Simulation and analysis of in vitro DNA evolution

We study theoretically the in vitro evolution of a DNA sequence by binding to a transcription factor. Using a simple model of protein-DNA binding and available binding constants for the Mnt protein, we perform large-scale, realistic simulations of evolution starting from a single DNA sequence. We identify different parameter regimes characterized by distinct evolutionary behaviors. For each regime we find analytical estimates which agree well with simulation results. For small population sizes, the DNA evolutional path is a random walk on a smooth landscape. While for large population sizes, the evolution dynamics can be well described by a mean-field theory. We also study how the details of the DNA-protein interaction affect the evolution.Comment: 11 pages, 11 figures. Submitted to PNA

Codes and Supersymmetry in One Dimension

Adinkras are diagrams that describe many useful supermultiplets in D=1 dimensions. We show that the topology of the Adinkra is uniquely determined by a doubly even code. Conversely, every doubly even code produces a possible topology of an Adinkra. A computation of doubly even codes results in an enumeration of these Adinkra topologies up to N=28, and for minimal supermultiplets, up to N=32.Comment: 48 pages, a new version that combines arXiv:0811.3410 and parts of arXiv:0806.0050, for submission for publicatio

Effective Symmetries of the Minimal Supermultiplet of N = 8 Extended Worldline Supersymmetry

A minimal representation of the N = 8 extended worldline supersymmetry, known as the `ultra-multiplet', is closely related to a family of supermultiplets with the same, E(8) chromotopology. We catalogue their effective symmetries and find a Spin(4) x Z(2) subgroup common to them all, which explains the particular basis used in the original construction. We specify a constrained superfield representation of the supermultiplets in the ultra-multiplet family, and show that such a superfield representation in fact exists for all adinkraic supermultiplets. We also exhibit the correspondences between these supermultiplets, their Adinkras and the E(8) root lattice bases. Finally, we construct quadratic Lagrangians that provide the standard kinetic terms and afford a mixing of an even number of such supermultiplets controlled by a coupling to an external 2-form of fluxes.Comment: 13 Figure

On Graph-Theoretic Identifications of Adinkras, Supersymmetry Representations and Superfields

In this paper we discuss off-shell representations of N-extended supersymmetry in one dimension, ie, N-extended supersymmetric quantum mechanics, and following earlier work on the subject codify them in terms of certain graphs, called Adinkras. This framework provides a method of generating all Adinkras with the same topology, and so also all the corresponding irreducible supersymmetric multiplets. We develop some graph theoretic techniques to understand these diagrams in terms of a relatively small amount of information, namely, at what heights various vertices of the graph should be "hung". We then show how Adinkras that are the graphs of N-dimensional cubes can be obtained as the Adinkra for superfields satisfying constraints that involve superderivatives. This dramatically widens the range of supermultiplets that can be described using the superspace formalism and organizes them. Other topologies for Adinkras are possible, and we show that it is reasonable that these are also the result of constraining superfields using superderivatives. The family of Adinkras with an N-cubical topology, and so also the sequence of corresponding irreducible supersymmetric multiplets, are arranged in a cyclical sequence called the main sequence. We produce the N=1 and N=2 main sequences in detail, and indicate some aspects of the situation for higher N.Comment: LaTeX, 58 pages, 52 illustrations in color; minor typos correcte

Fractional Loop Group and Twisted K-Theory

We study the structure of abelian extensions of the group $L_qG$ of $q$-differentiable loops (in the Sobolev sense), generalizing from the case of central extension of the smooth loop group. This is motivated by the aim of understanding the problems with current algebras in higher dimensions. Highest weight modules are constructed for the Lie algebra. The construction is extended to the current algebra of supersymmetric Wess-Zumino-Witten model. An application to the twisted K-theory on $G$ is discussed.Comment: Final version in Commun. Math. Phy

A Superfield for Every Dash-Chromotopology

The recent classification scheme of so-called adinkraic off-shell supermultiplets of N-extended worldline supersymmetry without central charges finds a combinatorial explosion. Completing our earlier efforts, we now complete the constructive proof that all of these trillions or more of supermultiplets have a superfield representation. While different as superfields and supermultiplets, these are still super-differentially related to a much more modest number of minimal supermultiplets, which we construct herein.Comment: 13 pages, integrated illustration