A model for the dynamics of expanded CAG repeat alleles: ATXN2 and ATXN3 as prototypes

Background: Spinocerebellar ataxia types 2 (SCA2) and 3 (SCA3/MJD) are diseases due to dominant unstable expansions of CAG repeats (CAGexp). Age of onset of symptoms (AO) correlates with the CAGexp length. Repeat instability leads to increases in the expanded repeats, to important AO anticipations and to the eventual extinction of lineages. Because of that, compensatory forces are expected to act on the maintenance of expanded alleles, but they are poorly understood. Objectives: we described the CAGexp dynamics, adapting a classical equation and aiming to estimate for how many generations will the descendants of a de novo expansion last. Methods: A mathematical model was adapted to encompass anticipation, fitness, and allelic segregation; and empirical data fed the model. The arbitrated ancestral mutations included in the model had the lowest CAGexp and the highest AO described in the literature. One thousand generations were simulated until the alleles were eliminated, fixed, or 650 generations had passed. Results: All SCA2 lineages were eliminated in a median of 10 generations. In SCA3/MJD lineages, 593 were eliminated in a median of 29 generations. The other ones were eliminated due to anticipation after the 650th generation or remained indefinitely with CAG repeats transitioning between expanded and unexpanded ranges. Discussion: the model predicted outcomes compatible with empirical data - the very old ancestral SCA3/MJD haplotype, and the de novo SCA2 expansions -, which previously seemed to be contradictory. This model accommodates these data into understandable dynamics and might be useful for other CAGexp disorders.