José Eduardo M. Hornos

SYMMETRY AND SYMMETRY BREAKING: AN ALGEBRAIC APPROACH TO THE GENETIC CODE

We give a comprehensive review of the algebraic approach to the genetic code originally proposed by two of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution. We present the relevant background material from molecular biology and from mathematics, including the representation theory of (semi) simple Lie groups/algebras, together with considerations of general nature.

Symmetry Preservation in the Evolution of the Genetic Code

The standard genetic code is found to exhibit an exact symmetry under a finite group of order 4 known in mathematics as the Klein group. The same symmetry is also present in almost all non‐standard codes, mitochondrial as well as nuclear. Analysis of the phylogenetic tree for the evolution of the mitochondrial codes reveals that all changes along the main line of evolution preserve this symmetry, with a tendency towards symmetry enhancement. In the side branches of the evolutionary tree, the majority of changes also respect the symmetry. The few exceptional cases where it is broken correspond to reassignments that appear to be unstable or incomplete. Since the Klein group emerges naturally from the symplectic model for the prebiotic evolution that has led to the standard code, we interpret these results as lending support to the hypothesis that this symmetry has been selected during the evolution of the genetic code, not only before but also after establishment of the standard code. IUBMB Life, 56: 125‐130, 2004

The Morse oscillator under time-dependent external fields

A method to solve the equations for the Morse oscillator under intense time-dependent external fields is presented. Exact analytical formulas for the dipole matrix elements are calculated by the use of the hypergeometric algebra. The continuum is described by an expansion using Laguerre functions. The full algorithm for the calculation of wave functions can be controlled by the convergence of series and by the errors of a first order integration method. We apply our technique to the selective preparation of high overtones by femtosecond laser pulses. The population of the target state is optimized as a function of the intensity and frequency. Introducing a second simultaneous laser, we study the effects of relative frequency and phase over the target state population and dissociation channels. The calculations exhibit a rich interference pattern showing the enhancement and the suppression of the target population by varying the laser parameters.

EXTENDING THE SEARCH FOR SYMMETRIES IN THE GENETIC CODE

We report on the search for symmetries in the genetic code involving the medium rank simple Lie algebras

ON AMINO ACID AND CODON ASSIGNMENT IN ALGEBRAIC MODELS FOR THE GENETIC CODE

B_6 = {\mathfrak{so}}(13)

Multimodality and flexibility of stochastic gene expression.

and

The overtone spectrum of monofluoracetylene in the algebraic approach

D_7 = {\mathfrak {so}}(14)

Stochastic model for gene transcription on Drosophila melanogaster embryos.

, in the context of the algebraic approach originally proposed by one of the present authors, which aims at explaining the degeneracies encountered in the genetic code as the result of a sequence of symmetry breakings that have occurred during its evolution.

The overtone spectrum of monochloroacetylene (HCCCl) in the algebraic approach

We give a list of all possible schemes for performing amino acid and codon assignments in algebraic models for the genetic code, which are consistent with a few simple symmetry principles, in accordance with the spirit of the algebraic approach to the evolution of the genetic code proposed by Hornos and Hornos. Our results are complete in the sense of covering all the algebraic models that arise within this approach, whether based on Lie groups/Lie algebras, on Lie superalgebras or on finite groups.

Dynamical symmetry in the vibrational overtone spectrum of monofluoroacetylene (HCCF)

We consider a general class of mathematical models for stochastic gene expression where the transcription rate is allowed to depend on a promoter state variable that can take an arbitrary (finite) number of values. We provide the solution of the master equations in the stationary limit, based on a factorization of the stochastic transition matrix that separates timescales and relative interaction strengths, and we express its entries in terms of parameters that have a natural physical and/or biological interpretation. The solution illustrates the capacity of multiple states promoters to generate multimodal distributions of gene products, without the need for feedback. Furthermore, using the example of a three states promoter operating at low, high, and intermediate expression levels, we show that using multiple states operons will typically lead to a significant reduction of noise in the system. The underlying mechanism is that a three-states promoter can change its level of expression from low to high by passing through an intermediate state with a much smaller increase of fluctuations than by means of a direct transition.