Did Sequence Dependent Geometry Influence the Evolution of the Genetic Code?

The genetic code is the function from the set of codons to the set of amino acids by which a DNA sequence encodes proteins. Since the codons also influence the shape of the DNA molecule itself, the same sequence that encodes a protein also has a separate geometric interpretation. A question then arises: How well-duplexed are these two "codes"? In other words, in choosing a genetic sequence to encode a particular protein, how much freedom does one still have to vary the geometry (or vice versa). A recent paper by the first author addressed this question using two different methods. After reviewing those results, this paper addresses the same question with a third method: the use of Monte Carlo and Gaussian sampling methods to approximate a multi-integral representing the mutual information of a variety of possible genetic codes. Once again, it is found that the genetic code used in nuclear DNA has a slightly lower than average duplexing efficiency as compared with other hypothetical genetic codes. A concluding section discusses the significance of these surprising results.Comment: To appear in Contemporary Mathematic

Wave energy localization by self-focusing in large molecular structures: a damped stochastic discrete nonlinear Schroedinger equation model

Wave self-focusing in molecular systems subject to thermal effects, such as thin molecular films and long biomolecules, can be modeled by stochastic versions of the Discrete Self-Trapping equation of Eilbeck, Lomdahl and Scott, and this can be approximated by continuum limits in the form of stochastic nonlinear Schroedinger equations. Previous studies directed at the SNLS approximations have indicated that the self-focusing of wave energy to highly localized states can be inhibited by phase noise (modeling thermal effects) and can be restored by phase damping (modeling heat radiation). We show that the continuum limit is probably ill-posed in the presence of spatially uncorrelated noise, at least with little or no damping, so that discrete models need to be addressed directly. Also, as has been noted by other authors, omission of damping produces highly unphysical results. Numerical results are presented for the first time for the discrete models including the highly nonlinear damping term, and new numerical methods are introduced for this purpose. Previous conjectures are in general confirmed, and the damping is shown to strongly stabilize the highly localized states of the discrete models. It appears that the previously noted inhibition of nonlinear wave phenomena by noise is an artifact of modeling that includes the effects of heat, but not of heat loss.Comment: 22 pages, 13 figures, revision of talk at FPU+50 conference in Rouen, June 200

MODELING THERMAL EFFECTS ON NONLINEAR WAVE MOTION IN BIOPOLYMERS BY A STOCHASTIC DISCRETE NONLINEAR SCHRÖDINGER EQUATION WITH PHASE DAMPING

Abstract. A mathematical model is introduced for weakly nonlinear wave phenomena in molecular systems like DNA and protein molecules that includes thermal effects: exchange of heat energy with the surrounding aqueous medium. The resulting equation is a stochastic discrete nonlinear Schrödinger equation with focusing cubic nonlinearity and “Thermal ” terms modeling heat input and loss: PDSDNLS. New numerical methods are introduced to handle the unusual combination of a conservative equation, stochastic, and fully nonlinear terms. Some analysis is given of accuracy needs, and the special issues of time step adjustment in stochastic realizations. Numerical studies are presented of the effects of thermalization on solitons, including damping induced self-trapping of wave energy, a discrete counterpart of single-point blowup. 1. Introduction. Puls

Regularization and Control of Self-Focusing in the 2D Cubic Schrödinger Equation by . . .

The self-focusing singularity of the attractive 2D Cubic Schrödinger Equation arises in nonlinear optics and many other situations, including certain models of Bose-Einstein condensates. This 2