A new molecular evolution model for limited insertion independent of substitution

We recently introduced a new molecular evolution model called the IDIS model for Insertion Deletion Independent of Substitution [13,14]. In the IDIS model, the three independent processes of substitution, insertion and deletion of residues have constant rates. In order to control the genome expansion during evolution, we generalize here the IDIS model by introducing an insertion rate which decreases when the sequence grows and tends to 0 for a maximum sequence length nmax. This new model, called LIIS for Limited Insertion Independent of Substitution, defines a matrix differential equation satisfied by a vector P(t) describing the sequence content in each residue at evolution time t. An analytical solution is obtained for any diagonalizable substitution matrix M. Thus, the LIIS model gives an expression of the sequence content vector P(t) in each residue under evolution time t as a function of the eigenvalues and the eigenvectors of matrix M, the residue insertion rate vector R, the total insertion rate r, the initial and maximum sequence lengths n0 and nmax, respectively, and the sequence content vector P(t0) at initial time t0. The derivation of the analytical solution is much more technical, compared to the IDIS model, as it involves Gauss hypergeometric functions. Several propositions of the LIIS model are derived: proof that the IDIS model is a particular case of the LIIS model when the maximum sequence length nmax tends to infinity, fixed point, time scale, time step and time inversion. Using a relation between the sequence length l and the evolution time t, an expression of the LIIS model as a function of the sequence length l=n(t) is obtained. Formulas for 'insertion only', i.e. when the substitution rates are all equal to 0, are derived at evolution time t and sequence length l. Analytical solutions of the LIIS model are explicitly derived, as a function of either evolution time t or sequence length l, for two classical substitution matrices: the 3-parameter symmetric substitution matrix [12] (LIIS-SYM3) and the HKY asymmetric substitution matrix[9] (LIIS-HKY). An evaluation of the LIIS model (precisely, LIIS-HKY) based on four statistical analyses of the GC content in complete genomes of four prokaryotic taxonomic groups, namely Chlamydiae, Crenarchaeota, Spirochaetes and Thermotogae, shows the expected improvement from the theory of the LIIS model compared to the IDIS model.