameter, and a nonspherical equilibrium shape. Our calculations are based on a phenomenological model with a gauge-like coupling between ψ and curvature, and our analysis follows closely the Abrikosov treatment of a type II superconductor just below H c2 .

We express the Crow-Kimura and Eigen models of quasispecies theory in a functional integral representation. We formulate the spin coherent state functional integrals using the Schwinger Boson method. In this formulation, we are able to deduce the long-time behavior of these models for arbitrary replication and degradation functions. We discuss the phase transitions that occur in these models as a function of mutation rate. We derive for these models the leading order corrections to the infinite genome length limit.