J.D. Bashford

Codon and nucleotide assignments in a supersymmetric model of the genetic code

Abstract The supersymmetric model we developed for the evolution of the genetic code is elaborated. Energy considerations in nucleic acid strand modelling, using sl(2) polarity spin and sl( 2 1 ) family box quartet symmetry, lead for the case of codons and anticodons to assignments of codons to 64-dimensional sl( 6 1 ) ⋍ A(5,0) multiplets.

Path integral formulation and Feynman rules for phylogenetic branching models

A dynamical picture of phylogenetic evolution is given in terms of Markov models on a state space, comprising joint probability distributions for character types of taxonomic classes. Phylogenetic branching is a process which augments the number of taxa under consideration, and hence the rank of the underlying joint probability state tensor. We point out the combinatorial necessity for a second-quantized, or Fock space setting, incorporating discrete counting labels for taxa and character types, to allow for a description in the number basis. Rate operators describing both time evolution without branching, and also phylogenetic branching events, are identified. A detailed development of these ideas is given, using standard transcriptions from the microscopic formulation of non-equilibrium reaction–diffusion or birth–death processes. These give the relations between stochastic rate matrices, the matrix elements of the corresponding evolution operators representing them, and the integral kernels needed to implement these as path integrals. The ‘free’ theory (without branching) is solved, and the correct trilinear ‘interaction’ terms (representing branching events) are presented. The full model is developed in perturbation theory via the derivation of explicit Feynman rules which establish that the probabilities (pattern frequencies of leaf colourations) arising as matrix elements of the time evolution operator are identical with those computed via the standard analysis. Simple examples (phylogenetic trees with two or three leaves), are discussed in detail. Further implications for the work are briefly considered including the role of time reparametrization covariance.

Quantum field theory and phylogenetic branching

A calculational framework is proposed for phylogenetics, using nonlocal quantum field theories in hypercubic geometry. Quadratic terms in the Hamiltonian give the underlying Markov dynamics, while higher degree terms represent branching events. The spatial dimension L is the number of leaves of the evolutionary tree under consideration. Momentum conservation modulo 2×L in L←1 scattering corresponds to tree edge labelling using binary L-vectors. The bilocal quadratic term allows for momentum-dependent rate constants - only the tree or trees compatible with selected nonzero edge rates contribute to the branching probability distribution. Applications to models of evolutionary branching processes are discussed.

A base-pairing model of duplex formation. I. Watson-Crick pairing geometries.

We present a base‐pairing model of oligonucleotide duplex formation and show in detail its equivalence to the nearest‐neighbor dimer methods from fits to free energy of duplex formation data for short DNA–DNA and DNA–RNA hybrids containing only Watson–Crick pairs. For completeness, the corresponding RNA–RNA parameters are included. In this approach, the connection between rank‐deficient polymer and rank‐determinant oligonucleotide parameter sets for DNA duplexes is transparent. The method is generalized to include RNA–DNA hybrids where the rank‐deficient model with 11 dimer parameters in fact provides slightly improved predictions relative to the standard method with 16 independent dimer parameters (ΔG mean errors of 4.5 and 5.4%, respectively).